s

Stability of switched systems

Existence of an attractor Stability of a closed orbit Stability of a switched equilibrium

Dwell-time

Lixian Zhang Harbin Institute of Technology, China

Reading:

T. Alpcan, T. Başar, A stability result for switched

systems with multiple equilibria. Dyn. Contin.

Discrete Impuls. Syst. Ser. A Math. Anal. 17

(2010), no. 6, 949–958. Phung

S. Mastellone, D. M. Stipanovic, M. W. Spong,

Stability and convergence for systems with

switching equilibria, 46th IEEE Conference on

Decision and Control 1-14 (2007) 4989-4996.

C. Perez, V. Azhmyakov, A. Poznyak, Practical

stabilization of a class of switched systems: dwell-

time approach. IMA J. Math. Control Inform. 32

(2015), no. 4, 689–702.

L. Zhang, S. Zhuang, R. D. Braatz; Switched

model predictive control of switched linear

systems: Feasibility, stability and robustness.

Automatica J. IFAC 67 (2016), 8–21.

Stability with respect

to multi-valued perturbations

Reading:

P. E. Kloeden, S. Siegmund, Bifurcations and continuous transitions of attractors

in autonomous and nonautonomous systems. Internat. J. Bifur. Chaos Appl. Sci.

Engrg. 15 (2005), no. 3, 743–762.

P. E. Kloeden, V. S. Kozyakin, The inflation of attractors and their discretization:

the autonomous case. Lakshmikantham's legacy: a tribute on his 75th birthday.

Nonlinear Anal. 40 (2000), no. 1-8, Ser. A: Theory Methods, 333–343.

G. Colombo, M. Fečkan, B. M. Garay, Multivalued perturbations of a saddle

dynamics. Differ. Equ. Dyn. Syst. 18 (2010), no. 1-2, 29–56.

Dither

perturbations

Maksim Arnold University of Texas at Dallas

Reading:

L. Iannelli, K. H. Johansson, U. T. Jonsson, F. Vasca,

Averaging of nonsmooth systems using dither, Automatica

42 (2006), no. 4, 669-676.

J. Piotrowski, Smoothing Dry Friction by Medium

Frequency Dither and Its Influence on Ride Dynamics of

Freight Wagons, in “Non-smooth Problems in Vehicle

Systems Dynamics”, Proceedings of the Euromech 500

Colloquium, 189-194.

M. Arnold, V. Zharnitsky, Pinball Dynamics: Unlimited

Energy Growth in Switching Hamiltonian Systems,

Communications in Mathematical Physics 338 (2015), no.

2, 501-521.

Stable convex

Combination

Reading:

P. Bolzem, W. Spinelli, Quadratic stabilization of a switched

affine system about a nonequilibrium point, Proceeding of the

2004 American Control Conference, June 30. July 2, 2004,

3890-3895.

Improving stability

Michael Posa (Massachusetts Institute of Technology, USA)

Edward Hooton (University of Texas at Dallas)

Reading:

J. Guckenheimer, A robust hybrid stabilization strategy for equilibria. IEEE Trans. Automat. Control 40 (1995), no. 2, 321–326.

A.S. Shiriaev, J. Perram, C. Canudas-de-Wit, Constructive tool for orbital stabilization of underactuated nonlinear systems: virtual constraints approach, IEEE

Trans. Automat. Control 50 (2005), no. 8, 1164–1176.

K. Pyragas, Control of chaos via extended delay feedback, Phys. Lett. A 206 (1995), no. 5-6, 323-330. E. Hooton

B. Fiedler, V. Flunkert, M. Georgi, Refuting the odd-number limitation of time-delayed feedback control. Physical Review Letters 98 (2007) 114101. E. Hooton

J.M. Gonçalves, Regions of stability for limit cycle oscillations in piecewise linear systems. IEEE Trans. Automat. Control 50 (2005), no. 11, 1877–1882.

I.R. Manchester, U. Mettin, F. Iida, Stable dynamic walking over uneven terrain. International Journal of Robotics Research 30 (2011), no. 3, 265-279.

V. Andrieu, B. Jayawardhana, L. Praly, On transverse exponential stability and its use in incremental stability, observer and synchronization. IEEE 52nd Annual

Conderence on Decision and Control (2013), 5915-5920.

M. Posa, M. Tobenkin, R. Tedrake, Stability Analysis and Control of Rigid-Body Systems with Impacts and Friction, IEEE Transactions on Automatic Control, doi:

10.1109/TAC.2015.2459151.

A. P. Dani, S.-J. Chung, S. Hutchinson, Observer design for stochastic nonlinear systems via contraction-based incremental stability. IEEE Trans. Automat. Control

60 (2015), no. 3, 700–714.

I. R. Manchester, J.-J. Slotine, Transverse contraction criteria for existence, stability, and robustness of a limit cycle, Systems Control Lett. 63 (2014) 32-38.

Chattering

Petri Piiroinen (National University of Ireland)

Harry Dankowicz (University of Illinois, USA)

Andrew Lamperski (University of Minnesota, USA)

Reading:

K.H. Johansson, A.E. Barabanov, K.J. Åström, Limit cycles with chattering in relay feedback systems. IEEE Trans. Automat. Control 47 (2002), no. 9, 1414–1423.

D.R.J. Chillingworth, Dynamics of an impact oscillator near a degenerate graze. Nonlinearity 23 (2010), no. 11, 2723–2748.

C. Budd, F. Dux, Chattering and related behavior in impact oscillators. Phil. Trans. Royal Soc. A 347 (1994), no. 1683, 365-389.

A. Nordmark, H. Dankowicz, A. Champneys, Friction-induced reverse chatter in rigid-body mechanisms with impacts. IMA J. Appl. Math. 76 (2011), no. 1, 85–119.

A.B. Nordmark, P.T. Piiroinen, Simulation and stability analysis of impacting systems with complete chattering. Non-linear Dynamics 58 (2009), no. 1-2, 85-106.

J. Zhang, K.H. Johansson, J. Lygeros, S. Sastry. Dynamical systems revisited: hybrid systems with Zeno executions. In Hybrid Systems: Computation and Control

(HSCC '00), Springer-Verlag, