International workshop on Far-From-Equilibrium Dynamics · P35 Akiyasu Tomoeda Starting-Wave of...
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FFED11 Far-From-Equilibrium Dynamics January 4 - January 8, 2011 Research Institute for Mathematical Sciences International workshop on Kyoto University
Transcript of International workshop on Far-From-Equilibrium Dynamics · P35 Akiyasu Tomoeda Starting-Wave of...
FFED11
Far-From-Equilibrium DynamicsJanuary 4 - January 8, 2011
Research Institute for Mathematical Sciences
International workshop on
Kyoto University
International Workshop on
”Far-From-Equilibrium Dynamics”
dedicated to the 60th birthday of Prof. Yasumasa Nishiura
Scope
The conference ”Far-From-Equilibrium Dynamics 2011”will take place in Kyoto, Japan onthe occasion of the 60th birthday of Prof. Yasumasa Nishiura. Topics include a variety ofproblems in nonlinear science such as pattern dynamics in dissipative systems, material science,mathematical biology, brain theory, and fluid dynamics as well as dynamical system thory andits applications. Altogether 25 invited talks will be presented and more than 50 posters will beposted.
Conference Program
Tuesday, January 4th – Saturday, January 8th, 2011Research Institute for Mathematical Sciences and Shiran-Kaikan, Kyoto University
Tuesday, January 4th, Room 420, RIMS (4F)
9:20 – 9:30 Opening Remarks9:30 – 10:10 L01 Peter Bates (Michigan State University)
Global dynamics of particles driven by Allen-Cahn dynamicson the boundary of a smooth domain
10:20 – 10:50 L02 Yoshihito Oshita (Okayama University)Coarsening and stabilization in micro phase separation
11:00 – 11:40 L03 Kenneth Showalter (West Virginia University)Dynamical Quorum Sensing and Synchronization
Lunch break
13:30 – 14:10 L04 Philip K. Maini (Oxford University)Modelling dynamical spread of tumour cells
14:20 – 15:00 L05 Ryo Kobayashi (Hiroshima University)Toward understanding the locomotion of animals
Tea break
15:30 – 16:10 L06 Arnd Scheel (University of Minnesota)Coherent structures and Liesegang pattern formation
16:20 – 17:00 L07 Hisashi Okamoto (Kyoto University)Pattern formations at high Reynolds numbers: Navier-Stokes,Kolmogorov, and · · ·
Wednesday, January 5th, Room 420, RIMS (4F)
9:30 – 10:10 L08 Ichiro Tsuda (Hokkaido University)A Mathematical Model for the Formation of Dynamic Memory in the Brain
10:20 – 11:20 Poster Flash Talks I (Odd-numbers)
Lunch break
13:30 – 14:10 L09 Arjen Doelman (Leiden University)An explicit theory for pulses in two component, singularly perturbed,reaction-diffusion equations
14:20 – 15:00 L10 Tasso J. Kaper (Boston University)Incorporating fluctuations in continuous models of fronts using cut-offs
Tea break
15:30 – 16:10 L11 Bjorn Sandstede (Brown University)Localized planar patterns
16:20 – 17:20 Poster Flash Talks II (Even-numbers)
Thursday, January 6th, Shiran-Kaikan
9:30 – 10:10 L12 Christoper K. R. T. Jones (University of North Carolina)Mathematical challenges of climate research: data assimilation and uncertainty
10:20 – 11:00 L13 Tomohiko Yamaguchi(National Institute of Advanced Industrial Science and Technology)
Reversible Gray-Scott Model as a Tool of Thermodynamic Investigationin Non-Equilibrium Chemical Systems
11:10 – 11:50 L14 Odo Diekmann (Utrecht University)On Delay Equations and Population Dynamics
Photo & Lunch break
14:00 – 15:00 L15 Yasumasa Nishiura (Hokkaido University)Dynamics of spatially localized patterns
15:00 – 16:30 Poster Session P01 – P5116:40 – 17:40 Movie Session
18:00 – Party
Friday, January 7th, Room 420, RIMS (4F)
9:30 – 10:10 L16 Michael J. Ward (University of British Columbia)The Stability and Dynamics of Localized Spot Patternsin the Two-Dimensional Gray-Scott Model
10:20 – 10:50 L17 Hiroshi Suito (Okayama University)Vortex dynamics in aorta with torsion
11:00 – 11:40 L18 Shin-Ichiro Ei (Kyushu University)Dynamics of pulses in two dimensional thin domain
Lunch break
13:30 – 14:10 L19 Jack Xin (University of California at Irvine)Asymptotic Front Speeds of Viscous G-Equation in Cellular Flows
14:20 – 14:50 L20 Takashi Sakajo (Hokkaido University)Point vortex dynamics in multiply connected domains
Tea break
15:20 – 16:00 L21 Yoshihisa Morita (Ryukoku University)Localized patterns in a reaction-diffusion system with conservation of a mass
16:10 – 16:50 L22 Hiroshi Matano (University of Tokyo)Traveling waves in a sawtoothed cylinder and their homogenization limit
Saturday, January 8th, Room 420, RIMS (4F)
9:30 – 10:10 L23 Konstantin Mischaikow (The State University of New Jersey)A combinatorial framework for nonlinear dynamics
10:20 – 11:00 L24 Hiroshi Kokubu (Kyoto University)Topological bifurcation theorems for Morse decompositions
11:20 – 12:00 L25 James P. Keener (University of Utah)Dynamics of Swelling of polyelectrolyte gels
Poster Presentations List
P01 Yuyu Peng TBAP02 Kaname Matsue Dynamical approach in rigorous verification
of equilibria for evolutionary equationsP03 Marcel Horning Control of pacing sites by electrical far-field stimulation
in discrete excitable mediaP04 Andrei Giniatoulline On the relationship between the essential spectrum of internal waves
and non-uniqueness of the limit amplitude for the dynamicsof stratified fluid
P05 Syuji Miyazaki Relaxation scaling laws in a periodically forced Swift-Hohenberg systemP06 Seirin Lee Gene Expression Time Delays and Turing Pattern FormationP07 Tomoyuki Miyaji Bifurcation analysis to the Lugiato-Lefever equation
in one space dimensionP08 Katsuhito Matsui Fluctuation-spectrum approach to few-
and large-degrees-of-freedom chaotic systemsP09 Takashi Miura Modeling lung branching morphogenesisP10 Shinji Nakaoka Periodic oscillations in hematopoietic diseasesP11 Yoshitaka Saiki Reconstruction of chaotic saddles and classification of
unstable periodic orbits of the Kuramoto-Sivashinsky equationP12 Mitsusuke Tarama Breathing instability versus drift instability
in a two-component reaction diffusion systemP13 Kunihiko Kaneko Itinerant Dynamics in Gene Expression Implies PluripotencyP14 Tetsuya J. Kobayashi Optimal Information Processing and Stochastic Bifurcation
in Biological NetworksP15 Kensuke Yokoi Numerical studies of droplet impacting and splashingP16 Taichi Haruna Complex Networks from Dual Point of ViewP17 Tsuyoshi Mizuguchi Symmetry Restoration Process and Unstable Symmetric ToriP18 Yasuaki Kobayashi Evolutionary design of robust oscillatory genetic networksP19 Ralf Tonjes Synchronization transition of identical phase oscillators
in a directed small-world networkP20 Hiroyuki Kitahata Interfacial tension driven motion of a droplet coupled
with chemical reactionP21 Masako Matsubara Analysis of synchronization of coupled damped oscillatorsP22 Hiroshi Kori Collective enhancement of regularity in networks of noisy oscillatorsP23 Yu Kurokawa Multi-dimensional pyramidal traveling fronts in the Allen-Cahn equationsP24 Ken H. Nagai Noise-induced synchronization of coupled oscillatorsP25 Tetsuya Ishiwata Motion of non-closed planar polygonal curves
by crystalline curvature flow with a driving forceP26 Yumino Hayase Thermal convection of binary mixtures by temperature rampP27 Natsuhiko Yoshinaga Self-propulsion of a drop driven by Marangoni flowP28 Yousuke Tonosaki Nonlinear dissipative wave under external forcing and feedback controlP29 Takeshi Takaishi Bifurcation behaviour of the shape of cracksP30 Hiroya Nakao Phase reduction approach to interacting traveling pulsesP31 Toshiya Kazama A Mathematical Approach to Explain in vitro Amoeba Locomotion
P32 Mayuko Iwamoto Mathematical Modeling of Asymmetric Wavesin Crawling Movement of Abalone
P33 Masaaki Yadome Onset of pulse generator patterns in a three-component systemP34 Nariya Uchida Hydrodynamic Synchronization of Active Microfluidic RotorsP35 Akiyasu Tomoeda Starting-Wave of Pedestrians and its Application for MarathonP36 Naoto Nakano On Steady Solutions for a Continuum Model
with Density Gradient-Dependent StressP37 Kei Nishi Behaviors of a front-back pulse in some bistable
reaction-diffusion system with heterogeneityP38 Kentaro Ito Network formation in spatio-temporally varying field
by true slime moldP39 Keita Iida Mathematical studies on the two-dimensional self-motion
of camphor scrapingsP40 Elliott Ginder The discrete Morse flow for volume-controlled membrane motionsP41 Fumito Mori Reliable Time in Biological ClocksP42 Xiaohui Yuan Heterogeneity-induced spot dynamics
for a three-component reaction-diffusion systemP43 Asaki Saito True Orbit Computation Using Integer ArithmeticP44 Hirofumi Notsu An application of the characteristics finite element method
to a thermal convection problem with the infinite Prandtl numberP45 Shu-ichi Kinoshita Comparison of Boolean dynamics in complex networksP46 Tohru Wakasa On a simplified tumor growth model with contact-inhibitionP47 Ayuki Sekisaka Topological and computational method for eigenvalue problem
of 1 dim shrodinger operatorP48 Tsubasa Masui Arch Structure in the Escape PanicP49 Masahiro Yamaguchi A mesh generator using a self-organizing mechanism
of a reaction-diffusion systemP50 Atsunari Katsuki Size distribution of barchan dunes with a cellular modelP51 Shingo Iwami Computational virology and immunology
- Quantification system of viral dynamics -
All posters are displayed in Yamauchi Hall at January 6. Poster presenters can mount theirposters on the board according to their poster abstract number from the noon of January 6.The poster board is W 90 x H 210 cm. Drawing pins (thumbtacks) and Scotch tapes areprovided. No printer is available in Yamauchi Hall.Poster presenters are scheduled for 2 min to present their work during Poster Flash Talk sessionsat January 5.
Session I for Odd-numbers: 10:20 – 11:20 of January 5Session II for Even-numbers: 16:20 – 17:20 of January 5
Presentation files ( 2-pages PDF file, landscape format ) have to be brought to the conferencereception by 9:30 of January 5. Poster presenters are requested to stand in front of their postersin the session: 15:00 – 16:30 of January 6.All posters have to be removed by 17:00 of January 6. All remaining posters will be removedby the conference organizers.
Organizing Comittee
Hisashi Okamoto (Chair, Kyoto University),Toshiyuki Ogawa (Secretary General, Osaka University),Makoto Iima, Ryo Kobayashi, Masaharu Nagayama, Shunsaku Nii, Takashi Sakajo,Hiromasa Suzuki, Takashi Teramoto, Kei-Ichi Ueda, Daishin Ueyama, Tatsuo Yanagita
Sponsor
Research Institute for Mathematical Sciences, Kyoto UniversityKyoto University Global COE Program(Fostering top leaders in mathematics -broadening the core and exploring new ground)
Contact Address
Toshiyuki Ogawa (Osaka University) and Kei-Ichi Ueda (Kyoto University)EMAIL: [email protected], URL: http://www.ffed11.org
Research Institute for Mathematical Sciences (RIMS) , Kyoto UniversityKita-Shirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502FAX: +81-75-753-7272
Shiran-Kaikan, Kyoto University11-1 Ushinomiya-cho, Yoshida, Sakyo-ku, Kyoto 606-8302TEL: +81-75-751-2713, FAX: +81-75-752-4015
Invited Talks ( L01 - L25 )
Global dynamics of particles driven by Allen-Cahn dynamics on theboundary of a smooth domain
Peter BatesMichigan State University
We consider particles described as spatially localized solutions to a singularly perturbed nonlin-ear parabolic partial differential equation. These arise by taking the gradient flow of an energyfunctional but not in the vicinity of the global minimizer of the energy, or near any equilibriumfor that matter. Nevertheless, the basic structure of the solution has a well-defined shape andit is natural to view particles as dynamic peak-like solutions far from equilibrium. By provingan abstract theorem about the existence of a true invariant manifold in the neighborhood ofan approximately invariant, approximately normally hyperbolic invariant manifold, we are ableto demonstrate the existence of these solutions for all positive and negative time, giving theirglobal dynamics in some detail.
L01
Coarsening and stabilization in micro phase separation
Yoshihito [email protected]
Department of Mathematics, Okayama University
Ohta–Kawasaki introduced a free energy functional describing micro phase separation ofdiblock copolymers [3]. Starting with the pioneering work [2], where the Ohta–Kawasaki theoryis formulated on a bounded domain as a singularly perturbed problem and the limiting sharpinterface problem as ε → 0 is identified, there has been a bulk of analytical work.
We study the free boundary problem describing the micro phase separation of diblock copoly-mer melts in two and three dimensions in the regime that one component has small volumefraction such that micro phase separation results in an ensemble of small spheres of one compo-nent. On some time scale, the evolution is dominated by coarsening and subsequent stabilizationof the radii of the spheres. Starting from the free boundary problem restricted to spheres (par-ticles) we rigorously derive the effective equations describing the dynamics in this time regimecalled mean-field models [1]. Depending on the parameter scalings, we have two models. One isthe spatially uniform mean-field models in the dilute case, where the self-interaction of particlesis dominated, and the other is the models for the joint distribution of particle radii and centersin the screened case, which is an inhomogeneous extension of the dilute case. Our analysis isbased on passing to the homogenization limit in the variational framework of a gradient flow.
Furthermore the dilute mean-field models for particle volumes are studied. We identify allthe steady states and their stabilities, and show the convergence of solutions. We see that thesteady states are of the form of the sum of at most two Dirac deltas except for disappearedparticles.
——–
REFERENCES[1] B. Niethammer and Y. Oshita, A rigorous derivation of mean-field models for diblock copolymer melts. Calc. Var.Partial Differential Equations 39, 273–305 (2010).[2] Y. Nishiura and I. Ohnishi, Some Aspects of the Micro-phase Separation in Diblock Copolymers. Physica D, 84,31–39 (1995).[3] T. Ohta and K. Kawasaki, Equilibrium Morphology of Block Copolymer Melts, Macromolecules 19, 2621–2632(1986).
1
L02
Dynamical Quorum Sensing and Synchronization
Kenneth Showalter West Virginia University
From the periodic firing of neurons to the flashing of fireflies, the synchronization of rhythmic activity plays a vital role in the functioning of biological systems. Synchronization often occurs by global coupling, where each oscillator is connected to every other oscillator through a common mean field. A distinctly different type of transition to synchronized oscillatory behavior is observed in suspensions of yeast cells. Relaxation experiments demonstrate that slightly below a critical cell density the system is made up of a collection of quiescent cells, whereas slightly above this density the cells oscillate in nearly complete synchrony. This type of transition is much like quorum-sensing transitions in bacteria populations, where each member of a population undergoes a sudden change in behavior with a supercritical increase in the concentration of a signaling molecule (autoinducer) in the extra-cellular solution. We have studied large, heterogeneous populations of discrete chemical oscillators (~100,000) to characterize the two different types of density-dependent transitions to synchronized oscillatory behavior. For different chemical exchange rates between the oscillators and the surrounding solution, we find with increasing oscillator density (1) the gradual synchronization of oscillatory activity or (2) the sudden "switching on" of synchronized oscillatory activity. We have also studied spatially distributed groups of excitable particles that diffusively exchange activator and inhibitor species with the surrounding solution. All particles are nonoscillatory when separated from the other particles; however, spatiotemporal oscillations spontaneously appear in groups above a critical size. A. F. Taylor et al., Science 323, 614 (2009). M. R. Tinsley et al., Phys. Rev. Lett. 102, 158301 (2009).
L03
Modelling dynamical spread of tumour cells
Philip K. MainiOxford University
L04
Toward understanding the locomotion of animals
Hiroshima University
The most characteristic and common feature of animals is that they locomote in the spaceby themselves. They actually exhibit very supple and agile motion, which is achieved by theelegant control of very large degree of freedom embedded in their body. In addition, they canmanage to respond to uncertain environments in real time. Such amazing ability is consideredto be brought off by an autonomous decentralized control, while almost all robots of the dayare controlled by fully centralized manner. Our goal is to understand the locomotion of animalsfrom both view points of mechanics and control, and make a soft robot which can move likeanimals. We will present a part of our work, especially concerning to crawling animals, whichis performed by the team consisting of biologists, mathematicians and roboticists.
L05
Coherent structures and Liesegang pattern formation
Arnd ScheelUniversity of Minnesota
Liesegang rings form in strikingly simple experimental setups. The resulting patterns, some-times regular, sometimes irregular, are ubiquitous in nature — yet mathematically poorlyunderstood. We present models and results that lead to a conceptual understanding of exis-tence and creation of such patterns, based on existence, stability, and bifurcation results forcoherent structures in our models. A crucial role is played by pushed and pulled fronts inspinodal decomposition scenarios.
L06
Pattern formations at high Reynolds numbers:Navier-Stokes, Kolmogorov, and · · ·
Hisashi [email protected]
Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan 606-8602
We consider Kolmogorov’s problem for the two-dimensional (2D) Navier-Stokes equations. Sta-bility of and bifurcation from the trivial solution are studied numerically. More specifically, wecompute solutions of the Navier-Stokes equations with a family of prescribed external forces ofincreasing degree of oscillation. We found in [3] that, whatever the external force may be, astable steady-state of simple geometric character exits for sufficiently large Reynolds numbers.We thus observe a kind of universal appearance of the solutions, which is independent of theexternal force. We have some reasons to believe that this phenomenon is peculiar to the 2DNavier–Stokes equations.
This observation is reinforced further by an asymptotic analysis of a simple equation calledthe generalized Proudman-Johnson equation, which was introduced in [1].
REFERENCES[1] H. Okamoto & J. Zhu, Some similarity solutions of the Navier-Stokes equations and related topics, Taiwanese J.Math., 4 (2000), 65–103.[2] Sun-Chul Kim & Hisashi Okamoto, Bifurcations and inviscid limit of rhombic Navier-Stokes flows in tori, IMA J.Appl. Math., 68 (2003), 119–134.[3] Sun-Chul Kim & Hisashi Okamoto, Vortices of large scale appearing in the 2D stationary Navier–Stokes equationsat large Reynolds numbers, Japan J. Indust. Appl. Math., 27 (2010), 47–71.[4] Sun-Chul Kim & Hisashi Okamoto, The generalized Proudman-Johnson equation at large Reynolds numbers, inpreparation.
L07
A Mathematical Model for the Formation of Dynamic
Memory in the Brain
Ichiro Tsuda
1 Research Institute for Electronic Science, Hokkaido University
&
Research Center for Integrative Mathematics, Hokkaido University
Kita-12, Nishi-6, Kita-ku, Sapporo
060-0812 Hokkaido, Japan
Abstract The hippocampus in the mammalian brain has been considered responsible for the
formation of episodic memory. It has also been pointed out that the hippocampus plays an important
role in imagination and the sense of presence, the former of which is related to the future and the
latter related to now. The fact that atrophy of the hippocampus could lead to Alzheimer’s disease
implies that the network structure of the hippocampus may provide fields for the creation of internal
time corresponding to the past, present, and future. We present a mathematical model, based on the
hippocampal neural networks and the related biological conditions, such that the hippocampus plays a
role in the formation of episodic memory, producing chaotic and fractal dynamics.
References
1. Tsuda,I., Towards an interpretation of dynamic neural activity in terms of chaotic dynamical systems. Behav.
Brain Sci. 24 (2001) 793‐847.
2. Tsuda, I. and Kuroda, S., Cantor coding in the hippocampus. Japan J. Indust. Appl. Math. 18 (2001) 249‐258.
3. Tsuda, I., Dynamic link of memories ― chaotic memory map in nonequilibrium neural networks. Neural
Netw. 5 (1992) 313‐326.
4. Tsuda, I., Chaotic itinerancy as a dynamical basis of hermeneutics of brain and mind. World Futures 32 (1991)
167‐185.
5. Tsuda, I., Hypotheses on the functional roles of chaotic transitory dynamics. Chaos 19 (2009) 015113-1 ‐
015113-10.
6. Sasaki, T., Matsuki, N., and Ikegaya, Y.: Metastability of active CA3 networks. J. Neurosci. 17 (2007) 517‐
528.
L08
7. Pinsky, P. F. and Rinzel, J., Intrinsic and network rhythmogenesis in a reduced traub model for CA3 neurons. J.
Comput. Neurosci. 1 (1994) 39‐60.
8. Yamaguti, Y., Kuroda, S. and Tsuda, I. A mathematical model for Cantor coding in the hippocampus. To appear
in Neural Networks 24 (2011)43-53.
9. Fukushima, Y., Tsukada, M., Tsuda, I., Yamaguti, Y., & Kuroda, S. Spatial clustering property and its
self-similarity in membrane potentials of hippocampal CA1 pyramidal neurons for a spatio-temporal input
sequence.Cognitive Neurodynamics, 1(4), (2007) 305–16.
L08
An explicit theory for pulses in two component, singularlyperturbed, reaction-diffusion equations
Arjen DoelmanLeiden University
In recent years, methods have been developed to study the existence, stability and bifurcationsof pulses in singularly perturbed reaction-diffusion equations in one space dimension, in thecontext of a number of model problems, such as the Gray-Scott and the Gierer- Meinhardtequations. Although these methods are in principle of a general nature, their applicabilitya priori relies on the characteristics of these models. For instance, the slow reduced spatialproblem is linear in the models considered in the literature. Moreover, the nonlinearities inthe fast reduced spatial problem are of a very specific, polynomial, nature. These propertiesare crucially used, especially in the stability and bifurcation analysis. In fact, these propertiesare responsible for quite specific – i.e. non-generic – behavior: the recently discovered ‘Hopfdance’ performed by the boundary of the Busse balloon for large wavelength spatially periodicpatterns in Gray-Scott/Gierer- Meinhardt type models originates from the linear structureof the slow reduced equations. In this talk, we present and discuss a significantly extendedexplicit theory for pulses in two-component singularly perturbed reaction-diffusion equationsin a general setting.
L09
Incorporating fluctuations in continuous models of fronts usingcut-offs
Tasso J. KaperBoston University
L10
Localized planar patterns
Bjorn SandstedeDivision of Applied Mathematics, Brown University
In this talk, I will discuss localized stationary 1D and 2D structures such as hexagon patches,localized radial target patterns, and localized 1D rolls in the Swift-Hohenberg equation andother models. Some of these solutions exhibit snaking: in parameter space, the localized stateslie on a vertical sine-shaped bifurcation curve so that the width of the underlying periodicpattern, such as hexagons or rolls, increases as we move up along the bifurcation curve. Inparticular, snaking implies the coexistence of infinitely many different localized structures. Iwill give an overview of recent analytical and numerical work in which localized structures andtheir snaking or non-snaking behavior is investigated.
L11
Mathematical challenges of climate research: data assimilation anduncertainty
Christopher K. R. T. JonesUniversity of North Carolina
Data assimilation (DA), whose methods grew out of control theory, has not been used in climatestudies anything like to the extent it has in weather prediction. I will discuss whether this islikely to change and argue that DA has a lot to offer climate research, particularly when cast ina Bayesian framework. Motivating examples from paleoclimate and ocean studies will be giventhat will serve to outline the major challenges arising in DA when it is used to tackle climateproblems.
L12
Reversible Gray-Scott Model as a Tool of Thermodynamic Investigation in Non-Equilibrium Chemical Systems
Tomohiko Yamaguchi and Hitoshi Mahara
[email protected] Nanotechnology Research Institute, AIST, JAPAN
Pattern dynamics is an issue of evolution of order, and order is often evaluated by entropy. Thermodynamically, entropy is a state variable. Since the change in entropy stands for the dissipation of free energy, it can be a measure of dissipation in a system of concern. We demonstrate a practical way to discuss the issue of entropy in a reaction-diffusion system.
Any reaction-diffusion system, of which boundary conditions are kept constant, is regarded as an open system (here, “open” means that the boundary is open to the exchange of mass and heat). There are three factors by which the concentration of each species in the system changes with time: reaction, diffusion in the system, and the mass exchange at the boundary. Therefore, the net change of entropy (dS/dt) is given as the sum of three terms: the entropy production due to the reaction (σ reaction) and the diffusion (σ diffusion), and the divergence of entropy flow at the boundary (div Js).
∫ −+= dVJdtdS sdiffusionreaction div )(/ σσ , where V is the volume of the system.
The divergence of entropy, even in a model system, can be calculated by using the newly introduced chemical potential. This chemical potential is defined based on the virtual equilibrium state of the system. This virtual equilibrium state is time-dependent, and is calculated by setting Js = 0 at every time in the course of calculation for the real system. This treatment is equivalent to assume for the present open system to be isolated from its environment. The time evolution of this isolated system is then calculated by the reaction-diffusion equations to obtain its steady equilibrium state. This approach has been applied to the one-dimensional reaction-diffusion system composed of the reversible Gray-Scott model, which is a reversible variant of Gray-Scott model.
REFERENCES [1] H. Mahara et al., JCP 121, 8968(2004). [2] H. Mahara et al., Chaos 15, 047508 (2005). [3] H. Maharaet al., Phys. Rev. E 78, 066210:1-6 (2008). [4] H. Mahara and T. Yamaguchi, Physica D 238, 729(2010).
L13
On Delay Equations and Population Dynamics
Odo DiekmannUtrecht University
A delay equation is a rule for extending a function of time towards the future, on the basisof the known past. Renewal Equations prescribe the current value, while Delay DifferentialEquations prescribe the derivative of the current value. With a delay equation one can associatea dynamical system by translation along the extended function.
I will illustrate how such equations arise in the description of size-structured populations.These population models are traditionally formulated as first order PDE with non-local bound-ary conditions. The delay formulation amounts to restriction to a forward invariant attractingset and so the information that is lost concerns transient behaviour only.
As an application, I will briefly discuss catastrophic collapse due to an emergent Allee effect,as found by Andre de Roos and Lennart Persson in their work on size-structured models ofexploited fish populations.
Next I will indicate how sun-star calculus for adjoint semigroups yields a convenient frame-work for developing the qualitative theory of delay equations.
The lecture is based on joint work with Mats Gyllenberg, Hans Metz and many others.
L14
Dynamics of spatially localized patterns
Yasumasa Nishiura [email protected]
Laboratory of Nonlinear Studies and Computation RIES, Hokkaido University
Spatially localized patterns are ubiquitous such as chemical blobs, discharge patterns, morphology patterns, and binary convection cells [5]. They have their own internal dynamics and can propel by themselves, for instance, via drift bifurcation. They collide each other, interact strongly and emit various outputs depending on the velocity and other parameters such as incident angle. Typical outputs are annihilation, rebound, and merging, however more exotic dynamics such as spatio-temporal chaos [8] and rotational motions [3] can also be observed. It has been clarified at least numerically that a network of unstable patterns called scattors forms a backbone for large deformations from input to output [7]. One of the origins for such a network could be an unfolding of some singularity of high-codimension. The above viewpoint turns out to be quite useful to understand the dynamics of traveling patterns in heterogeneous media. When the media is not uniform, there appear various types of defects depending on the shape of heterogeneity [4][6]. Motion in such a media can be regarded as an interaction process between the traveling pattern and a network of defects. Even the simplest type of heterogeneity like a jump along line produces a rich dynamics as discussed in [2]. Recently a new mathematical model to describe the adaptive behaviors of plasmodium slime molds to chemical repellent is proposed [1] and it is expected that the underlying mathematical mechanism for the responses such as penetration, rebound, and splitting could fall in our framework. This talk is based on the long-term collaboration with Takashi Teramoto, Kei-Ichi Ueda, Makoto Iima, and Yuan Xiaohui. REFERENCES [1] K.-I. Ueda, S.Takagi, Y.Nishiura, and T.Nakagaki, Mathematical model for contemplative amoeboid locomotion, to appear in Phys. Rev. E. [2] X. Yuan, T.Teramoto, and Y. Nishiura, Heterogeneity-induced spot dynamics for a three-component reaction-diffusion system, to appear in Comm. Pure Appl. Anal. [3] T.Teramoto, K.Suzuki, and Y.Nishiura, Rotational motion of traveling spots in dissipative systems, Phys. Rev. E 80, 046208 (2009) [4] T. Teramoto, X.Yuan, and Y. Nishiura, Onset of unidirectional pulse propagation in an excitable medium with asymmetric heterogeneity, Phys. Rev. E 79, 046205 (2009) [5] M. Iima and Y. Nishiura, Unstable periodic solution controlling collision of localized convection cells in binary fluid mixture, Physica D 238 (2009) 449-460 [6] Y. Nishiura, T. Teramoto, X. Yuan and K. Ueda, Dynamics of traveling pulses in heterogeneous media, Chaos, 17(3) : 037104 (2007) [7] Y. Nishiura, T. Teramoto and K. Ueda, Scattering of traveling spots in dissipative systems, Chaos, 15: 047509-047519 (2005) [8] Y. Nishiura, T. Teramoto and K. Ueda, Dynamic transitions through scattors in dissipative systems, Chaos, 13(3): 962-972 (2003)
L15
The Stability and Dynamics of Localized Spot Patterns in theTwo-Dimensional Gray-Scott Model
Wan Chen and Michael J. [email protected] [email protected]
Wan Chen (Oxford Centre Collaborative Applied Mathematics, Oxford, UK); Michael J. Ward (Dept. ofMathematics, UBC, Canada)
The dynamics and stability of multi-spot patterns to the Gray-Scott (GS) reaction-diffusionmodel in a two-dimensional domain is studied in the singularly perturbed limit of small dif-fusivity ε of one of the two solution components. A hybrid asymptotic-numerical approachbased on combining the method of matched asymptotic expansions with the detailed numericalstudy of certain eigenvalue problems is used to predict the dynamical behavior and instabilitymechanisms of multi-spot quasi-equilibrium patterns for the GS model in the limit ε → 0. Forε → 0, a quasi-equilibrium k-spot pattern is constructed by representing each localized spot asa logarithmic singularity of unknown strength Sj for j = 1, . . . , k at unknown spot locationsxj ∈ Ω for j = 1, . . . , k. A formal asymptotic analysis is then used to derive a differentialalgebraic ODE system for the collective coordinates Sj and xj for j = 1, . . . , k, which char-acterizes the slow dynamics of a spot pattern. Instabilities of the multi-spot pattern due tothe three distinct mechanisms of spot self-replication, spot oscillation, and spot annihilation,are studied by first deriving certain associated eigenvalue problems by using singular perturba-tion techniques. From a numerical computation of the spectrum of these eigenvalue problems,phase diagrams representing in the GS parameter space corresponding to the onset of spotinstabilities are obtained for various simple spatial configurations of multi-spot patterns. Inaddition, it is shown that there is a wide parameter range where a spot instability can be trig-gered only as a result of the intrinsic slow motion of the collection of spots. The constructionof the quasi-equilibrium multi-spot patterns and the numerical study of the spectrum of theeigenvalue problems relies on certain detailed properties of the reduced-wave Green’s function.The hybrid asymptotic-numerical results for spot dynamics and spot instabilities are validatedfrom full numerical results computed from the GS model for various spatial configurations ofspots.
REFERENCES[1] W. Chen, M. J. Ward, The Stability and Dynamics of Localized Spot Patterns in the Two-Dimensional Gray-ScottModel, submitted, SIADS, (2010).
L16
Vortex dynamics in aorta with torsion
Hiroshi Suito*† and Takuya Ueda†† [email protected]
*Department of Environmental and Mathematical Sciences, Okayama University, Okayama, 700-8530, Japan †PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan
††Department of Radiology, Chiba University Hospital, Chiba, 260-8670, Japan
Thoracic endovascular aortic repair (TEVAR), or stent–graft treatment, has become widely accepted as an important option of thoracic aortic diseases. Many studies have proven the safety and efficacy of TEVAR with satisfactory short-term to mid-term outcomes. Even if the initial TEVAR treatment technically succeeds, some patients show recurrence and progression of disease many years after treatment. Based on long-term follow-up examinations, such long-term morphological change and effects of hemodynamic flow apparently interact synergically. Constant pulsatile hemodynamic effects from blood flow apparently induce degeneration of the underlying aorta to cause its morphological change and to induce minor morphological changes that alter the hemodynamic state. These changes ultimately engender long-term adverse events.
The morphology of the thoracic aorta varies widely among individuals in terms of distributions of radius, curvature, torsion, and so on. Therefore, the flow fields within them show very different characteristics from a fluid dynamical perspective. Figure 1 portrays one numerical result, depicting contour surfaces of streamwise vorticity at the late diastole of the cardiac cycle. Some patients with aneurysms or strongly tortuous vessels show swirling flows that remain even at the late diastole. Streamwise vortex cores, which are initially generated at the systole of the cardiac cycle, merge to form a single vortex core.
Therefore, we check the effects of curvature and torsion on the formation of the swirling flow. Simple spiral tubes are used for this purpose which torsions are varied while their curvatures are kept constant. Secondary flows in systole phase are almost same while they are completely different in diastole phase. Figure 2 show the secondary flows in non-zero torsion case in systole and diastole phases. Swirling flow in the diastole phase can be seen also in this simplified case. This phenomenon seems to depend on both Dean’s number
€
De = Re d /2rc and Womersley number
€
Wo = d ω /4ν , where Re, d,
€
rc ,
€
ω and
€
ν are Reynolds number, diameter of the tube, radius of curvature, angular frequency of the pulsation and kinetic viscosity, respectively. This phenomenon apparently affects the long-term morphological change of the aorta and long-term adverse events. REFERENCES [1] J. R. Womersley, Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known, Journal
of Physiology, 127, 3 (1954) 553–563. [2] S. A. Berger, L. Talbot and L. S. Yao, Flow in curved pipes, Annual Review of Fluid Mechanics, 15 (1983) 461-512. [3] H. Suito and T. Ueda, Numerical simulation of blood flow in thoracic aorta, Medical Imaging Technology, 28, 3 (2010) 175-180.
Fig. 1 Streamwise vorticity.
Peak systole Late diastole Fig. 2 Secondary flows in non-zero torsion case
L17
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L18
Asymptotic Front Speeds of Viscous G-Equation in Cellular Flows
Jack XinUniversity of California at Irvine
Inviscid G-equation is a well-known Hamilton-Jacobi level-set equation in turbulent combustion.The large time front speed c* depends on the streamline structure of the flow field. In case ofperiodic array of vortices (cellular flows), c* of the inviscid G equation scales as O(A/log A),for large A, the amplitude of cellular flow. Viscous G-equation arises from several scenariossuch as numerical approximation, viscous regularization, or the so called Markstein diffusivityin flame stability analysis. A striking phenomenon is that for any small viscosity, the viscousfront speed c* is uniformly bounded in large A. The proof of this speed bending effect is basedon analysis of traveling front equation or cell problem of homogenization theory.
L19
Point vortex dynamics in multiply connected domains
Takashi [email protected]
Department of mathematics, Hokkaido UniversityPRESTO, Japan Science and Technology
We consider incompressible fluid flows in two-dimensional multiply connected domains, whichare regarded as simple mathematical models for biofluids such as insect-flights, fish swimmingand falling leaves. It has recently been recognized that the vortex-boundary interaction is afundamental mechanism to realize an efficient flight of insects[5] and a slow falling of a plantseed[4]. In slightly viscous fluids, vortices detach from the boundaries after they are createdby the viscous effect due to the formation of boundary layers in the neighborhood of theboundaries. Then the detached vortices begin interacting with the boundaries, which generatesadditional forces. Thus theoretical study of such vortex-boundary interaction is becoming anmathematical challenge.
Here we suppose that the fluid is inviscid. The effect of viscosity is neglected for the timebeing, since we are interested in the theoretical understanding of the interaction between theboundaries and the vortex located far from the boundaries. Then the vorticity is a conservedquantity along the path of a fluid particle according to Kelvin’s circulation theorem. For thesake of further simplicity, we assume that the vorticity is confined in a discrete point, which iscalled a point vortex. Hence, we have only to investigate the evolution of point vortices thatexist at the initial moment, which makes the theoretical study easier.
In the present talk, I will provide two kinds of mathematical formulations to describe theinteractions between boundaries and point vortices. First, we consider the unit circle in thecomplex plane that contains many circular boundaries, which is regarded as a canonical circulardomain. The equation of motion for point vortices in the circular domain is given by Sakajo[6]based on an analytic formula of the Green function due to Crowdy et al[3]. For a given multiplyconnected domain, the equation for point vortices is derived from the canonical equation byconstructing a conformal mapping from the target domain to the canonical circular domain.Second, we propose a numerical method to approximate the evolution of point vortices usingthe numerical conformal mapping method developed by Amano et al[1,2]. This method can beapplied to multiply connected domains with complex boundaries.
Moreover, using the mathematical formulations, we discuss stationary configurations of pointvortices in multiply connected domains. They shed lights on fundamental aspects of the vortex-boundary interaction. Besides, it is known that the stable vortex equilibria are closely relatedto the most statistically probable states in the system of many point vortices.
REFERENCES[1] K. Amano, A charge simulation method or the numerical conformal mapping of interior, exterior anddoubly-connected domains, J. Comp. Appl. Math. 53 (1994), 353–370.[2] K. Amano, A charge simulation method for numerical conformal mapping onto circular and radial slit domains,SIAM J. Sci. Comput. 19 (4) (1998), 1169–1187.[3] D. Crowdy and J. Marshall, Analytical formulae for the Kirchhoff-Routh path function in multiply connecteddomains, Proc. Roy. Soc. A 461 (2005), pp.2477–2501.[4] D. Lentink, W. B. Dickson, J. L. van Leeuwen and M. H. Dickinson, Leading-edge vortices elevate lift ofautorotating plant seeds, Science 324 (2009), pp.1438–1440.[5] M. Iima and T. Yanagita, Is a two-dimensional butterfly able to fly by symmetric flapping?, J. Phys. Soc. Japan 70(2001), pp.5–8.[6] T. Sakajo, Equation of motion for point vortices in multiply connected circular domains, Proc. Roy. Soc. A 465(2009), pp. 2589–2611.
1
L20
Localized patterns in a reaction-diffusion system with conservationof a mass
Yoshihisa [email protected]
Department of Applied Mathematics and Informatics, Ryukoku University
We consider the following two-component system of reaction-diffusion equationsut = d1∆u− g1(u, v) + g2(u, v),
vt = d2∆v + g1(u, v)− g2(u, v),
in a bounded domain Ω with the homogeneous Neumann boundary conditions. This systemhas conservation of a mass, namely, the integral of u + v over the domain is preserved for t.With an appropriate choice of g1, g2 this system allows a Turing-type instability for a constantsteady state and it induces a localized pattern. We discuss the mechanism of this patternformation. In particular, if g1 = f(u), g2 = v, the Morse index of the steady state solutionof the system coincides with the one of an associate scalar reaction-diffusion equation witha nonlocal term. It thereby tells why the stable localized pattern emerges. We also provideanother case of nonlinear terms, which allows a time periodic motion. This talk is partiallybased on the collaboration work with Professor Toshiyuki Ogawa.
L21
Traveling waves in a sawtoothed cylinder and their homogenizationlimit
Hiroshi [email protected]
University of Tokyo
My talk is concerned with a curvature-dependent motion of plane curves in a two-dimensionalinfinite cylinder with spatially undulating boundary. The domain is given in the form
Ω = (x, y) ∈ R2 | −H − g−(y) < x < H + g+(y),where H > 0 is a constant and g±(y) ≥ 0 are bounded smooth functions which are assumedto be recurrent. (This class includes periodic, quasi-periodic and almost periodic functions asspecial cases.) The curve γt moves by the law V = κ+A, where V denotes the normal velocity,κ the curvature and A is a positive constant. The two ends of curve γt meet the boundaries ofΩ perpendicularly. Such an equation appears as a singular limit of the Allen-Cahn equation.
The goal is to study how the average speed of the traveling wave depends on the geometryof the domain boundary. For this purpose we consider the homogenization problem as theboundary undulation becomes finer and finer. More precisely, we replace g±(y) by εg±(y/ε),and determine the homogenization limit (as ε → 0) of the average speed and the limit profile ofthe traveling waves. Quite surprisingly, this homogenized speed depends only on the maximalopening angles of the domain boundary and no other geometrical features are relevant.
Next we consider the special case where the boundary undulation is quasi-periodic with mindependent frequencies. More precisely, we assume that
g±(y) = f(ω1y, ω2y, . . . , ωmy),
where f(z1, . . . , zm) is periodic with period 1 in each component, and the coefficients ω1, . . . , ωm
are linearly independent over Q. Under suitable assumptions on f we show that the rate ofconvergence of the average speed cε to its homogenization limit c0 depends on this number m.
More specifically, the convergence rate is roughly given by O(ε2
m+3 ). In the special case whereg± are periodic (that is, m = 1), this rate amounts to O(
√ε), but it is slower if m ≥ 2.
This is joint work with Bendong Lou and Ken-Ichi Nakamura.
L22
A combinatorial framework for nonlinear dynamics
Konstantin MischaikowRutgers, The State University of New Jersey
Invariant sets are the focal point of much of nonlinear dynamics. Using biological models asexamples I will argue that there are situations in which this information is much too pre-cise. With this as motivation I will describe new computational tools that provided rigorouscomputational results for multiparameter systems. I will conclude with a discussion of recentefforts to develop a theoretical framework for nonlinear dynamics that is explicitly tied to acomputational framework.
L23
Topological bifurcation theorems for Morse decompositions
Hiroshi KokubuDepartment of Mathematics, Kyoto University / JST CREST
Inspired by recent development of topological-computational methods for global dynamics, Iwill discuss a few results on ”bifurcations” associated to changes of Morse decompositions thatcan be observed by these computational methods.
L24
Dynamics of Swelling of polyelectrolyte gels
James P. KeenerUniversity of Utah
It has long been known that the swelling dynamics of mucin vesicles is much different thanthat of most other polymer gels. For example, mucin released from goblet cells can swellmore than 100-fold in size in a few seconds, while the swelling of most ”typical” gels takesmany hours. The reason for this drastic difference is that mucin is a polyelectrolyte gel whosenegative charges can be shielded by monovalent (hydrogen, sodium, potassium, for example)or divalent ions (calcium). When shielded by divalent ions the gel can be highly condensed,while when shielded by monovalent ions, condensation is not possible. Thus, a change in therelative concentration of monovalent to divalent ions in the solvent bath can lead to dramaticand rapid changes in the condensation state of the gel.
In this talk, I will present the first comprehensive theory for the kinetics of swelling anddeswelling of polyelectrolyte gels using the framework of multiphase fluid flow. I will showhow to incorporate the Gibbs free energy into the kinetic equations, and how the free energyis influenced by the relative amounts of monovalent and divalent ions. Further, I will showthat the equations of motion satisfy a minimum energy dissipation rate principle, similar to theminimum energy dissipation rate principle for a Stokes flow established by Helmholz. Finally, Iwill show that the swelling and deswelling of a polyelectrolyte gel takes place in two steps, thefirst in which crosslinking changes rapidly with little change in volume fraction, and the secondin which crosslinking and gel expansion and contraction take place on the same time scale.
This model is relevant for many other biological gels where calcium is used to regulatecontraction or expansion.
L25
Poster Presentations ( P01 - P51 )
TBA
Yuyu PengUniversity of California at Irvine
P01
Dynamical approach in rigorous verification of equilibria forevolutionary equations
Kaname [email protected]
Department of Mathematics, Kyoto University.
We present a dynamical theoretical approach combined with rigorous numerics for proving theexistence of stationary solutions of partial differential equations. If we want to understandphenomena, it is important to study the behavior of systems, say, the global attractor fordissipative systems, and the study of dynamics around equilibria is realized as the beginningof the study of global dynamics.
We use a topological tool called the Conley-Rybakowski index ([5]), an extension of Conley’soriginal index, and the finite element method. Our method can be applied to study dynamicsnear equilibria of dissipative partial differential equations with various boundary conditions.We consider the following type of partial differential equations:
(1) utt + αAut + Au = f(u, ut), (t, x) ∈ [0,∞)× Ω,
where α > 0, A is a sectorial operator, f is a nonlinearity and Ω ⊂ Rn is a bounded domain.
Our approach can be realized as the extension of Nakao’s method (e.g.[3]) and the ZM-method(i.e. method by Zgliczynski and Mischaikow ([6])) in the following sense.
• We can verify the existence of invariant sets containing equilibria of (1) and their sta-bility.• We can also verify the local uniqueness and hyperbolicity of equilibria.• We can apply our verification method to (1) with various boundary conditions such as
the zero Dirichlet or zero Neumann conditions.• We can apply our verification method to (1) on (not necessarily convex) bounded do-
mains.
The basic idea of our method is based on Nakao’s method and the ZM-method. Thereforeour idea has a potential for being applied to studying global dynamics for a broad class ofdissipative partial differential equations.
REFERENCES
[1] J.K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, 25. AMS,Providence, 1988.[2] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, 1981.[3] N. Yamamoto and M.T. Nakao, Numerical verification of solutions for elliptic equations in nonconvex polygonaldomains, Numer. Math., 65(1993), 503–521.[4] M.T. Nakao, K. Hashimoto and Y. Watanabe, A Numerical Method to Verify the Invertibility of Linear EllipticOperators with Applications to Nonlinear Problems, Computing, 75(2005), 1–14.[5] K.P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer-Verlag, 1987.[6] P. Zgliczynski and K. Mischaikow, Rigorous numerics for partial differential equations: The Kuramoto-SivashinskyEquation, Found. Comput. Math. 1(2001), 255–288.
1
P02
Control of pacing sites by electrical far-field stimulation in discreteexcitable media
Marcel Horning1, Seiji Takagi3, and Kenichi Yoshikawa1,2
[email protected] Department of Physics, Graduate School of Science, Kyoto University, Japan,2 Spatio-Temporal Order Project, ICORP JST, Kyoto, 606-8502, Japan, and
3 Research Institute for Electronic Science, Hokkaido University, Sapporo 060-0812, Japan
Cardiac arrhythmias, a precursor of fibrillation-like states in the beating heart, are associatedwith spiral waves and spatiotemporal chaos of wave propagation, respectively. Far-field pacing(FFP) also known as wave emission on heterogeneities is a promising method for terminatingsuch waves by using heterogeneities in the tissue as internal pacing sites. It is known that theminimum electrical field decreases for increasing obstacle sizes [1]. In this study we investigatedthe role of multiple obstacles and their interaction during FFP. It is shown that a secondarynearby obstacle can significantly modulate the minimum electrical field [1]. Further, we in-vestigate in the role of cell alignment and their influence of the number of pacing sites duringFFP. It is shown that excitable media with large cell angle dissipation require significant lowerelectrical fields to utilize FFP.Results are obtained by means of numerical simulations of a bidomain model on the one hand,and on the other hand are confirmed by experiments of cardiac tissue culture. The presentedresults open a way for a multisite pacing method in real beating heart that was not possiblewith conventional methods.
REFERENCES
[1] A. Pumir, V. Nikolski, M. Horning, A. Isomura, K. Agladze, K. Yoshikawa, R. Gilmour, E. Bodenschatz and V.Krinsky, Physical Review Letters, 99, 208101, 2007[2] M. Horning, S. Takagi, and K. Yoshikawa, Physical Review E, 82, 021926, 2010
1
P03
On the relationship between the essential spectrum of internal waves and non-
uniqueness of the limit amplitude for the dynamics of stratified fluid
A.Giniatoulline, E. Mayorga [email protected], [email protected]
Los Andes University, Colombia, South America
We consider the non-homogeneous system which describes the dynamics of exponentially
stratified 3-dimensional fluid in the gravity field:
=∂
∂+
∂
∂+
∂
∂
=−∂
∂
=∂
∂++
∂
∂
=∂
∂+
∂
∂
=∂
∂+
∂
∂
0
),(
),(
),(
),(
3
3
2
2
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434
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x
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x
v
x
v
txFvt
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txFx
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txFx
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1. For the Cauchy problem and the boundary value problem for the half-space 3+R we prove
the existence and uniqueness of the solutions in pL class.
2. For the half-space problem we establish the boundary-layer effect near the bottom of the
Ocean: in the proximity of the bottom of the Ocean, the asymptotical decay of the velocity
field is slower then in the interior domains.
3. We prove that for ∞→t , for the case when the right side depends on time harmonically
( tiexftxF ω−= )(),( ), the stabilization takes place and the solution describes the mode of
the induced vibrations with frequency ω .
4. We also study the relationship between the spectrum of the normal vibrations and the
uniqueness of the limit amplitude. The spectrum of normal vibrations contains bifurcation
points and characterizes the appearance of turbulent flows. We prove that, if the spectral
parameter belongs to the essential spectrum, then the limit amplitude loses the uniqueness.
REFERENCES [1] A.Giniatoulline, On the essential spectrum of the operators generated by PDE systems of stratified fluids, Internat.J.Computer
Research, 12,2003,p.63-72
[2] A.Giniatoulline, C.Rincón, On the spectrum of normal vibrations for stratified flows, Computational Fluid Dynamics
J.,13(2),2004,p.273-281.
[3] T.Kato, Perturbation theory for linear operators, Springer, Berlin, 1966
[4] V.Maslennikova, A.Giniatoulline, Spectral properties of operators for systems of hydrodynamics, Siberian
Math.J.,29,1988,p.812-824
[5] A.Giniatouline, O.Zapata. On some qualitative properties of stratified flows, RIMS Kôkyûroku Bessatsu, Series B1, 2007,
p.145-157.
[6] A.Giniatouline, E.Mayorga. On some properties of the solutions of the problem modelling stratified ocean and atmosphere
flows in the half-space, Rev.Colombiana Mat., 43,2009,p.43-54
P04
Relaxation scaling laws in a periodically forced Swift-Hohenberg systemSyuji Miyazaki†, Kai Morino1 and Katsuya Ouchi2
Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan1Graduate School of Information Science and Technology, University of Tokyo, Tokyo 113-8656, Japan
2Kobe Design University, Kobe, 651-2196, Japan
Many studies have been compiled on pattern dynamics far from equilibrium such as thermal convection,chemical reaction and biological pattern formation, etc.[1, 2, 3] One of the examples of such phenomena isthe onset of Rayleigh-Benard convection and the formation of roll pattern. It is well known that the systemis modeled by the Swift-Hohenberg (SH) equation[4]. It is derived by considering the Navier-Stokes equationin the Boussinesq approximation and the thermal diffusion equation, the boundary conditions of which aregiven by two infinite horizontal plates with temperature T0 and T0 + ∆T . When the value of Rayleigh numberR, which is a dimensionless temperature difference ∆T , is larger than a critical value Rc, convection rollsemerge. The linearized equations give two stable eigenvalues corresponding to two horizontal directions ofthe velocity field and two unstable ones corresponding to the vertical direction and the other direction of thetemperature field. Neglecting the stable modes and considering wavelengths near the most unstable wavelength
k0 only, which can be set unity without loss of generality, we have the following SH equation∂s(r, t)∂t
=[ϵ − (∇2 + k2
0)2]s(r, t) − s(r, t)3, where s(r, t) is a scalar state valuable as a function of two-dimensional
position r and time t, and ϵ is a control parameter being proportional to the distance from the critical pointR − Rc. When ϵ is positive, the asymptotic spatial structure of s(r, t) is characterized by a stripe pattern,which represents convection rolls. The generalized Swift-Hohenberg equation was also introduced, which showspattern dynamics different from those of the original ones.[5] Let S(k, t) be an intensity of the Fourier transformof s(r, t) defined by S(k, t) ≡ |
∫s(r, t) exp (ik · r) dr|2, where k is a two-dimensional vector of wave number.
The amplitude k and the angle θ are given by k = (k cos θ, k sin θ) as cylindrical coordinates. Because S(k, t)depends not very on the angle θ but strongly on the amplitude k due to an isotropy of the system, we introduceS(k, t), where is referred to as structure factor in the following, by averaging S(k, t) over the angle directionas: S(k, t) = 2
π
∫ π/2
0S(k, θ, t)dθ, which has a sharp peak around k0. Not only in a convective system but
also in a uniaxial ferromagnetic garnet thin film, various spatial patterns are observed. The films withoutexternal field show labyrinthine magnetic domain structures. Added a temporally oscillating magnetic fieldperpendicular to the film, the domain structure changes to various patterns, depending on the amplitude andthe frequency of the applied field.[6, 7] Noting that garnet films have the magnetic easy axis perpendicular to thefilm plane, we use a two-dimensional continuous Ising spin system in order to describe the domain structures.Noting that when no magnetic field is applied, the system has a spin-flip symmetry and the domain structureis labyrinthine, Tsukamoto and his coworkers considered the pattern dynamics of the following periodically
forced SH equation:[8]∂s(r, t)∂t
=[ϵ − (∇2 + k2
0)2]s(r, t) − s(r, t)3 + h sin(Ωt). They found that a stripe
pattern, a polka-dot pattern and a spatially uniform state with a temporal oscillation are observed, dependingon the values of the amplitude h and the angular frequency Ω of the external field[8]．We assume the followingscaling law of the structure factor S(k, t) = tαfh,Ω
((k0 − k)tα
)introduced by Elder and his coworkers, who
estimated the scaling exponent as α = 1/5 for the original SH equation, and α = 1/4 for the SH equationwith an additional noise in the case of ϵ = 0.25.[9, 10] Hou and his coworkers found a logarithmic behaviorS(k0, t) ∝ log t for ϵ = 0.75 at zero noise.[11] Note that the scaling function in our case is not universal anddepends on the amplitude h and the angular frequency Ω of the external field. In this presentation, we studyrelaxation and hysteresis in a periodically forced Swift-Hohenberg equation, in which we obtain different scalingexponents for a stripe pattern and a polka-dot patterns and compare those with the result of the unforced case,and we observe a bistability of stripe and polka-dot patterns.
References
[1] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65 (1993), 851.[2] P. Manneville, Dissipative Structures and Weak Turbulence (Academic Press, Boston, 1990).
[3] G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems (Wiley, New York, 1977).[4] J. Swift and P. C. Hohenberg, Phys. Rev. A 15 (1977), 319.[5] M’ F. Hilali, S. Metens, P. Borckmans and G. Dewel, Phys. Rev. E 51 (1995), 2046.[6] G. S. Kandaurova, Physics-Uspekhi 45 (2002), 1051.
[7] M. Mino, S. Miura, K. Dohi and H. Yamazaki, J. Magn. Magn. Mater. 226-230 (2001), 1530.[8] N. Tsukamoto, H. Fujisaka and K. Ouchi, Prog. Theor. Phys. Suppl. 161 (2006), 372.[9] K. R. Elder, J. Vinals and M. Grant, Phys. Rev. A 46 (1992), 7618.
[10] K. R. Elder, J. Vinals and M. Grant, Phys. Rev. Lett. 68 (1992), 3024.
[11] Q. Hou, S. Sasa and N. Goldenfeld, Physica A 239 (1997), 219.
1
P05
Gene Expression Time Delays and Turing Pattern Formation
S. Seirin Leea and E.A. Ganeyb
aGraduate School of Mathematical SciencesThe University of Tokyo, JAPAN
[email protected]; [email protected]
bCentre for Mathematical BiologyMathematical Institute, University of Oxford, UK
There are numerous examples of morphogen gradients controlling long range signalling in developmen-tal and cellular systems. The prospect of two such interacting morphogens instigating long range self-organisation in biological systems via a Turing bifurcation has been explored, postulated or implicatedin the context of numerous developmental processes. However, modelling investigations of cellular sys-tems typically neglect the inuence of gene expression on such dynamics, even though transcription andtranslation are observed to be important in morphogenetic systems. Critically, the previous study [1] ob-served that even these relatively small gene expression delays substantially amplify patterning lags, thusdemonstrating the potential for gene expression dynamics to exert enormous temporal inuence on patternformation.
The investigations of our study demonstrate that the behaviour of Turing models profoundly changes onthe inclusion of gene expression dynamics and is sensitive to the sub-cellular details of gene expression [2].Furthermore, they also highlight that domain growth can no longer ameliorate the excessive sensitivityof Turing's mechanism in the presence of gene expression time delays [3, 4]. These results also indicatethat the behaviour of Turing pattern formation systems on the inclusion of gene expression time delaysmay provide a means of distinguishing between possible forms of interaction kinetics, and also emphasisesthat sub-cellular and gene expression dynamics should not be simply neglected in models of long rangebiological pattern formation via morphogens. Finally, our study calls into question the plausibility ofthe Turing mechanism for pattern formation in biology where robustness is a key requirement. Turing'smechanism would generally require a novel and extensive secondary mechanism to control reaction diusionpatterning.
Keywords Pattern Formation, Cellular Dynamics, Time Delays, Reaction-Diusion System
References
[1 ] E.A. Ganey, N.A.M. Monk, Gene expression time delays and Turing pattern formation systems, Bull.Math.Bio. (2006)68: 99130.
[2 ] S. Seirin Lee, E.A. Ganey, N.A.M. Monk,The Inuence of Gene Expression Time Delays on Gierer-Meinhardt Pattern
Formation Systems, Bull.Math.Bio. (2010) 72: 2139?2160.
[3 ] S. Seirin Lee, E.A. Ganey, Aberrant Behaviours of Reaction Diusion Self-organisation Models on Growing Domains
in The Presence of Gene Expression Time Delays, (2010) 72: 2161?2179.
[4 ] S. Seirin Lee, E.A. Ganey, R.E. Baker, An Investigation of Pattern Formation on Morphogen-controlled Growing
Domains with Gene Expression Time Delays and Domain Response Delays, Submitted
P06
Bifurcation analysis to the Lugiato–Lefever equation in one space dimension
T. Miyajia, I. Ohnishi
b and Y. Tsutsumi
c
a Research Institute for Mathematical Sciences, Kyoto University
b Department of Mathematical and Life Sciences, Hiroshima University
c Department of Mathematics, Kyoto University
We study the stability and bifurcation of steady states for a certain kind of damped driven nonline-
ar Schrödinger equation (NLS) with cubic nonlinearity and a detuning term in one space dimen-
sion, mathematically in a rigorous sense. The equation is derived by Lugiato and Lefever [1] as a
model equation for pattern formation in nonlinear optics. They pointed out that the Turing-type
instability is responsible for the pattern formation in optical cavity. It is known by numerical sim-
ulations that the system has a solitary wave solution as a stationary solution. Unlike the (conserva-
tive) NLS, the Lugiato-Lefever equation defines a dissipative system. Therefore the solitary wave
solution should be distinguished from so-called solitons. Moreover the system shows lots of coex-
isting spatially localized structures as a result of subcritical Turing bifurcation [2]. In addition, the-
se coexisting states are connected in the extended phase space and form snaking bifurcation struc-
tures, on which branches of stationary solutions undergo successive fold bifurcations [3].
Since the equation does not have a variational structure, unlike the conservative case, we
cannot apply a variational method for capturing the ground state. The authors have analyzed the
equation from a viewpoint of bifurcation theory [4].
In the case of a finite interval with periodic boundary conditions, we prove the fold bifurca-
tion of nontrivial stationary solutions around the codimension two bifurcation point of the trivial
equilibrium point by exact computation of a fifth-order expansion on a center manifold reduction.
In addition, we analyze the steady-state mode interaction and prove the bifurcation of mixed-mode
solutions. The mixed-mode solutions are all unstable. However they are important because they
will be a germ of snaking bifurcation structures on a finite interval. Finally, we study the corre-
sponding problem on the entire real line by use of spatial dynamics. We obtain a small solitary
wave solution bifurcated adequately from the trivial equilibrium.
REFERENCES [1] L.A. Lugiato, R. Lefever, Spatial dissipative structures in passive optical systems, Phys. Rev. Lett. 58 (1987) 2209–2211.
[2] A.J. Scroggie, W.J. Firth, G.S. McDonald, M. Tlidi, R. Lefever, L.A. Lugiato, Pattern formation in a passive kerr cavity, Chaos
Solitons Fractals 4 (1994) 1323–1354.
[3] D. Gomila, A. Jacobo, M.A. Matías, P. Colet, Phase-space structure of two-dimensional excitable localized structures, Phys.
Rev. E 75 (2007) 026217.
[4] T. Miyaji, I. Ohnishi, Y. Tsutsumi, Bifurcation analysis to the Lugiato–Lefever equation in one space dimension, Physica D
239 (2010) 2066–2083.
P07
Fluctuation-spectrum approach tofew- and large-degrees-of-freedom chaotic systems
Katsuhito Matsui†, Syuji Miyazaki, Miki U. Kobayashi1,Katsuya Ouchi2 and Takehiko Horita3
†[email protected] School of Informatics, Kyoto University, Kyoto 606-8501, Japan
1Research Institute for Mathematical Science, Kyoto University, Kyoto 606-8502, Japan2Kobe Design University, Kobe, 651-2196, Japan
3Department of Mathematical Sciences, Osaka Prefecture University, Osaka 599-8531, Japan
In few-degrees-of-freedom chaotic dynamical systems, local expansion rates which evaluatean orbital instability fluctuate largely in time, reflecting a complex structure in the phase space.Its average is called the Lyapunov exponent, whose positive sign is a practical criterion of chaos.There exist numerous investigations based on large deviation statistics in which one considersdistributions of coarse-grained expansion rates (finite-time Lyapunov exponent) in order toextract large deviations caused by non-hyperbolicities or long correlations in the vicinity ofbifurcation points [1]. It is recently shown[2] that the Lorenz system[3] has both hyperbolicand non-hyperbolic parameter regions by use of covariant Lyapunov vectors[4]. The Lorenzplot also reflect a difference between hyperbolicity and non-hyperbolicity. In this presentation,we show that the fluctuation spectra (rate functions) of the local expansion rate distinguishthe both. In this presentation, we also characterize spatio-temporal intermittency in a coupledsystems of chaotic elements[5] and turbulence modeled by a shell model[6] using statisticalproperties of not only the largest but also all other Lyapunov exponents such as variances andrate functions as well as Lyapunov dimensions.
30
32
34
36
38
40
42
44
46
48
30 32 34 36 38 40 42 44 46 48
z(n+
1)
z(n)
(a) r = 28 (hyperbolic)
40
45
50
55
60
65
70
40 45 50 55 60 65 70
z(n+
1)
z(n)
(b) r = 40 (non-hyperbolic)
Figure 1. Lorenz plots of the Lorenz equation (x, y, z) = (−10x + 10y,−xz +rx − y, xy − 8z/3)
References
[1] H. Mori and Y. Kuramoto, Dissipative Structures and Chaos, Springer, Berlin (1998).[2] Y. Saiki and M. U. Kobayashi, JSIAM Letters 2 (2010) pp. 107-110[3] E. N. Lorenz, Journal of the Atmospheric Sciences, 20 (1963) pp. 130-141.[4] F. Ginelli et al., Phys. Rev. Lett. 99, 130601 (2007) [4 pages].[5] K. Kaneko, Prog. Theor. Phys. 74 (1985) pp. 1033-1044.[6] M. Yamada and K. Ohkitani, J. Phys. Soc. Jpn. 56 (1987) pp. 4210-4213
1
P08
Modeling lung branching morphogenesis
Takashi Miura [email protected]
Department of Anatomy and Developmental Biology Kyoto University Graduate School of Medicine
Vertebrate lung forms a tree-like structure to increase surface area for gas exchange. This structure is generated by repeated splitting of the epithelial tube during development via complex interaction between epithelial and mesenchymal tissues. Numerous molecules are known to be involved in this process during lung development, but how interaction of these molecules results in branched structure is not well understood.
At first we looked for simplest possible experimental system to understand the mechanism and utilize mesenchyme-free culture system [1]. In this system, lung epithelium is isolated from mesenchyme and embeded in Matrigel and cultured under the presence of fibroblast growth factor (FGF) molecule. Then the lung epithelium undergoes branching morphogenesis in vitro without mesenchyme tissue. This system is in many aspects similar to bacteria colony experimental system in which consumption of nutrient by the bacterial cell induce interface instability. We compare these two systems and formulate a simple reaction-diffusion based model. We experimentally verified the model and conclude this model captures the essence of the pattern formation in vitro [2]. Next, we extrapolate this result to in vivo situation using avian lung. In avian lung, branched structure and cystic structure (air sac) is generated simultaneously. We can predict that cystic structure is generated when diffusion of FGF is very fast or concentration of FGF is very high. We experimentally show that fast FGF diffusion is responsible for the cystic structure in avian lung [3].
Despite all these effort, mechanism of branching morphogenesis in vivo still remain to be
elucidated. Recently a model was proposed which incorporate the regulation of FGF expression and tissue geometry regulate tip splitting during branching morphogenesis [4]. To test this model, we developed a reconstruction culture in which mesenchyme tissue is isolated to single cell level and reconstructed using centrifugation. This system has an ability to generate branched structure, indicating the interaction between epithelium and mesenchyme has a self-organizing property. We show that previous descriptive model [4] actually have an ability to generate interface instability. REFERENCES [1] Nogawa, H. and Ito, T. ,Branching morphogenesis of embryonic mouse lung epithelium in mesenchyme-free Culture, Development 1995, 121(4) 1015-22. [2] Miura, T. and Shiota, K., Depletion of FGF acts as a lateral inhibitory factor in lung branching morphogenesis in vitro, Mech Dev 2002, 116(1-2), 29-38. [3] Miura T. et al., The cyst-branch difference in developing chick lung results from a different morphogen diffusion coefficient. Mech Dev 2008 , 126, 160-172. [4] Hirashima et al., Mechanisms for split localization of Fgf10 expression in early lung development. Dev Dyn 2009, 238(11) 2813-2822
P09
Far-From-Equilibrium Dynamics, January, 4-8, 2011, KYOTO, JAPAN
Periodic oscillations in some hematopoietic diseases
Shinji [email protected]
Graduate School of Mathematical Sciences, The University of Tokyo
Hematopoietic cells such as red blood cells, granulocytes, platelets and white blood cells arisefrom a hematopoietic stem cell by cell differentiation. Neutrophils are one of hematopoietic cellsand the most abandant white blood cells (40-70%). Neutrophils exhibit several anti-microbialfunctions to protect the host from infection. Neutropenia is characterized by an abnormallow number of neutrophils (below 500 cells/mL, 200–7500 cells/mL for normal individuals).Under the condition of neutropenia, a patient has a high propensity to bacterial infection.Severe congenital neutropenia (SCN) is an autosomal genetic disorder which occurs soon afterborn and continues chronically. It has already been identified that mutation in ELA2 gene isrelated to about 40-60% of patients with SCN or cyclic neutropenia (CN), periodic oscillationsin neutrophil number. Although recombinant human G-CSF (rhG-CSF) can be used as atherapeutic way to temporally recover the number of neutropnils, it is not possible to completelycure congenital neutropenia. Moreover, any mouse models which resemble SCN/CN in humanhas not been developed yet.
In order to understand and control neutropenia, mathematical theory and techniques wouldhelp us to reveal dynamical aspects and mechanisms underlying the cause of neutropenia. Herewe present recent progress of our mathematical study on neutrophil differentiation. Our mathe-matical model is formulated in terms of delay equations, a coupled system of renewal equationsand delay differential equations [1,2]. It is shown that a mathematical model studied in [3]can be equivalently reformulated by delay equations with some modification. We show someanalytical and numerical computation results for stability and bifurcation properties of the lin-earized system around an equilibrium, and some numerical simulations which represent periodicoscillations of neutrophils. Our model can be used quantitatively to incorporate experimentalknowledge as well as qualitatively to simulate long-term behavior of neutrophil development,which is in most cases difficult to track by experiments only.
REFERENCES
[1] Diekmann O, van Gils SA, Verduyn Lunelc SM,Walther H-O (1995) Delay equations: functional, complex, andnonlinear analysis, vol 110 of Applied Mathematical Sciences. Springer-Verlag[2] Diekmann O, Getto P, Gyllenberg M (2007) Stability and bifurcation analysis of Volterra functional equations inthe light of suns and stars. SIAM J Math Anal 39:1023–1069[3] C. Foley and M. C. Mackey. Mathematical model for G-CSF administration after chemotherapy. J Theor Biol, 257,pp.27–44, (2009).
1
P10
Reconstruction of chaotic saddles and classification of unstable
periodic orbits of the Kuramoto-Sivashinsky equation
Yoshitaka Saiki1, Michio Yamada2, Abraham C.-L. Chian3
1) Hokkaido University, 2) RIMS, Kyoto University, 3) INPE, Brazil
Chaotic saddles are responsible for important nonlinear phenomena, such as chaotic transients,chaotic scattering, and fractal basin boundaries. The chaotic saddle lies on the intersection ofits stable and unstable manifolds. In the past literature, chaotic saddles are usually identifiedby the sprinkler method[1] the PIM triple method[2], and the stagger-and-step algorithms[3].
Large-scale invariant sets such as chaotic attractors can undergo bifurcations as a controlparameter is varied. These bifurcations include sudden changes in the size and/or type ofthe set. An attractor-merging crisis (AMC) is an example of sudden changes in which two ormore chaotic attractors merge to form one single chaotic attractor[4,5]. At the AMC, chaoticattractors simultaneously touch the boundary separating their basins of attraction, and collidewith one or more unstable periodic orbits on the basin boundary.
In this work unstable periodic orbits embedded in a chaotic attractor of the Kuramoto-Sivashinsky equation after an AMC are classified into three types and employed to characterizechaotic saddles which continue from pre-AMC regime. It is found that in the post-AMC regimechaotic saddles with gaps are reconstructed by unstable periodic orbits which also exist inpre-AMC regime and that unstable periodic orbits emerging after the crisis fill the gaps. Thelatter type of unstable periodic orbits originate from saddle node bifurcations accompaniedwith periodic windows after the AMC or period doubling bifurcations of the node orbits. Itis conjectured from the number of detected periodic orbits that periodic windows with longerperiods accumulate toward the AMC point.
REFERENCES
[1] H. Kantz and P. Grassberger, Physica D , 17 75 (1985).[2] H. E. Nusse and J. A. Yorke, Physica D , 36 137 (1989).[3] D. Sweet, H. E. Nusse, and J. A. Yorke, Phys. Rev. Lett., 86 2261 (2001).[4] C. Grebogi, E. Ott, F. Romeiras and J. A. Yorke, Phys. Rev. A., 36 5365 (1987).[5] E. L. Rempel and A. C.-L. Chian, Phys. Rev. E , 71, 016203 (2005).
1
P11
Breathing instability versus drift instabilityin a two-component reaction diffusion system
Mitsusuke Tarama1∗, Takao Ohta1, and Len M. Pismen2
∗[email protected] of Physics, Kyoto University, Kyoto, 606-8502, Japan
2Department of Chemical Engineering and Minerva Center for Nonlinear Physics of Complex Systems,Technion-Israel Institute of Technology, 32000 Haifa, Israel
For more than two decades, formation and stability of domains in excitable reaction-diffusionsystems have been studied both theoretically and experimentally. In two-component excitablereaction-diffusion equations, it has been known that a motionless localized domain loses itsstability when the bias and coupling parameters are varied. Among the possibilities, there area static shape deformation around a circular shape in two dimensions (spherical shape in threedimensions), and a breathing motion, where the domain radius undergoes an oscillation whilethe center of mass is time-independent. If the model equation has a mechanism such that thebreathing motion is not allowed, there is another bifurcation called a drift bifurcation where thedomain undergoes a translational motion. As an example of such mechanism, there is a globalcoupling which makes the domain area (volume in three dimensions) constant in time. Anotherway to realize a stable propagating domain is to consider three-component reaction-diffusionequations which consist of one short-range activator and two long-range inhibitors [3]. L. M.Pismen (2001) developed an interfacial approach to two-component reaction-diffusion equationswithout any breathing-prohibit mechanism, and concluded that stable propagating domains doexist in the basic activator-inhibitor model, even without additional long-range variables.
We have investigated the stability of an excited domain in the same model and derived thedynamical phase diagram of the domain stability (2010). Our conclusion is that a motionlesscircular domain undergoes either a breathing instability or a shape deformation instabilitybefore a drift instability would occur, which is qualitatively different from the previous Pismen’sresult (2001). Actually, Pismen has also reached the similar conclusion in his book (2006),which corrects his previous result. However, the description in his book is again not free fromsome deficiency, which is serious enough to change the stability diagram significantly and, inparticular, reordering the drift, breathing, and shape instabilities.
REFERENCES[1] L. M. Pismen, Phys. Rev. Lett. 86, 548 (2001).[2] L. M. Pismen, Patterns and Interfaces in Dissipative Dynamics (Springer, Berlin, 2006).[3] Y. Nishiura, T. Teramoto, and K. I. Ueda, Chaos 15, 047509 (2005).[4] Mitsusuke Tarama, Takao Ohta, and Len M. Pismen, submitted to Phys. Rev. E (2010).
1
P12
Itinerant Dynamics in Gene Expression Implies Pluripotency:
Kunihiko Kaneko, Narito Suzuki (Univ of Tokyo), and
Chikara Furusawa (Osaka Univ)
During normal development, cells undergo a unidirectional course of differentiation
that progressively decreases the number of cell types they can potentially become.
pluripotent stem cells can differentiate into several types of cells, but terminally
differentiated cells cannot differentiate any further. A fundamental problem in stem
cell biology is the characterization of the difference between pluripotent stem cells
and terminally differentiated cells.
To address the problem, we developed a dynamical systems theory for cell
differentiation by using intracellular dynamics with cell-cell interactions. We have
carried out extensive simulations of a model of interacting cells with intracellular
gene-expression dynamics and cell divisions. We have carried simulations of more
than hundred million gene-regulation networks, and selected those networks that
show pluripotency, i.e., both proliferation and cell-differentiation. The selected cells
show oscillatory gene expression dynamics with itinerancy over states. The
oscillations, at a certain stage, exhibit transient or stationary chaotic oscillations. With
differentiation, such chaotic oscillations are lost successively, leading to a loss of
pluripotency. Gene regulation networks that show hierarchical differentiation to
several cell types through the increase in cell number by cell division are naturally
designed based on this itinerant oscillatory dynamics. Recent single-cell-level
measurements in stem cells support such oscillatory dynamics in gene expression.
Based on these numerical, theoretical, and experimental studies we propose the
following hypothesis: Itinerant oscillation dynamics in the expression of some genes
leads to cell pluripotency and affords cellular state heterogeneity, which is supported
by transitions over quasi-stable states. Differentiation stabilizes these states, leading
to a loss of pluripotency.
Our hypothesis suggests a feasible route to recover the potential to differentiate, i.e.,
by increasing the variety of expressed genes to restore chaotic expression dynamics,
as is consistent with the recent generation of induced pluripotent stem (iPS) cells.
P13
Optimal Information Processing and Stochastic Bifurcation inBiological Networks
Tetsuya J. [email protected]
Institute of Industrial Science, the University of Tokyo
The signal transduction pathways in a cell typically respond to the environmental signals.The pathway have to produce the proper output that facilitates the survival of the cell inunpredictably changing environment. Because of the existence of intrinsic and extrinsic noise,however, such pathways have to be robust to the noise that undermines the information ofthe signals. In other words, the pathway have to extract (decode) necessary information forsurvival from the noise. Even with recent knowledge about the molecular details of pathways,we still do not know what kind of intracellular reactions are responsible for such decoding ofthe information from noise.
To address this problem theoretically, I derive the optimal decoding kinetics by a cell byusing the theory of Bayesian inference[1]. I also show that an auto-phosphorlation auto-dephosphorlation cycle (aPadP cycle) is the simplest implementation of the decoding kineticsby intracellular kinetics. By investigating this aPadP cycle as a stochastic dynamical system,its dynamical properties will further be revealed. Of particular interest is that the aPadP cyclealso has an noise-induced bifurcation structure that can be seen only in the existence of noise.
Cellular systems can operate reliably even with substantial intrinsic and extrinsic noise insideand outside of the cell. To clarify the underlying principle of this robustness, the usual approachusing stochastic dynamics methods is not sufficient. As illustrated in this work, to challengethat problem, we need to develop the integrated theory of information processing with those ofstochastic dynamics and stochastic dynamical systems.
REFERENCES[1] T. J. Kobayashi, Implementation of Dynamic Bayesian Decision Making by Intracellular Kinetics, Phys. Rev. Lett.,104, 228104, 2010.
1
P14
Numerical studies of droplet impacting and splashing
Kensuke [email protected]
Cardiff University
We numerically study various types of droplet impact phenomena such as droplet impacts into a thin liquid layer, droplet depositions onto dry surfaces (homogeneous and heterogeneous), droplet splashing on dry surfaces and multiple droplet impacting. The numerical framework [1,2] consists of a CLSVOF (coupled level set and volume-of-fluid) framework, CIP-CSL (constraint interpolation profile-Conservative Semi-Lagrange) method, VSIAM3 (volume/surface integrated average based multi-moment method) and a CSF (continuum surface force) model. Our numerical results show almost quantitative agreements with these experiments of droplet impacts. I also talk about relationships between dynamic contact angle and droplet behaviors on dry surfaces. Figures 1 and 2 show numerical results of single droplet splashing and multiple droplet impacting.
Fig. 1. Snapshots of a droplet impacting onto a hydrophobic surface. 200x200x100 grids were used.
Fig. 2. Five droplet impacting onto the super hydrophobic substrate. REFERENCES[1] Kensuke Yokoi, "A numerical method for free-surface flows and its application to droplet impact on a thin liquid layer" , Journal of Scientific Computing, 35, 372 (2008).[2] Kensuke Yokoi, Damien Vadillo, John Hinch and Ian Hutchings, "Numerical studies of the influence of the dynamic contact angle on a droplet impacting on a dry surface", Physics of Fluids, 21, 072102 (2009).
P15
Complex Networks from Dual Point of View
Taichi [email protected]
Graduate School of Science, Kobe University/PRESTO, JST
A system of interacting elements can be represented by a directed graph so that elements arenodes and interaction between two elements is an arc. In real view on directed graphs each nodeis just a point, each arc represents some kind of interaction between two nodes and nothing morein so far as we consider network topology. However, in many real systems, each element has itsown intra-node process. Hence we can interpret interaction between two elements as interfacebetween two intra-node processes. This dual view on directed graphs can be formulated inthe framework of category theory. In particular, we use left Kan extensions to construct amathematical expression for dual view. We show that a new notion of connectedness whichwe call lateral connectedness (LC) emerges as a canonical structure obtained from dual view.We discuss applications of LC to the study of complex networks. Generalization for other datastructures than directed graph is also presented.
P16
Symmetry Restoration Process and Unstable Symmetric Tori
Tsuyoshi Mizuguchi†‡, Makoto Yomosa∗
† [email protected], ∗[email protected]†∗ Department of Mathematical Sciences, Osaka Prefecture Universtiy
‡ PRESTO, Japan Science and Technology Agency(JST)
In the study of dynamical systems unstable solutions play important roles in various situationssuch as control of chaotic behaviour, characterization of chaos and turbulence, and so on.Among them, we focus on unstable symmetric solutions which act as finger posts of the flowin the phase space. And roles and searching methods of these solutions are investigated.
As an example, symmetry restoration process at a bifurcation point of attractor mergingcrises (AMC) is investigated. When the system approaches to the bifurcation point of theAMC from the symmetric side, the symmetry is broken via an intermittency. If we approachto it from the asymmetric side, how the symmetry is restored? The restoration process is wellcharacterized by a closeness to a specific unstable symmetric solution including a limit cycle ora torus. Here, a method using time series analysis is suggested, which does not require concretefunction form of the specific solution. The minimum distance between a reference state on theorbit and the image of the corresponding transformation is measured to characterize a degreeof the symmetry restoration. Several multi-stable chaotic systems are analyzed.
REFERENCES
[1] M. U. Kobayashi, T. Mizuguchi, “Chaotically oscillating interfaces in a parametrically forced system”, Phys. Rev.E, 73 (2006) 016212.[2] M. U. Kobayashi, T. Mizuguchi, “Chaotic interfaces in a parametrically forces system”, Prog. Theor. Phys.Supplement, 161 (2006) 228–231.[3] Y. Morita, N. Fujiwara, M. U. Kobayashi and T. Mizuguchi, “Scytale decodes chaos: A method for estimatingunstable symmetric solutions”, Chaos, 20 (2010) 013126.
1
P17
Evolutionary design of robust oscillatory genetic networks
Yasuaki Kobayashi1, Tatsuo Shibata2,3, Yoshiki Kuramoto4,Alexander S. Mikhailov1
[email protected] of Physical Chemistry, Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6,
D-14195 Berlin, Germany2Department of Mathematics and Life Sciences, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima,
739-8526, Japan3PRESTO, Japan Science and Technology Agency (JST), 4-1-8 Honcho Kawaguchi, Saitama, Japan
4Research Institute for Mathematical Science, Kyoto University, Kyoto 606-8502, Japan
The present study is devoted to the design and statistical investigations of dynamical geneexpression networks. In our model problem, we aim to design genetic networks which wouldexhibit stable periodic oscillations with a prescribed temporal period. While no rational solutionof this problem is available, we show that it can be effectively solved by running a computerevolution of the network models. In this process, structural rewiring mutations are applied tothe networks with inhibitory interactions between genes and the evolving networks are selecteddepending on whether, after a mutation, they closer approach the targeted dynamics. We showthat, by using this method, networks with required oscillation periods, varying by up to threeorders of magnitude, can be constructed by changing the architecture of regulatory connectionsbetween the genes. Statistical properties of designed networks, including motif distributionsand Laplacian spectra, are considered. Also, we show that the same evolutionary optimizationmethod can be used to increase the robustness of the networks against several types of noise.
REFERENCES[1] Y. Kobayashi, Kobayashi, Y., T. Shibata, Y. Kuramoto, and A.S. Mikhailov, Eur. Phys. J. B 76, 167-178 (2010)
1
P18
Synchronization transition of identical phase oscillators in adirected small-world network
Ralf [email protected]
Ochadai Academic Production, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan
In [1] we study the synchronization transition of identical phase oscillators in a unidirectionalrandom coupling network. When the mean in-degree of the network is increased we nd adiscontinuous transition of the order parameter from r = 0, signifying an incoherent state ofuniformly distributed phases, to complete synchronization r = 1 or partial synchronization0 < r < 1. Using a dynamic parameter α of the oscillators as control parameter we can tracethe bifurcation line r(α) of the order parameter at stable as well as unstable equilibria. Thetransition from partial to complete synchronization is understood as a non-equilibrium phasetransition into an absorbing state.
While the numerical analysis in [1] draws a detailed picture of the complex behavior in theseemingly simple system, an analytic description is still missing. In this poster we presentthree open questions as an invitation and challenge to solve.
REFERENCES[1] R. Toenjes, N. Masuda and H. Kori, Chaos 20, 033108 (2010)
1
P19
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P20
Analysis of synchronization of coupled damped oscillators
Masako Matsubara1,*, Ken Nagai2, Tetsuo Deguchi1, Hiroshi Kori1,3,# 1 Department of Physics, Ochanomizu University, Tokyo, Japan
2 Division of Advanced Sciences, Ochadai Academic Production, Ochanomizu University, Tokyo, Japan 3 PRESTO, Japan Science and Technology Agency, Kawaguchi, Japan
*Email address: [email protected], #Email address: [email protected]
In the past, the application of synchronization theory was limited to the field of coupled self-sustained oscillators such as limit-cycle or chaotic oscillators. We thought that almost all biological rhythms could be described as limit-cycle oscillators. However, recent experimental research on biological clocks has demonstrated that the rhythms of individual cell clocks in the brain are synchronized, damping in parallel [1, 2]. This suggests that cell clocks are not self-sustained oscillators. However, no theoretical frameworks exist for synchronization of non-self-sustained oscillators. In the present study, we aim to theoretically clarify the synchronization behavior of coupled damped oscillators using a linear system model that is considered to be the most simple. Suppose that each of two oscillators in motion is subjected to a frictional force in proportion to the velocity of the other. Motion equations of the two objects can be given as follows;
!!x1+ !x
1+ kx
1= ! !x
2,
!!x2+ !x
2+ (k + d)x
2= ! !x
1.
Here, d is the difference in eigen frequency, and ε is coupling strength. We can analytically solve this model, but its solutions are incredibly complex. Therefore it is difficult to determine the behavior of this model. We therefore next defined the amplitude and phase of each oscillator properly, and numerically observed time variations in amplitude ratio and phase differences between the two oscillators (Fig.1). Over time, amplitude ratio and phase difference between the two oscillators converged within a definite range (frequency synchronization). To analytically calculate the solution for synchronization, we derived dynamical equations for the amplitude ratio and the phase difference using coordinate transformation and time-averaging. The analytical solution agreed excellently with the numerical results. We therefore demonstrated that coupled damped oscillators are synchronized.
0
2
4
6
8
10
0 500 1000 1500
ampl
itude
ratio
time
-8
-4
0
0 500 1000 1500
phas
e di
ffere
nce
[rad
]
time
Fig.1: Time series of amplitude ratio (left) and phase difference (right).
REFERENCES [1] H. Fukuda, N. Nakamichi, M. Hisatsune, and T. Mizuno: Phys. Rev. Lett. 99, 098102 (2007). [2] Shun Yamaguchi et al.: Science, Vol. 302, p. 1408, 2003.
P21
Collective enhancement of regularity in networks of noisy oscillators
Hiroshi Kori1,2,*, Yoji Kawamura3, Naoki Masuda4,2 1 Division of Advanced Sciences, Ochadai Academic Production, Ochanomizu University, Tokyo, Japan
2 PRESTO, Japan Science and Technology Agency, Kawaguchi, Japan 3 Institute for Research on Earth Evolution, Japan Agency for Marine-Earth Science and Technology, Yokohama,
Japan 4 Graduate School of Information Science and Technology, the University of Tokyo, Tokyo, Japan
*Email address: [email protected]
Biological oscillators, such as heart pacemaker cells and circadian clock cells, are inevitably subject to external or intrinsic noise, and their phase dynamics thus fluctuate with time. Consequently, cycle-to-cycle periods in an oscillator also fluctuate. However, precision in oscillations is an important requirement for proper biological functioning. One may expect that a larger population of oscillators tends to suppress fluctuations, resulting in more precise oscillations. Actually, there is experimental [1] and numerical evidence supporting this idea. However, as A. Winfree pointed out, the mathematical essence behind such an improvement is still unrevealed. In this presentation, we propose a general theoretical framework for dealing with fluctuations in the phase dynamics (called phase diffusion) and the precision of cycle-to-cycle periods in networks of noisy oscillators [2,3]. Our framework is based on phase models, which are known to approximate networks of real oscillators. By confining ourselves to longtime behavior or a strong coupling case, we obtain a concise relationship between an inherent phase diffusion constant (that is of an isolated oscillator) and a collective phase diffusion constant (that is shared in a synchronizing network). We then further relate them to the temporal precision in oscillations. It turns out that the temporal precision strongly depends on the network structure, e.g., network size N and detailed topology. In particular, we find (i) in undirected networks, predictions improves with N as the same manner as the law stated by the central limit theorem, and (ii) in directed networks the precision improves slower with N than the case of undirected networks. Further examples using mathematical models of biological oscillators are also presented. Our approach provides a clear answer in certain cases to the classical problem, i.e., the collective enhancement of precision.
REFERENCES [1] J.R. Clay and R.L. DeHaan: Rhythmic heart-cell cluster: role of membrane voltage, Biophys. J. 28 (1979). [2] N. Masuda, Y. Kawamura, H. Kori: Collective fluctuations in networks of noisy components, New J. Physics 12, 093007 (2010). [3] H. Kori, Y. Kawamura, N. Masuda: in preperation.
P22
Multi-dimensional pyramidal traveling fronts in the Allen-Cahnequations
Yu Kurokawa and Masaharu [email protected] [email protected]
Tokyo Institute of Technology
In this report, we consider the Allen-Cahn (Nagumo) equation:
∂u
∂t= ∆u + f(u) x ∈ RN , t > 0,
u|t=0 = u0 x ∈ RN .
Here a given function u0 is of class BU(RN). The Laplacian ∆ stands for∑N
i=1 ∂2/∂x2i . The
following are the assumptions on f :
(A1) f is of class C1[−1, 1] with f(1) = 0, f(−1) = 0, f ′(1) < 0 and f ′(−1) < 0.
(A2)∫ 1
−1f > 0 holds true.
(A3) There exists Φ(µ) that satisfies
−Φ′′(µ) − kΦ′(µ) − f(Φ(µ)) = 0, −∞ < µ < ∞,
Φ(−∞) = 1, Φ(∞) = −1.
for some k ∈ R.
To construct N -dimensional pyramidal traveling fronts for N ≥ 4 is the aim of this report.Multi-dimensional traveling fronts in the Allen-Cahn equation equation have been studied
by many researchers. For example, two-dimensional V-form front solutions have been stud-ied by Ninomiya and Taniguchi (2005, 2006), Hamel, Monneau and Roquejoffre (2005, 2006)and Haragus and Scheel (2006). Cylindrically symmetric traveling fronts have been studied byHamel, Monneau and Roquejoffre (2005, 2006) and Chen, Guo, Hamel, Ninomiya and Roque-joffre (2007). Three-dimensional traveling fronts with pyramidal shapes have been studied in2007 and 2009.
To study all traveling front solutions in RN will be an interesting and difficult question. Thisproblem will become more interesting and more difficult as the space dimension N becomeslarger. As an approach this problem, the authors introduce pyramidal traveling fronts for anyspace dimensions including N ≥ 4.
To obtain the main result in this report, we make a supersolution and a subsolution, andconstruct a pyramidal traveling front solution between them. For the construction of a super-solution we use a multi-scale method.
1
P23
Noise-induced synchronization of coupled oscillators
Ken H. Nagai1, Hiroshi Kori1,2
[email protected] of Advanced Sciences, Ochadai Academic Production, Ochanomizu Univeristy, Tokyo 112-8610,
Japan, 2PRESTO, Japan Science and Technology Agency, Kawaguchi 332-0012, Japan
Since real systems are inevitably subject to noise, it is important to understand the effectof noise on the synchronization of periodic oscillators. There are many situations in which asingle noise process, such as that originating from environmental fluctuations, acts on an entiresystem. Whether such common noise enhances or inhibits synchronization remains unclear.The effect of common noise on uncoupled oscillators has been extensively studied for periodicoscillators, and rigorous theoretical frameworks have been proposed [1, 2]. In contrast, for alarge population of coupled oscillators, the theoretical treatment is still an open and challengingproblem.
We study a large population of globally coupled phase oscillators, known as the Sakaguchi-Kuramoto model [3], subject to common white Gaussian noise
dθi
dt=ωi +
K
N
N∑j=1
sin(θj − θi + β) + p(t) sin θi, 〈p(t)〉 = 0, 〈p(t)p(s)〉 = 2Dδ(t − s),
where θi and ωi are the phase and the natural frequency, respectively, of the i-th oscillator,K > 0 is the coupling strength, β is a parameter of the coupling function (−π/2 < β < π/2),and p(t) is a common external random force. Utilizing the anzatz recently proposed by Ott andAntonsen [4], we analytically show that the addition of common noise leads to a decrease ( ofsize D ) in critical coupling strength for synchronization transition as compared to that in theoriginal Kuramoto transition. Our prediction is corroborated by direct numerical simulationsof the model. We also numerically confirm that globally coupled limit-cycle oscillators showthe same dependence on common noise [5].
REFERENCES[1] J. Teramae and D. Tanaka, Phys. Rev. Lett., 93, 204103 (2004).[2] K. Nagai, et al., Phys. Rev. E, 71, 036217 (2005).[3] H. Sakaguchi and Y. Kuramoto, Prog. Theor. Phys., 3, 576 (1986).[4] E. Ott and T. M. Antonsen, Chaos, 18, 037113 (2008).[5] K. H. Nagai and H. Kori, Phys. Rev. E, 81, 065202(R) (2010).
0
0.02
0.04
0.06
1.96 1.98 2 2.02 2.04 2.06 2.08 2.1
Am
ax
K
D=0
D=0.02
D=0.04
Figure 1. The order parameter as a function of K. Critical coupling strengthdecreases with increase in D.
1
P24
Motion of non-closed planar polygonal curves by crystallinecurvature flow with a driving force
Tetsuya [email protected]
Department of Mathematical Sciences, Shibaura Institute of Technology, Japan.
In this presentation we consider the motion of non-closed planar polygonal curves governedby generalized crystalline curvature flow with a driving force:
(1) β(Nj)Vj = U − g(Hj),
where Vj, Nj and Hj denote an outward velocity, an outward normal vector and a crystallinecurvature of the j-th facet of solution curve, respectively. Here, “facet” means a lines segmentof solution curve Γ(t) and Fj denotes the j-th facet of Γ(t). The positive function β andU describe an anisotropy of the mobility and a driving force, respectively. In this paper weconsider the case that U > 0 and the function g = g(λ) is odd, locally Lipschitz continuous,monotone increasing in λ and has linear or superlinear growth for λ ∼ ±∞.
The equation (1) with g(λ) = λ is a simple model of an interface motion of crystals (See [1]and [6]). In the case when the solution region is bounded, many authors discuss the behaviorof the solutions for several types of crystalline motions (See [2],[3],[7] and their references). Onthe other hands, in the case when the solution region is unbounded, there are a few results.Marutani et al. [5] shown the existence of traveling “V-shaped” solutions for the case g(λ) = λas the limit of the smooth solution for weighted curvature flow. In [4], we consider the simplecase β ≡ 1 and g(λ) = λ and show that the solution with non-V-shaped initial region eventuallybecome V-shaped under some conditions. This kind of monotonicity phenomena are alreadyknown for the case when the solutions are bounded. The most famous result is shown byM. Grayson in 1987 for smooth solution curves to curve-shortening flow. He showed that thesolution curves from non-convex initial curve becomes convex in finite time. For crystallinecase, it is shown in [3] for the case U = 0 that crystalline curvature of all facets of the solutionpolygons from non-convex initial polygon becomes non-negative in finite time.
In this presentation we show the extended results in [4] for generalized motion (1) for thecase that the solution is unbounded.
REFERENCES[1] S. Angenent and M.E. Gurtin, Multiphase thermomechanics with interfacial structure, 2. Evolution of an isothermalinterface, Arch. Rational Mech. Anal. 108 (1989), 323–391.[2] Y. Giga and M. E. Gurtin, A comparison theorem for crystalline evolution in the plane, Quart. J. Appl. Math. LIV(1996), 727–737.[3] T. Ishiwata, Motion of non-convex polygons by crystalline curvature and almost convexity phenomena, JapanJournal of Industrial and Applied Mathematics, 25 (2008), no. 2, 233–253.[4] T. Ishiwata, On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect,to appear in Discrete and Continuous Dynamical Systems, Series S.[5] Y. Marutani, H. Ninomiya and R. Weidenfeld, Traveling curved fronts of anisotropic curvature flows, Japan Journalof Industrial and Applied Mathematics, 23 (2006), no. 1, 83–104.[6] J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry, Proceedings of the Conferenceon Differential Geometry, Rio de Janeiro, Pitman Monographs Surveys Pure Appl. Math. 52 (1991) 321–336, PitmanLondon.[7] S. Yazaki, Point-extinction and geometric expansion of solutions to a crystalline motion, Hokkaido Math. J. 30(2001), 327–357.
1
P25
Far-From-Equilibrium Dynamics, January, 4-8, 2011, KYOTO, JAPAN
Thermal convection of binary mixtures by temperature ramp
Y. Hayase, E.M. Sam, D. Vollmer, H. Pleiner, G. K. [email protected]
Max Planck Institute for Polymer Research, 55128 Mainz, Germany
We investigate the convection under continuous cooling of mixed solutions. When a liquid iscooled simultaneously from the top and bottom boundaries, a temperature difference betweenthe boundary and the center arises. Even in a horizontal layer of a liquid, convection can ariseif the cooling rate is fast enough. This convection is driven by upper half of the system and hasthe same origin as Rayleigh-Benard convection. When the system is cooled below the phaseseparation temperature, nucleation is induced and produces latent heat. Consequently, thetemperature difference between the center and the boundaries increases further. To describethis, we investigate numerically the coupled system of the Navier-Stokes equation and the heatequation.
We consider a system with a stable single phase for T > Tc, and two phases for T < T
c. We
apply the Boussinesq approximation for the system. The relevant variables of the system arethe flow velocity ~v, and temperature T .
∇ · ~v = 0,(1)
∂~v
∂t+ ~v · ∇~v = ν∇2~v − ρ−1
0∇p + αg(T − T0)ez
,(2)
∂T
∂t+ ~v · ∇T = κ∇2T + q.(3)
Here, ν is the kinematic viscosity, κ is the thermal diffusivity, ρ0 is the density, α is the thermalexpansion coefficient, g is the acceleration due to gravity. Eq. (1) is a continuity equation, eq.(2) is the Navier-Stokes equation, and Eq. (3) is the heat equation. The term q in equation (3)is the latent heat source term. For simplicity, q is taken to be a step function between 0 andQ(> 0) at T = T
c.
We numerically observe a convection instability when the system is cooled from the boundary.By adding the latent heat effects, we obtain three difference scenarios of the pattern dynamicsdepending on the cooling rate. 1. For slow cooling rate, there is no convection. 2. Forintermediate cooling rate, we observe a regular pattern. 3. For fast cooling rate, complexpatterns arise.
Recently, E. M. Sam et al. [1] found convection patterns in a thin horizontal layer of amixture with temperature ramp. They prepared a methanol-hexane-ethanol mixture in theone-phase region and continuously cooled into the two-phase region. After crossing the binodalcurve, the sample shows a hexagonal pattern with droplets. For slower cooling rate, no patternis observed. For faster cooling rate, complex patterns are observed. The experimental resultssupport the results of our model equations very well.
REFERENCES
[1] E.M. Sam, Y. Hayase, G. K. Auernhammer, and D. Vollmer. in preparation
1
P26
Self-propulsion of a drop driven by Marangoni flow
1, K. Nagai2, Y. Sumino3 and H. Kitahata4 1Fukui Institute for Fundamental Chemistry, Kyoto University, Kyoto 606-8103, Japan
2Ochadai Academic Production, Ochanomizu University, Tokyo 112-8610, Japan 3Department of Applied Physics, the University of Tokyo, Tokyo 113-0033, Japan
4Department of Physics, Chiba University, Chiba 263-8522, Japan
Spontaneous motion or self-propulsion have been attracting attention in last decades for its potential application to biological problems such as cell motility and wound healing. Recently several model experiments showing spontaneous motion have been proposed [1-3]. The systems in these works consist of relatively simple ingredients for instance oil drops in water nevertheless the motion is as if the drops are alive. The key questions are why the particle moves without external force and why it breaks symmetry and chooses one direction.
The first point has been discussed in hydrodynamics of the Marangoni effect in which a liquid droplet is driven by a gradient of surface tension. The mechanism is that the gradient induces convective flow inside and outside of a drop, which leads to swimming motion of the drop itself. The second point was less discussed, but has been discussed in the field of nonlinear dynamics as drift instability. In fact, Ohta et al. derived the mode equations of drift instability and deformation with symmetry consideration and from specific reaction diffusion equations [4].
Thus far there are only few attempts to discuss the spontaneous symmetry breaking from hydrodynamics. In this work, we derive the nonlinear equations exhibiting drift instability. This is of importance because all the coefficients are determined with physical quantities.
FIG: Schematics of our system. Surfactants dissolve in the outer bulk and are absorbed onto the interface, resulting in reduction of the surface tension. (A) Isotropic distribution of the surfactants when a drop is stationary. (B) When the distribution breaks symmetry the flow (the red thin arrow) occurs and the drop starts to move (the black thick arrow). REFERENCES [1] T. Toyota, N. Maru, M. Hanczyc, T. Ikegami, and T. Sugawara, J. Am. Chem. Soc., 131, 5012 (2009). [2] Y. Sumino, N. Magome, T. Hamada, and K. Yoshikawa, Phys. Rev. Lett., 94, 068301 (2005). [3] K. Nagai, Y. Sumino, H. Kitahata and K. Yoshikawa, Phys. Rev. E , 71, 065301 (2005). [4] T. Ohta and Ohkuma, Phys. Rev. Lett., 102, 154101 (2009).
P27
Nonlinear dissipative waveunder external forcing and feedback control
Yousuke Tonosaki 1, Takao Ohta 1, Vladimir Zykov 2
[email protected]. Department of Physics, Kyoto University, Kyoto, Japan2. Institut fuer Theoretische Physik, Technische Universitaet Berlin, Berlin, Germany
We investigate the dynamics of propagating dissipative waves under external forcing and feedback control,based on the model system undergoing phase separation and chemical reactions as
(1)∂ψ
∂t= ∇2
[−∇2ψ − τψ + ψ3
]+ a1ψ + a2ϕ+ a3 + Γ,
(2)∂ϕ
∂t= b1ψ + b2ϕ+ b3 + Γ,
where ψ and ϕ are the order parameters related with the local concentrations of the chemical species [1]. Theparameters τ, ai, bi specify the phase separation and chemical reactions and Γ is the external force or feedbackcontrol.
In order to investigate synchronization and modulation of the traveling waves under external forcing, we considerthe case that Γ = ϵ cos(qfx−Ωt) where ϵ, 2π/qf and Ω is the magnitude, the spatial period and the frequency ofthe forcing, respectively. A phase diagram for the entrained and non-entrained states under the external forcingis obtained numerically. Theoretical analysis in terms of phase description of the traveling waves is carriedout to show that the transition between the entrained and the non-entrained states by changing the externalfrequency occurs either through a saddle-node bifurcation or through a Hopf bifurcation and that these twobifurcation lines are connected at a Bogdanov-Takens bifurcation point[2].
The feedback control is studied by putting Γ = F [ψ(x, t− δ)− ψ] or Γ = F [ψ(x− δ, t)− ψ] +F [ψ(x+ δ, t)− ψ]where the constant F is the strength of the feedback and ψ is the spatial average. By numerical simulations,we find various interesting behaviors of the propagating waves which can be understood by theoretical analysisbased on phase dynamics [3].
REFERENCES[1] T. Okuzono and T. Ohta, Phys. Rev. E., 67 (2003) 056211.[2] Y. Tonosaki, T. Ohta and V. Zykov, Physica D, 239 (2010) 1718 .[3] Y. Tonosaki, H. Tokuda, T. Ohta and V. Zykov, Europhysics Letters, 83 (2008) 50011.
1
P28
Bifurcation behaviour of the shape of cracks
1,∗)Takeshi Takaishi, 2)Masato Kimura∗[email protected]
1)Faculty of Information Design, Hiroshima Kokusai Gakuin University2)Faculty of Mathematics, Kyushu University
We proposed a phase field model that the mode III (anti-plane shear mode) crack growthon a plate is described as reaction-diffusion system that is consisted by the of the anti-planedisplacement and the phase field that describes the crack[2,3]. This system is derived from theenergy description introduced by Francfort and Marigo[1], and this reaction-diffusion systemmakes the computation of the crack problem easy. For t > 0, we consider the equations:
(1)
α1∂u
∂t= div
((1− z)2∇u
)x ∈ Ω
α2∂z
∂t=
(ε∆z − γ2
εz + |∇u|2(1− z)
)+
x ∈ Ω
u(x, t) = g(x, t) x ∈ ΓD
∂u
∂n= 0 x ∈ ΓN ,
∂z
∂n= 0 x ∈ Γ
where u(x, t) represents the small anti-plane displacement at the position x ∈ Ω and time t ≥ 0,and g(x, t) is a given anti-plane displacement on the boundary ΓD. z(x) is a phase field whichsatisfies z ≈ 1 around the crack and z ≈ 0 for the other region.
x1
x2
0 1
1
-1
-1
x2( , )0 0
-x1 -x2( , )0 0
x1
ΓD
ΓD
Figure 1. Initialcrack profile
(a) (b)
Figure 2. Spatial profile of z where initialcracks of lengths (a) 1.7 and (2) 1.8.
Now we set the initial crack at t = 0 as Fig.1, numerical results show that small differnece ofthe initial shape of cracks produces the difference of the shape when it is break down (Fig.2). Wefound the difference between the results of numerical simulations and the simple estimation ofsurface energy. It shows that the path of evolution of crack doesn’t always choose the minimumenergy state, and we remark that the process of crack-evolution is important for selection ofthe shape of crack.
REFERENCES
[1] G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys.Solids, 46 (1998),1319–1342.[2] T. Takaishi and M. Kimura, Phase field model for mode III crack growth, Kybernetika 45 (2009), 605–614.[3] T. Takaishi, Numerical simulations of a phase field model for mode III crack growth, Transactions of the JapanSociety for Industrial and Applied Mathematics 19 (2009), 351–369 (in japanese).
1
P29
Phase reduction approach to interacting traveling pulses
Hiroya [email protected]
Department of Physics, Kyoto University Kyoto 606-8502, Japan
Collective phase description for a traveling pulse on a ring, which can be regarded as a limitcycle in high-dimensional phase space, is developed. By locally approximating the isochrons ofthis limit cycle orbit, phase sensitivity function of the traveling pulse, which quantifies linearresponse of the traveling pulse to spatial perturbations, is derived. Based on this formulation,a pair of weakly interacting traveling pulses is analyzed. For example, a pair of mutuallyinteracting reaction-diffusion systems
∂
∂tX1(x, t) = F(X1) + D
∂2
∂x2X1 + ε(X2 − X1),
∂
∂tX2(x, t) = F(X2) + D
∂2
∂x2X2 + ε(X1 − X2),
can be reduced to the following coupled phase equations:
φ1(t) = ω + εΓ(φ2 − φ1),
φ2(t) = ω + εΓ(φ1 − φ2),
where X1(x, t) and X2(x, t) represent certain reaction-diffusion fields exhibiting stable travelingpulses, φ1 and φ2 represent the phases of the traveling pulses, i.e., their locations, and thefunction Γ is the phase coupling function that determines effective interaction between thepulses. It is shown that the pulses can mutually phase lock at multiple locations, reflectingcomplex functional shapes of the phase sensitivity function. Synchronization of traveling pulsesdue to common noisy forcing can also be analyzed within the same framework.
0 100 200 300 400x
-0.4
0.0
0.4
0.8
activ
ator
Figure 1. Phase locking of two traveling pulses
Collaborators: Tatsuo Yanagita (Hokkaido), Yoji Kawamura (JAMSTEC)
REFERENCES[1] F.C. Hoppensteadt and E.M. Izhikevich, “Weakly Connected Neural Networks”, Springer, New York, 1997.[2] Y. Kawamura, H. Nakao, and Y. Kuramoto, “Noise-induced turbulence in nonlocally coupled oscillators”, PhysicalReview E 75, 036209 (2007).[3] Y. Kawamura, H. Nakao, K. Arai, H. Kori, and Y. Kuramoto, “Collective Phase Sensitivity”, Physical ReviewLetters 101, 024101 (2008).
1
P30
A Mathematical Approach to Explain in vitro Amoeba Locomotion
T. Kazama1*, Y. Nishigami2, A. Taniguchi2,
K. Ito1, R. Kobayashi1 and S. Sonobe2 * [email protected]
1. Graduate School of Science, Hiroshima University 2. Graduate School of Life Science, University of Hyogo,
Small organisms consist of just one cell can crawl, such as amoebae, although there is no permanent part of the body designed for locomotion. This movement is achieved by pseudopods in a well-known manner [1]. Recently, Nishigami and Sonobe developed the semi-artificial organisms that move like amoeba, and they named it “in vitro Amoeba” [2]. When the crude myosin distilled from Amoeba proteus was injected into the amoeba extract, the crude myosin moved as follows: Initially the crude myosin was surrounded by some layers and contracted as a whole. Then a part of the crude myosin effused. The effused myosin was surrounded by some layers and it was contracted again, and a new effusion was induced. These contraction and effusion process occur multiple times, and these motions of the crude myosin are similar to the pseudopod formation of a living Amoeba proteus [2] (Fig.1). Moreover, they observed several locomotion patterns which depend on the extract conditions: When the concentration of Ca2+ in the extract is high, the period between an effusion and the next effusion is short, on the other hand, the period is long at the low concentration of Ca2+ in the extract [2]. The locomotion of in vitro Amoeba is interesting from not only the biological viewpoint but also mathematical viewpoint, however the mechanism of it is not well understood. In this study, we propose a hypothesis to explain the mechanism of the locomotion: At first, the membrane-like structure is formed at the interface between the extract and the crude myosin, because the amoeba extract forms actin mesh structure by the Ca2+ mediated ATPase activity [3]. As this reaction proceeds, the tension of the membrane-like-structure increases, and the internal pressure of the crude myosin is raised. When the internal pressure of the crude myosin increases up to some threshold, the membrane-like-structure destructs locally. Then the crude myosin effuses from the break point of the membrane-like-structure. In this poster, we examine this hypothesis using a mathematical approach. We constructed a simulation model describing the internal pressure dynamics of the crude myosin. Our model successfully reproduced the in vitro Amoeba locomotion; the multiple production of the contraction and effusion of crude myosin.
Figure 1. Snap shots of in vitro Amoeba. Allows and numbers indicate the effused crude myosin and created number of them, respectively.
REFERENCES [1] McNeill, A. R. (1992) Exploring biomechanics: Animals in motion, W.H.Freeman and Company, New York. [2] Nishigami, Y., Taniguchi, A., Shimmen, T. and Sonobe, S. (2009) In vitro amoeba. Proc.of The 61st Annual Meeting of the
Japan Sociaety of Cell Biology. [3] Condeelis, J. S. (1977) The isolation of microquantities of myosin from Amoeba proteus and Chaos carolinensis, Analytical
Biochemistry 78, pp. 374-394.
P31
Mathematical Modeling of Asymmetric Wavesin Crawling Movement of Abalone
*Mayuko IWAMOTO, T. Kazama, K. Ito, R. Kobayashi*[email protected]
Department of Mathematical and Life Sciences, Graduate School of Science, Hiroshima University
Gastropods such as snails and slugs crawlon the ground by translating contraction wavesalong their wide and flat foot. The wave run-ning to the same direction as the traveling direc-tion of the body is called ”direct wave”, whichis adopted by snails and slugs. On the contrary,the wave running to the opposite direction to thetraveling direction is called ”retrograde wave”.It is known that the sea hare move by this typeof wave. The above-mentioned species move byalmost symmetric waves as shown in the uppercolumn of Fig.1. Interestingly, several speciesof gastropod exhibit asymmetric waves, for ex-ample, limpets move by translating asymmetricdirect waves and patella by asymmetric retrograde waves[1] as shown in the lower column ofFig.1.In general, gastropods wear peculiar mucus on the their foot. An experiment on Banana slugsshowed that there is a threshold strength of force below which mucus behave like solid andabove which behave like liquid [2].To understand the motion of these gastropods, and especially to answer the question why somespecies break symmetry of waves, we selected abalone as an experimental animal. Based onthe careful observation, we constructed a mathematical model of the motion of abalone usinga spring mass system. We adopted RTS (Realtime Tunable Spring); a spring whose naturallength is controllable in real time. RTS has both of active and passive characters, and thuscan be a simple mimic of muscle. The simulation of one dimension model which takes thecharacteristic of mucus into account has indicated that the direction of the wave is related tothe threshold value of mucus and muscular force. In addition, an extended model to two di-mensions could represent the utility of symmetry breaking from the viewpoint of the efficiencyof motion. In this poster presentation, we would like to discuss the simulation result of one andtwo dimensional mathematical models.
REFERENCES[1] E.R.Trueman. Locomotion in molluscs. The Mollusca (ed. W. D. Russell-Hunter), 4, 155-194. Academic Press, NewYork, 1983.[2] M. W. Denny. The role of gastropod pedal mucus in locomotion. Nature, 285:160-161, 1980.
1
P32
Onset of pulse generator patterns in a three-component system
Masaaki Yadome, Yasumasa Nishiura and Takashi Teramoto†[email protected]
Research Institute for Electronic Science, Hokkaido UniversityChitose Institute of Science and Technology†
We have studied the pulse behaviors using the three-component reaction-diffusion system witha jump type spatial heterogeneity [1-4]. The heterogeneity causes the system to produce thelocalized structures, called the defects around the jump point. By varying the parameters,the defects lost stabilities and oscillating type defect solutions are emanated from the Hopfbifurcation points. The global bifurcation structure plays a key role in clarifying the underlyingmechanism [3].
In the parameter regions where the stable defects disappear, we observed the complicatedpulse behaviors (we call pulse generator hereafter), where traveling pulses are emitted periodi-cally from the jump region. By careful numerical simulations, it is shown that there are variousrhythms of generation depending on the parameters.
In this study, we detect such pulse generator patterns as periodic solutions and investigateits stability property to draw the global structures of pulse generator solutions. As a result,we find that the onset of pulse generators are related to the unfoldings of unstable periodicsolutions. The details will be given in the poster session.
REFERENCES
[1]Y.Nishiura, Y.Oyama and K.-I.Ueda, ”Dynamics of traveling pulses in heterogeneous media of jump type”, HokkaidoMath.J., 36 (2007).[2]Y.Nishiura, T.Teramoto, X.Yuan and K.-I.Ueda, ”Dynamics of traveling pulses in heterogeneous media”, chaos, 17(2007).[3]X.Yuan, T.Teramoto and Y.Nishiura, ”Heterogeneity-induced defect bifurcation and pulse dynamics ofr athree-component reaction-diffusion system”, Physical Review E, 75 (2007).[4]T.Teramoto, X.Yuan, Markus Bar and Y.Nishiura, ”Onset of unidirectional pulse propagation in an excitablemedium with asymmetric heterogeneity”, Physical Review E, 79 (2009).
P33
Hydrodynamic Synchronization of Active Microfluidic Rotors
Nariya Uchida* and Ramin Golestanian†
* Department of Physics, Tohoku University †The Rudolf Peierls Centre for Theoretical Physics, University of Oxford
Cells swimming in viscous environment often utilize coordinated motion of cilia and flagella to achieve efficient propulsion [1]. For example, the surface of a Paramecium is covered by about 5000 cilia that beat in synchrony to form a propagating wave on the surface (metachronal wave). An E. Coli has 6-10 flagella that are autonomously bundled in the swimming mode and unbundled in the tumbling mode. The emergent collective pumping by these active cell machineries has potential application to microfluidic devices, as recently demonstrated by the bacterial carpet [2]. It is a dense monolayer of flagellated bacteria that are softly attached to an elastomeric substrate with their heads, while their tails (flagella) can freely rotate in the fluid. Orientational ordering of the flagella result in non-trivial flow patterns ("whirlpools" and "rivers") that enhance fluid mixing near the surface. These systems provide interesting examples of synchronization mediated by long-range hydrodynamic interaction. To address this problem theoretically, we have introduced a simple yet generic model of microfluidic rotors arrayed on a flat substrate [3]. In our model, each rotor has a spherical bead that moves on a circular trajectory of radius b at constant height from the substrate. The rotor is driven by an active torque τ and exerts a radial pumping force F on the surrounding fluid. The dynamics of the rotors arrayed on the substrate is described by a coupled phase oscillator model with a long-range coupling that decays as the inverse cube of the distance (Oseen-Blake tensor). The ratio between the tangential force τ/b and the radial force F determines the phase delay δ, which controls geometrical frustration. For δ=0, the system attains global synchronization via coarsening of topological defects, while for δ = π/2 the system is fully frustrated and disordered. For intermediate values of δ, we find turbulent and self- proliferating spiral waves in the dynamic steady state (Fig.1).
REFERENCES Fig.1: Turbulent spiral waves for δ = π/4.
[1] Bray, D., Cell Movements: From Molecules to Motility (2nd Ed.), Garland Publishing, N.Y. (2001) . [2] Darnton, N. et al., Biophys. J., 86, 1863 (2004). [3] Uchida, N. and Golestanian, R., Phys. Rev. Lett., 104, 178103 (2010). [4] Uchida, N. and Golestanian, R., Europhys. Lett., 89, 50011 (2010).
We also consider a random distribution of δ around zero. As the randomness is increased, the system exhibits a gradualcrossover from the synchronized to desynchronized state [4]. This is in contrast to the sharp transition in globally coupled oscillators, and can be interpreted by spatial fluctuations of the frequencies of locally synchronized clusters.
P34
Starting-Wave of Pedestrians and its Application for Marathon
Akiyasu [email protected]
Meiji Institute for Advanced Study of Mathematical Sciences, Meiji UniversityCollaborators : Daichi Yanagisawa(JSPS/Univ. Tokyo), Takashi Imamura(Univ. Tokyo), Katsuhiro Nishinari(Univ. Tokyo/JST)
Jamming phenomenon, modeled as a system of interacting particles driven far from equilib-rium, offers the possibility to study various fundamental aspects of non-equilibrium systems.The investigations of recent years about jamming formation tell us that one of the most impor-tant factor to cause the jamming phenomena is the time delay of reaction to the stimulus. Asan example, if the time delay of reaction is extremely small, jamming phenomena have neveroccured by adjusting the behavior immediately to the forward movement.
Definitely the reaction time of pedestrians is important toward smooth movement in crowdscene. Especially, we would like to point out that the pedestrians’ successive reaction, so-calledstarting wave, plays a significant role for the waiting time of queuing systems and evacua-tion time, since the decreasing required time to start walking accomplishes the more smoothmovement in crowd scene. In this contribution, we have investigated the propagation speedof starting wave of pedestrians in a line by using numerical simulations and real experiments.Furthermore, taking into account this result, we have found the optimal density of runners tominimize the travel time at the initial stage in marathon.
Figure 1. Which is the optimal density as an initial distribution?
ACKNOWLEDGMENTSWe thank Kozo Keikaku Engineering Inc. in Japan for the assistance of the experiments. The author (AT) issupported by the Meiji University Global COE Program ”Formation and Development of MathematicalSciences Based on Modeling and Analysis”. We acknowledge the support of Japan Society for the Promotionof Science and Japan Science and Technology Agency.
REFERENCES
[1] D. Chowdhury et al., Phys. Rep. 329 (2000), 199-329.[2] D. Helbing, Rev. Mod. Phys. 73 (2001), 1067-1141.[3] M. Bando, et al., Phys. Rev. E, 51, 1035 (1995).[4] A. Seyfried, et al., J. Stat. Mech., 10002 (2005).[5] B. D. Greenshields, in Proceedings of the Highway Research Board, Washington, D. C., 14 (1935), 448.
1
P35
On Simple Shear Flows of a Continuum Model withDensity-gradient Dependent Stress
Naoto [email protected]
Shibaura Institute of Technology
Here, the steady state of flows of a continuum model with density-dependent gradient stressis studied. The governing system of a simple shear flow down on an inclined plane is given asfollows.
(1)
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
(a3(%(y))u0(y))0/2 + %(y)g sin θ = 0 for 0 < y < h,
−p(%(y)) + (a(%(y)))(%0(y))20 − %(y)g cos θ = 0 for 0 < y < h,
a3(%(h))u0(h)/2 = 0,
−p(%(h)) + a(%(h))(%0(h))2 = −pe,u(0)− ka3(%(0))u0(0)/2 = 0.
Here, v is the velocity vector field; % is the density; p(%) is the pressure of barotropic type;T is the Cauchy stress tensor; b is the external body forces; g is the acceleration of gravity;(∇ · T)i =
P3j=1
∂Tij∂xj; D(v) = 1
2(∇v + [∇v]T) ; M = ∇%⊗∇%; aj(%) is material moduli.
Rajagopal and Massoudi [1] introduced the constitutive equation for granular flows whosemotion depends on the inhomogeneity of the body taking the form T = T(%,D,M).From this model the typical density and velocity profile are caputured, due to the appearance
of terms concerning the density gradient. In the poster we will show the mathematical resultson existence theorem and some properties of solutions to the simple shear model, and also showthe profiles obtaind by numerical computations.
REFERENCES
[1] K. R. Rajagopal and M. Massoudi: A method for measuring material moduli of granular materials: flow in anorthogonal rheomer. Topical Report U.S. Department of Energy DOE/PETC/TR-90/3 (1990).
1
P36
Behaviors of a front-back pulse in some bistable reaction-diffusionsystem with heterogeneity
K. Nishi 1, T. Teramoto 2, Y. Nishiura 3
[email protected],3 Hokkaido University, 2 Chitose Institute of Science and Technology
We study the behavior of a localized moving pattern arising in a two-component reaction-diffusion system with heterogeneity. It is well known that front solutions typically appear inbistable systems. When the nonlinearity of the system is slightly perturbed from the oddsymmetry, such front solutions are glued together to form a spatially localized pulse (front-back pulse). The front-back pulse exhibits not just traveling but also oscillatory motions viathe Hopf bifurcations depending the parameters. The global behaviors of such pulse solutionshave already been studied numerically and analytically [1][2].
In this study, we consider the traveling pulse behaviors in the jump-type heterogeneousmedia, in which one constant parameter value jumps up (or down) to another constant valueat one point in space. By carrying out numerical simulations, it was found that there occur fivedifferent behaviors through the collision between the pulse and the heterogeneity; penetration,travelling-breather, oscillation rebound, annihilation and decomposition.
We deal with the singular limit system (we call hybrid system hereafter), in which one of theequations of the system is replaced by the equations of motion for interfaces by employing thetraveling wave solution to a scalar reaction-diffusion equation. It is numerically confirmed thatthis reduced hybrid system reproduces the dynamical behaviors observed in the original PDEs.By using the center manifold theory, we also derived finite dimensional ODEs to investigatethe pulse dynamics around the penetration-rebound boundaries analytically. The details willbe shown in the poster session.
REFERENCES[1] M.Mimura, M. Nagayama, H. Ikeda, T. Ikeda: Hiroshima Math Journal Vol. 30 (2000).[2] T.Ikeda, H. Ikeda, M. Mimura,: Methods and Appl. Analysis Vol. 7 (2000),165-194.
1
P37
Network formation in spatio-temporally varying field by true slimemold
Kentaro ItoDepartment of mathematical and life sciences, Hiroshima University
David SumpterMathematics Department, Uppsala University
Toshiyuki NakagakiDepartment of Complex and Intelligent Systems, Future University Hakodate
Revealing how lower organisms solve complicated problems is a challenging research area, whichcould reveal the evolutionary origin of biological information processing. Here we report on theability of a single-celled organism, true slime mold, to find a smart solution of risk managementunder spatio-temporally varying conditions.
We designed test conditions under which there were three food-locations at vertices of equi-lateral triangle and a toxic light illuminated the organism on alternating halves of the triangle.We found that the organism behavior depended on the period of the repeated illumination,even though the total exposure time was kept the same. Long periods of lighting leads to aseparation of the organism, while shorter periods produce a highly connected structure.
A simple mathematical model for the experimental results is proposed from a dynamicalsystem point of view, and the mathematical model reproduced the basic tendency of the finalpattern with respect to the period P. To understand the dynamical behavior of the model, wetransfer the continuous system into a discrete dynamical system. We found that the discretedynamical system showed the saddle-node bifurcation when we changed the illumination periodP, which together with a dependency on initial conditions explains the experimental outcome.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
P=1.1 P=2.2 P=3.3
Du Du
Ds
Ds
D1(2nP)
D1(
(2n
+2
)P)
D1(2nP)
D1(
(2n
+2
)P)
D1(2nP)
D1(
(2n
+2
)P)
REFERENCES[1] Kentaro Ito, David Sumpter and Toshiyuki Nakagaki. ”Risk management in spatio-temporally varying field by trueslime mold” . Nonlinear Theory and Its Applications, IEICE. Vol. 1, No. 1, p.26-36 . (2010)
1
P38
Mathematical studies on the two-dimensional self-motion ofcamphor scrapings
Keita Iida,1,∗) Hiroyuki Kitahata,2,5)
Satoshi Nakata3) and Masaharu Nagayama4,5)
∗) [email protected]) Graduate School of Natural Science and Technology, Kanazawa University,
2) Graduate School of Science, Chiba University,3) Graduate School of Science, Hiroshima University,
4) Institute of Science and Engineering, Kanazawa University,5) PRESTO, Japan Science and Technology Agency
There are several types of autonomous motors known to exhibit various manners of self-motion at the air/water interface. Particularly, we focus here on the self-motion of surfactantsthat have been studied experimentally and theoretically. Certain types of surfactants maintaintheir motion at the water surface for a long time under almost isothermal and nonequilibriumconditions. Hence, they have been considered as a novel efficient chemo-mechanical transducerwhich mimics living organisms [1]. Examples are camphor scrapings (diameter, ca. 1 mm) thatexhibit unidirectional translation, rotation and revolution on pure water [2].
Theoretical analyses on the one- and two-dimensional self-motion of “camphor discs” havebeen carried out on the basis of the mathematical model and numerical calculations [3, 4].The next step we discuss below is to consider both translational and rotational motion inducedby the anisotropic shape of the camphor, for the camphor disc cannot rotate in the theory.Although a mathematical model consisting of translational and rotational equations coupledwith a reaction-diffusion equation has been proposed in 2005 [5], the characteristics of thetwo-dimensional camphor motion depending on its shape has not yet been clarified.
To characterize the relationship between the mode of self-motion and the shape of the cam-phor scraping with the revised model, we reproduced the various manners of motion withnumerical simulations. For example, disc and s-shaped camphors exhibited only unidirectionaltranslation and rotation, respectively, while the camphor with the shape of a “comma (,)”exhibited both of these motions. In fact, these are good examples to verify the validity of themathematical model by comparing to experimental results. However, when considering the el-liptical shape, we faced sensitive questions; which direction does the camphor move, in the longaxis or in the short axis? Or does the elliptical camphor rotate? We found it difficult to inves-tigate the above question by the brief experiment and simulation due to difficulties originatingfrom the elliptical shape (e.g., we have to compute the elliptic integral in the model).
Therefore, we apply the mathematical analyses and conduct proper experiments to determinethe mode of motion of elliptical camphor scraping. In the poster session, we will introduce ournumerical findings and analytical results.
REFERENCES[1] A. Mikhailov and D. Meinkohn, Self-Motion in Physico-Chemical Systems Far from Thermal Equilibrium, Springer,
Berlin Heidelberg, 484, (1997).[2] S. Nakata and Y. Iguchi, S. Ose, M. Kuboyama, T. Ishii and K. Yoshikawa, Langmuir, 13, 4454–4458 (1997).[3] M. Nagayama, S. Nakata, Y. Doi and Y. Hayashima, Physica D (Amsterdam), 194, 151–165 (2004).[4] X. Chen, S. Ei and M. Mimura, Networks and Heterogeneous Media, 4, 1–18 (2009).[5] H. Kitahata and K. Yoshikawa, Physica D (Amsterdam), 205, 283–291 (2005).
P39
The discrete Morse ow for volume-controlled membrane motions
Elliott [email protected]
Kanazawa UniversityGraduate School of Mathematics and Physics
We present a variational method for constructing weak solutions to parabolic and hyperbolicproblems with time-dependent volume constraints. The problems originate from within anevolutionary equation that expresses a model of droplet formation and motion under conden-sation. This evolutionary equation is a damped hyperbolic free boundary problem with anon-local term corresponding to the prescribed volume. The constituent problems also includenon-local terms which arise from the volume constraints, and energy estimates are obtainedby testing with special volume preserving perturbations. We discuss these topics, as well asintroduce the numerical implementation of our method and show the results of its applicationto our problems.
The case of a parabolic problem with constant volume constraint was treated in [3], and in[1] it was shown that the volume can be controlled in time by an L2-function. The hyperbolicproblem with constant volume constraint appeared in [4], and we have also analyzed the controlof the hyperbolic problem.
Similar studies have been made for analyzing constrained free boundary problems. A para-bolic free boundary problem with constant volume constraint appeared in [5], and for a hyper-bolic free boundary problem with constant volume constraint, see [2]. Various investigationsinto the numerical behaviors of such problems have also recently been made. The interestedreader can consult the cited references for further information.
Our method, known as the discrete Morse flow (DMF), discretizes time and produces anapproximating sequence of weak solutions to specied elliptic problems. It is similar in thisregard to the method of Rothe, but rst uses functional minimizations to construct solutions toelliptic problems. Moreover, this variational aspect of the DMF allows us to prescribe a volumeto each weak solution. We build approximate solutions by interpolating the elliptic problemsolutions in time, and we derive appropriate energy estimates. These estimates allow for thelimit passages to the stated denitions of weak solutions.
A remarkable aspect of the DMF is that it can be used to obtain both analytic and numericresults. Our functionals are approximated by means of the nite element method, and theirminimizations are realized by a gradient descent. The volume constraints are imposed byrestricting the gradient search to the set of admissible functions via a projection onto a volumeconstrained hyperplane at each iteration. This approach is numerically advantageous in thesense that penalties for the volume constraint are automatically satised.
REFERENCES[1] E. Ginder. Construction of solutions to heat-type problems with time-dependent volume constraints. Advances inMathematical Sciences and Applications, 20 (2010) (to appear).[2] E. Ginder, K. Svadlenka. A variational approach to a constrained hyperbolic free boundary problem. NonlinearAnalysis, 71 (2009), 1527-1537.[3] K. Svadlenka, S. Omata. Construction of Solutions to Heat-Type Problems with Volume Constraint via the DiscreteMorse Flow. Funkcialaj Ekvacioj, 50 (2007), 261-285.[4] K. Svadlenka, S. Omata. Mathematical modelling of surface vibration with volume constraint and its analysis.Nonlinear Analysis, 69 (2008), 3202-3212.[5] K. Svadlenka, S. Omata. Mathematical Analysis of a Constrained Parabolic Free Boundary Problem DescribingDroplet Motion on a Surface. Indiana University Mathematics Journal, 58 (2009), 2073-2102.
P40
Reliable Time in Biological Clocks
Fumito Mori1 and Hiroshi Kori1,2
[email protected] of Advanced Sciences, Ochadai Academic Production, Ochanomizu University, Tokyo 112-8610,
Japan2PRESTO, Japan Science and Technology Agency, Kawaguchi 332-0012, Japan
Various biological rhythms, such as heartbeats and circadian rhythms, are found in livingorganisms. Herzog et al. reported an interesting observation about circadian rhythms in mice[1]. They observed cycle-to-cycle variability of behavioral rhythms in mice kept in a darkroom, and found that the variability in the circadian periods between the onsets of activityis lower than that between the offsets. Thus, mice seem to recognize the onset time of dailybehavior more precisely than the offset time. The circadian clock in mammals is orchestratedby the suprachiasmatic nucleus (SCN), which consists of coupled clock cells. Herzog et al. alsoobserved that the onset time of electric activity in explant SCNs is more precise than the offsettime, which is consistent with the above finding in behavioral rhythms. In this sense, the onsettime kept by the biological clock is more reliable than the offset time. However, the reasonfor the changes in the reliability of the biological clock remains an open issue. Although thereare theoretical studies on the variability of periods in a phase-coupled oscillator model [2], thismodel can not show the time-dependence of the variability because of its rotational symmetry.
To deal with this issue theoretically, we consider a phase-coupled oscillator model withoutrotational symmetry given by
(1)dθi
dt= ω + κZ(θi)
∑j
Aij(h(θj) − h(θi)) + ξi(t),
where θi and ω are the phase and the intrinsic frequency of the ith oscillator, respectively; κ isthe coupling strength; and Z(θ) and h(θ) are 2π-periodic functions, called the phase responsefunction and the signal function, respectively. Aij is an element of an adjacency matrix A,where Aij = Aji = 1 if oscillators i and j interact with each other, and Aij = 0 otherwise.The term ξi(t) represents time-dependent noise. Note that this model is generally derived fromweakly coupled limit-cycle oscillators [3]. We assume that, in the absence of noise, the oscillatorssynchronize in phase. To quantify the variability of periods, the coefficient of variation (CV)
of periods is introduced as follows. The kth oscillation time of the ith oscillator, t(i)k , is defined
as the time at which θi passes 2kπ + θ0 for the first time, where θ0 is a check-point phase. The
oscillation period ∆t(i)k is defined by ∆t
(i)k = t
(i)k − t
(i)k1. The CV is defined by the standard
deviation of the period divided by the average period.By linearizing Eq.(1) around the synchronized state, we analytically obtained the CV as a
function of θ0, and thereby the effect of the interaction on the reliability becomes clear. Toconfirm our theoretical result, we performed numerical simulations of Eq.(1) with Z(θ) = cos(θ)and h(θ) = sin(θ) and calculated the CV. The CV obtained numerically agreed well with thetheoretical curve. Our study provides a design principle of reliable time in biological clocks.
REFERENCES[1] E. D. Herzog, S. J. Aton, R. Numano, Y. Sakaki and H. Tei, J. Biol. Rhythms 19, 35 (2004).[2] N. Masuda, Y. Kawamura and H. Kori, New. J. Phys. 12 093007 (2010); H. Kori, Y. Kawamura, N. Masuda: inpreparation.[3] A. T. Winfree, J. Theor. Biol., 16, 15 (1967); Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence(Springer, NewYork, 1984).
P41
Heterogeneity-induced spot dynamics for a three-componentreaction-diffusion system
Xiaohui Yuan and Takashi Teramoto and Yasumasa [email protected] [email protected] [email protected]
Northeast Institute of Geography and Agroecology, Chinese Academy of SciencesFaculty of Photonics Science, Chitose Institute of Science and Technology
Research Institute for Electronic Science, Hokkaido University
Heterogeneity is one of most important and ubiquitous types of external perturbation in dis-sipative systems. It causes the emergence of various types of heterogeneity-induced-ordered-patterns (HIOPs), which influence how two-dimensional traveling spots propagate in those sys-tems. We studied the dynamics of traveling spots in a heterogeneous media, in particular whenthey encounter a jump type line heterogeneity. The model system used is a three-componentreaction-diffusion system with monostable excitability. Since the heterogeneity is introducedin the kinetic part k1, the homogeneous background is no longer a rest state and various typesof nonuniform ordered patterns, called heterogeneity-induced-ordered-patterns (HIOPs), takeover. A variety of outputs emerge through the interaction between the HIOPs and the travel-ing spots, including rebound, penetration and trapped behaviors. Such rich behaviors originatefrom the wide variety of these interactions. The global bifurcation and eigenvalue behavior ofa heterogeneity-induced spot patterns (HISPs) clarify the underlying mechanisms for the tran-sitions among those dynamics. A reduction to a finite-dimensional system is presented here toextract the model-independent nature of the dynamics. Selected numerical techniques for thebifurcation analysis are provided.
REFERENCES
[1] Y. Nishiura, T. Teramoto and K.-I. Ueda, Dynamic transitions through scattors in dissipative systems, Chaos, 13(2003), 962-972.[2] Y. Nishiura, T. Teramoto and K.-I. Ueda, Scattering and separators in dissipative systems, Phys. Rev. E, 67(2003), 056210.[3] Y. Nishiura, T. Teramoto, X. Yuan and K.-I. Ueda, Dynamics of traveling pulses in heterogenous media, Chaos, 17(2007), 037104.[4] X. Yuan, T. Teramoto and Y. Nishiura, Heterogeneity-induced defect bifurcation and pulse dynamics for athree-component reaction-diffusion system, Phys. Rev. E, 75 (2007), 036220
1
P42
True Orbit Computation Using Integer Arithmetic
Asaki Saito∗ and Shunji Ito†∗[email protected]
Future University Hakodate, JapanPRESTO, JST, Japan†[email protected]
Kanazawa University, Japan
We introduce a method allowing accurate simulations of discrete-time dynamical systemsdefined by one-dimensional piecewise linear fractional maps. The salient feature of this methodis that it incorporates integer arithmetic to compute true orbits. In this presentation, we reportthe results for the case in which cubic surds were used to represent numbers and generate trueorbits of two maps, the Bernoulli map (i.e., the 2x modulo 1 map) and the following map T ,
T (x) =
⎧⎪⎪⎨⎪⎪⎩−xx− 1 , if x ∈ [0, 1/2)
2x− 1x
, if x ∈ [1/2, 1)
which presents f−1 power spectrum [1]. Standard computer simulations with floating-pointnumber representation have a fundamental difficulty in generating the orbits of these maps. Forexample, the inevitable round-off error can break down the property of the f−1 power spectrumin principle. In contrast, our method can successfully generate accurate orbits without sucherrors by utilizing integer coefficients of cubic equations to represent numbers. In fact, Figs. 1(a)and (b) show true orbits of the Bernoulli map and the map T , respectively, originating from thesame initial condition x0 ∈ (0, 1) satisfying x30 + x20 + x0 − 1 = 0. As shown in Figs. 1(a) and(b), the true orbits generated by our method clearly show chaotic and intermittent behaviors,respectively. We also show that the statistical properties obtained by the generated orbits wellcoincide with those of the typical orbits of the two maps, by demonstrating results concerninginvariant measures and the power spectrum.
ACKNOWLEDGMENT
This work was supported by JST PRESTO program and MEXT KAKENHI 21700256.
REFERENCES
[1] A. Saito, “Computational aspects of a modified Bernoulli map,” Prog. Theor. Phys. Suppl. 161 (2006) 328—331.
(a) (b)
Figure 1. (a) Iterates of the Bernoulli map starting from x0 satisfying x30 +
x20 + x0 − 1 = 0. (b) Iterates of the map T starting from the same x0.
P43
An application of the characteristics nite element method to athermal convection problem with the innite Prandtl number
Hirofumi [email protected]
Meiji Institute for Advanced Study of Mathematical Sciences
For a thermal convection problem with the infinite Prandtl number (e.g., see [3]), we present afinite element scheme based on the method of characteristics and show numerical results.
The idea of the method of characteristics is to consider the trajectory of the fluid particleand discretize the material derivative term along the trajectory. It enables us to construct asymmetric scheme and to use symmetric linear solvers[2].
The problem is governed by a nonlinearly coupled system of the Stokes equations and theconvection-diffusion equation. Roughly speaking, we solve the two equations one after theother. Applying the idea of the method of characteristics to the convection-diffusion equa-tion and employing a cheap element P1/P1/P1 with a pressure stabilization[1], we devise acharacteristics finite element scheme, which is useful for large scale computations.
REFERENCES[1] F. Brezzi and J. Douglas Jr., Stabilized mixed methods for the Stokes problem, Numerische Mathematik, Vol.53(1988), pp.225–235.[2] H. Notsu and M. Tabata, A single-step characteristic-curve nite element scheme of second order in time for theincompressible Navier-Stokes equations, Journal of Scientific Computing, Vol.38, No.1 (2009), pp.1–14.[3] M. Tabata and A. Suzuki, A stabilized nite element method for the Rayleigh-Benard equations with innite Prandtlnumber in a spherical shell, Computer Methods in Applied Mechanics and Engineering, Vol.190 (2000), pp.387–402.
P44
Comparison of Boolean dynamics in complex networks
Shu-ichi Kinoshita, Kazumoto Iguchi and Hiroaki [email protected] University, Japan
Dynamics of gene interactions in cell and robustness of cell still open problems. A most sim-plified model for such gene interactions is known as the Kauffman’s random Boolean network(RBN) model [1], where it is assumed that a gene is regulated by a certain fixed number ofother input genes. However, in real gene networks, the number of input genes that regulateeach gene is fluctuating among the genes. The characteristic of the fluctuation is various suchthat the distribution function is given as either an inverse power-low distribution or an expo-nential distribution or a Poisson distribution. Therefore, we compare the characteristics of theattractors in the Boolean dynamics on the scale-free network (SFRBN) with that on RBN.
A Boolean network consists of N nodes, each of which receives ki input degree. The numberki of in-degree at the i−th node is determined by the preferential attachment rule [2]. The stateσi of the nodes are synchronously updated in time step, according to the Boolean function fi
assigned for each node as,
(1) σi(t + 1) = fi(σi1(t), σi2(t), · · · , σiki(t)).
All trajectories starting at any initial state run into a certain number of attractors, i.e. points orcycles. In this system, the genetic states are represented in terms of the language of attractors.
From the results of numerical simulation, the behavior that each initial state goes to anattractor in SFRBN is different from that in RBN. We have obtained the following results.First, The attractor periods in SFRBN are much more widely distributed than in RBN. Thisresult mean that SFRBN has a lot of long period attractors compared with RBN. Second, theaveraged relative entropy for SFRBN is smaller than that for RBN. The small relative entropymeans the robustness of attractors. So, the attractors in SFRBN are more robust than thatin RBN. Third, we succeeded in taking out important a core network related to the dynamicsdirectly from network. The topology of core networks is difference between SFRBN and RBN.We found that the loop structures of core network an important role for dynamics in network.
In this presentation, we show some other quantities to clear the differences between SFRBNand RBN too. We focus on the relation between the network topology and the dynamics onnetwork.
REFERENCES[1] Kauffman, S. A., “Metabolic Stability and Epigenesis in Randomly Constructed Genetic Nets”, J. Theor. Biol. 22,437-467 (1969)[2] Barabasi, A.-L., and Albert, R., “Emergence of scaling in random networks”, Science 286, 509-512 (1999)[3] Iguchi, K., Kinoshita, S., and Yamada, H., “Boolean Dynamics of Kauffman Model with a Scale-Free Network”, J.Theor. Biol. 247, 138-151 (2007)[4] Kinoshita, S., Iguchi, K., and Yamada, H., “Intrinsic Properties of Boolean Dynamics in Complex Networks”, J.Theor. Biol. 256, 351-369 (2009)
P45
On a simplified tumor growth model with contact-inhibition
Michiel Bertsch, Masayasu Mimura, Yusaku Nagata and Tohru Wakasa
Meiji University
In the last two decades several mathematical models on tumor invasion has been discussedby many researchers (see [1], [3]-[5]). Among them, we refer to [3] by Chaplain-Graziano-Presiozi; they have proposed a reaction diffusion system for normal and abnormal cells, extracellular matrice generated by normall and abnormal cells, and matrix degrading enzyme. Froma viewpoint of contact-inhibition of cells, a simplified system of two cell populations, has beenintroduced in [1]. This system shows us contact-inhibition of cells in the following sense: Forsegregated initial data, the solution keeps segregated after the contact. Moreover, the behaviorof solutions with the special segregated initial data can be reduced to by a free boundaryproblem.
In order to understand simplified tumor invasion process with contact-inhibition, we willstudy the qualitative behavior of the solution of the above system. Here we will introduce anew positive parameter α, which represents the effect of physical differences (e.g., average sizeof cells) into the reaction between normal and abnormal cells, and will consider how α affectthe behavior of the solution.
In this presentation, we will show analytical and numerical results on 1-dimensional problemof the above reaction diffusion systems, which are obtained in [2].Numerical results suggest usthat
(i) if α is small, then population densities tends to segregated, and abnormal cells densitypropagate the region like traveling wave solutions,
(ii) if α is large, then population densities tends to overlapped, and a new steady-state ofcell population appears.
Also, we will explain that such differences of qualitative behavior is related with the velocityand stability of segregated traveling wave solution of free boundary problem as stated above.
REFERENCES
[1] M. Bertsch, R. Dal Passo, M. Mimura, A free boundary problem arising in a simplified tumour growth model ofcontact inhibition, Interfaces and Free Boundaries, 2, 235-250 (2010).[2] M. Bertsch, M. Mimura, Y. Nagata, T. Wakasa, Traveling waves for a 1-dimensional tumor growth model withcontact-inhibition, preprint.[3] M. A. J. Chaplain, L. Graziano, L. Preziosi, Mathematical modelling of the loss of tissue compression responsivenessand its role in solid tumour deveropment, Math. Med. Bio. 23, 197-229 (2006).[4] G. Garria-Ramos, F. Sanches-Garduno, P. K. Maini, Disparsal can sharpen parapatric boundaries on a spatiallyvarying environment , Ecology, 81, 749-760 (2000).[5] J. A. Sherratt, Wave front propagation in a competition equation with a new motility term modelling contactinhibition between cell populations, Proc. R. Soc. Lond. A, 456, 2365-2386 (2000).
P46
Topological and Computational method for Eigenvalue
problem of 1-dim shrodinger operator
Hokkaido Univ
Ayuki sekisaka
I constructed a rigorous computational method for finding a exis-
tence of eigenvalue of a 1-dim shrodinger operator which has a periodic
potential and compact perturbation term. This method is based on
topological index.
First, eigenvalue problem of the 1-dim shrodinger operator can be
transforme to a first order ordinary differential equations. Then，eigen-
value problem is equivalent to problem of existence of connecting orbit
for unstable subspace to stable subspace.
We focus to the angle of stable and unstable subspace. Therefore，in
the polor coordinate，the system reduction to the only angle equation.
Then, eigenvalue problem is equivalent to problem of existence of
heteroclinic orbit for the angle equation. Details of this method will
be describe at the poster contribution.
P47
Arch Structure in the Escape Panic
Tsubasa Masui and Daishin Ueyama
Graduate School of Science and Technology, Meiji University
Abstract One of disastrous forms of collective human behavior is crowd stampede induce by panic. Sometimes, the jam of people is triggered in life-threatening situations (i.e. Akashi walking bridge’s disaster in Japan). Many researchers are studying this problem by using several methods. We will use an escape panic model proposed by Helbing et al. [1]. They have studied the relationship between maximum velocities of each individual and escape time. Surprisingly, the escape time will increase as desired velocity increased. We claim that clogging around the exit is one of the important factors to understand the phenomenon. We will discuss about the arch structure around the exit from the viewpoint of global structure of stationary solutions of the model. [1] Dirk Helbing & Illés Farkas & Tamás Vicsek , Simulation
dynamical features of escape panic, Nature, Vol.407, p.487, 2000.
P48
A mesh generator using a self-organizing mechanism of a reaction-diffusion system
Masahiro Yamaguchi1*, Hirofumi Notsu2, Daishin Ueyama3 *[email protected]
1Graduate School of Science and Technology, Meiji University 2Meiji Institute for Advanced Study of Mathematical Sciences
3School of Science and Technology, Meiji University
Triangular or tetrahedral meshes are necessary for numerical calculations of partial differential equations. We develop a new mesh generator using a self-organizing mechanism of a reaction-diffusion system.
The procedure of the mesh generation is as follows. First, get the domain shape from a bitmap picture. Second, solve the Gray-Scott Model (GS) [1] on the domain, which is a reaction-diffusion system and gives spot patterns for some parameter regime. Third, detect peak points of the spots as nodes of the mesh. Fourth, to have a final mesh, do the Delaunay triangulation [2] for the obtained nodes.
High-quality solutions of GS are not necessary for the mesh generator. We need only spots given by the self-organizing mechanism. Therefore, considering the computational cost, we use coarse square mesh and a single precision in the computation. Fig. 1 shows a result. The left shows a bitmap picture of a cross-section shape of upper end of the human femur. The right is a mesh by the mesh generator. It can be seen that traces the complicated shape.
Fig. 1. A bitmap picture (left) and a generated mesh (right).
It is possible to control mesh sizes by changing diffusion constants locally in GS, and
would be applicable to moving boundary problems. Since it can be expected that the self-organizing mechanism works well in three dimension, we can get tetrahedral meshes by a similar procedure in the future.
This work is supported by the Meiji University Global COE Program “Formation and Development of Mathematical Sciences Based on Modeling and Analysis”. REFERENCES [1] J. E. Pearson, Complex Patterns in a Simple System, SCIENCE, Vol.261 (1993), pp.189 - 192. [2] T. Taniguchi, Automatic mesh generation for FEM – Application of Delaunay triangulation, Morikita, Tokyo, 1992 (in Japanese).
P49
Size distribution of barchan dunes with a cellular model
Astunari Katsuki
CST, Nihon University
Sand dunes are found in many places such as deserts, the sea bottom and the surface of Mars.
They are formed through interplay between sand and air flow or water flow. When a strong flow
blows, sand grains are dislodged from the sand surface. The entrained sand grains collide with the
ground and are sometimes deposited. This process takes place repeatedly, resulting in the
formation of a dune. The profile of the wind flow is modified by dune topography. Most
fascinated dune is barchan, which is crescent dune. We reproduced many barchans in numerical
simulations and investigate the dynamics.
In the model, the dune field is divided into square cells. Each cell is considered to represent
an area of the ground which is sufficiently larger than the sand grains. A field variable h(x,y,t)
which expresses the local surface height is assigned to each cell; t denotes the discrete time step
and the spatial coordinates x and y denote the positions of the center of a cell in the flow and the
lateral directions, respectively. The edge length of the cell is taken as a unit of length. The motion
of sand grains is realized by two processes: saltation and avalanche. Saltation is the transportation
process of sand grains by flow. The saltation length and saltation mass are denoted L and q,
respectively. Saltation occurs only for cells on the upwind face of dunes. The saltation length L
and the amount of transported sand q are modeled by the following rules,
L = a+bh(x,y,t)-ch2(x,y,t)
where a=1.0, b=1.0, and c=0.01 are phenomenological parameters. The last term is introduced for
L not to become too large. Note that L is used only in the range where L increases as a function of
h(x,y,t). The saltation mass is fixed at 0.1 for simplicity. In the avalanche process the sand grains
slide down along the locally steepest slope until the slope relaxes to be (or be lower than) the
angle of repose which is set to be 34
We reproduced a few hundred of barchans in numerical field by above model. Barchan releases
sand from tips of two horns. The downwind barchan can capture the sand stream. Also, barchans
sometimes collide each other. These direct and indirect interaction forms complex barchan fields.
The size distribution of a few thousand of barchans is fitted by lognormal distribution well. This
indicated that the small barchans exist around the large ones and the large barchans are around
small barchans. The average size of barchans increase as the amount of supplied sand do.
However the variation of the distribution is saturated as the amount of supplied sand increase.
Also the migration velocity of corridor is constant. Next, when two barchan corridors collide, the
size of barchan in the boundary between corridors has three type. Type (I) is not decoupling
distribution, which shows superposition of each distribution. Type (II) is a distribution of uniform
size. Through collision and inter-dune sand stream, the size of each barchan become uniform.
Type (III) shows a enhanced distribution of the barchan's size. The size in boundary region
between barchan corridors is lager locally. These results show that different distribution of
barchans can be coexistence.
P50
Computational virology and immunology - Quantification system ofviral dynamics -
Shingo [email protected]
JST PRESTO, The University of Tokyo, Kyoto University
In the early 1990s, some researchers who were specialist in mathematics and physics in-vestigated qualitative viral dynamics, in particular, about HIV infection using mathematicalmodels but many virologists and immunologists did not pay attention to the importance ofthese theoretical works.
However, historically two important papers published by X. Wei et al., and D. D. Ho etal., in same volume of Nature in 1995 changed their mind and mathematical modeling becameimportant weapon to understand HIV infection [1, 2]. In these two papers, they used a math-ematical model and estimated a very short half-life of HIV virion from HIV RNA data of thepatients with antiviral drug therapy. And also, further theoretical studies has revealed severalpuzzling features of the infection such as half-life of short and long lived productively infectedcells, immune cells, and so on.
Thus, mathematical modeling, which at one time was essentially ignored by the experimen-tal AIDS community, has in the last 15 years become an important tool, and almost all ofthe major experimental groups are now collaborating with a theorist. This is because mathe-matical modeling has provided several quantitative insights which cannot be obtained by onlyexperimental and clinical studies.
In the last three years, I am collaborating with Institute for Virus Research at Kyoto Uni-versity, which is one of the biggest AIDS research group in Japan, and trying to develop a newresearch area ”Computational Virology and Immunology” which is a next generation modelingresearch with experiments. Based on in vitro / in vivo experiments, I make a theory to obtainnovel information about the experiment and knowledge of the disease Furthermore, we alsodesign a new experiment based on my theory to reveal what we want to know. In this way, wemake ”experiments based theory” and conduct ”theory based experiment” to understand HIVinfection and propose an idea to develop an effective strategy against the infection.
Today, I am going to show our recent studies about ”Quantification system of viral dynamics”.
REFERENCES[1] Wei X, Ghosh SK, Taylor ME, Johnson VA, Emini EA, Deutsch P, Lifson JD, Bonhoeffer S, Nowak MA, Hahn BH,Saag MS and Shaw GM, Viral dynamics in human immunodeficiency virus type 1 infection, Nature, 373, 6510(1995),pp117-122.[2] Ho DD, Neumann AU, Perelson AS, Chen W, Leonard JM and Markowitz M, Rapid turnover of plasma virions andCD4 lymphocytes in HIV-1 infection, Nature, 373, 6510(1995), pp123-126.
P51
Map of Kyoto University Area
From Shijo-Karasuma area, take the bus No.201 (for Hyakumanben) or No.31 (for Iwakura)from Shijo-Takakura and go to Hyakumanben. From JR Kyoto Station, take the bus No.17(for Ginkakuji-temple) from Kyoto-Station and go to Hyakumanben. RIMS is a ten-minutewalk to the east from Hyakumanben. Shiran-Kaikan is a three-minute walk to the south fromHyakumanben.
Participants List
Toshio Aoyagi Kyoto Univ. Hitoshi Arai Univ. TokyoZin Arai Hokkaido Univ. Peter Bates Michigan State
Univ., USAOdo Diekmann Utrecht Univ., NL Arjen Doelman Leiden Univ., NLShin-Ichiro Ei Kyushu Univ. Elliott Ginder Kanazawa Univ.Andrei Giniatoulline Los Andes Univ., Taichi Haruna Kobe Univ.
COToshiaki Hattori Mitsubishi Rayon Yumino Hayase MPIP Mainz DEMarcel Horning Kyoto Univ. Takashi Ichinomiya Gifu Univ.Keita Iida Kanazawa Univ. Masato Iida Miyazaki Univ.Makoto Iima Hokkaido Univ. Hideo Ikeda Toyama Univ.Tsutomu Ikeda Ryukoku Univ. Hitoshi Imai Tokushima Univ.Hiroshi Ishikawa Tetsuya Ishiwata Shibaura Inst. Tech.Kentaro Ito Hiroshima Univ. Shingo Iwami PRESTO, JSTMayuko Iwamoto Hiroshima Univ. Christopher K. R. T. Jones Univ. North
Carolina, USAKunihiko Kaneko Univ. Tokyo Yukio Kan-on Ehime Univ.Tasso J. Kaper Boston Univ., USA Naoto KataokaAtsunari Katsuki Nihon Univ. Toshiya Kazama Hiroshima Univ.James P. Keener Univ. Utah, USA Shu-ichi Kinoshita Meiji Univ.Hiroyuki Kitahata Chiba Univ. Ryo Kobayashi Hiroshima Univ.Tetsuya J. Kobayashi Univ. Tokyo Yasuaki Kobayashi FHI Berlin, DEMiyuki Koiso Kyushu Univ. Hiroshi Kokubu Kyoto Univ.Hiroshi Kori Ochanomizu Univ. Motoko Kotani Tohoku Univ.Hideo Kubo Tohoku Univ. Yu Kurokawa Tokyo Inst. Tech.Masataka Kuwamura Kobe Univ. Bendong Lou Tongji Univ., CNPhilip K. Maini Oxford Univ., UK Tsubasa Masui Meiji Univ.Hiroshi Matano Univ. Tokyo Masako Matsubara Ochanomizu Univ.Katsuhito Matsui Kyoto Univ. Konstantin Mischaikow State Univ.
New Jersey, USATakashi Miura Kyoto Univ. Tomoshige Miyaguchi Osaka City Univ.Tomoyuki Miyaji Kyoto Univ. Yasuhito Miyamoto Tokyo Inst. Tech.Syuji Miyazaki Kyoto Univ. Tsuyoshi Mizuguchi Osaka
Prefecture Univ.Fumito Mori Ochanomizu Univ. Yoshihisa Morita Ryukoku Univ.Yukio Nagahata Osaka Univ. Ken H. Nagai Ochanomizu Univ.Masaharu Nagayama Kanazawa Univ. Tatsuyuki Nakaki Hiroshima Univ.Hiroya Nakao Kyoto Univ. Ken-ichi Nakamura Univ. Elec.-Comm.Naoto Nakano Shibaura Inst. Tech. Shinji Nakaoka Univ. TokyoShunsaku Nii Kyushu Univ. Kei Nishi Hokkaido Univ.
Yasumasa Nishiura Hokkaido Univ. Hirofumi Notsu Meiji Univ.Toshiyuki Ogawa Osaka Univ. Toshiko Ogiwara Josai Univ.Isamu Ohnishi Hiroshima Univ. Hiroe Oka Ryukoku Univ.Hisashi Okamoto Kyoto Univ. Seiro Omata Kanazawa Univ.Yoshihito Oshita Okayama Univ. Tohru Ozawa Waseda Univ.Yuyu Peng Univ. California, Yoshitaka Saiki Hokkaido Univ.
Irvine, USAAsaki Saito Future Univ. Munetaka Saito Hokkaido Univ.
HakodateTakashi Sakajo Hokkaido Univ. Kunimochi Sakamoto Hiroshima Univ.Tatsunari Sakurai Chiba Univ. Bjorn Sandstede Brown Univ., USAYuzuru Sato Hokkaido Univ. Arnd Scheel Univ. Minnesota, USAYoshihiro Shibata Waseda Univ. Kenneth Showalter West Virginia Univ.,
USAKokichi Sugihara Meiji Univ. Hiroshi Suito Okayama Univ.Hiromasa Suzuki Shiga Univ. Kanako Suzuki Tohoku Univ.Takashi Suzuki Osaka Univ. Riichi Takahashi Genesis Resear.
Inst., IncTakeshi Takaishi Hiroshima Kokusai Masaharu Taniguchi Tokyo Inst. Tech.
Gakuin Univ.Mitsusuke Tarama Kyoto Univ. Takashi Teramoto Chitose Inst. Sci. Tech.Atsushi Tero PRESTO, JST Sadayoshi Toh Kyoto Univ.Akiyasu Tomoeda Meiji Univ. Ralf Tonjes Ochanomizu Univ.Yousuke Tonosaki Kyoto Univ. Yoshihiro Tonegawa Hokkaido Univ.Ichiro Tsuda Hokkaido Univ. Toru Tsujikawa Miyazaki Univ.Nariya Uchida Tohoku Univ. Kei-Ichi Ueda Kyoto Univ.Daishin Ueyama Meiji Univ. Takeo Ushijima Tokyo Univ. Sci.Tohru Wakasa Meiji University Michael J. Ward Univ. British
Columbia, CAJack Xin Univ. California, Masaaki Yadome Hokkaido Univ.
Irvine, USAKazuyuki Yagasaki Niigata Univ. Masahiro Yamaguchi Meiji Univ.Tomohiko Yamaguchi AIST Yutaka Yamaguti Hokkaido Univ.Eiji Yanagida Tokyo Inst. Tech. Tatsuo Yanagita Hokkaido Univ.Kensuke Yokoi Cardiff Univ., UK Natsuhiko Yoshinaga Kyoto Univ.Xiaohui Yuan NEIGAE,CAS, CN
In the traditional Japanese calender, there are two types of year cycle, ”十干 (jyukkann)” and”十二支 (jyunishi)”, which originates from ancient Chinese thoughts. Each year is named bya combination of 10 fortune-telling symbols and 12 animals. For example, both years of 1950and 2010 are named as ”庚寅 (kanoe-tora)”. 60 is the lowest common multiple of 10 and 12, sothe name of year returns to the first one after 60 years. Japanese people celebrate one’s 60thbirthday, ”還暦 (kanreki)”, as a history cycle.
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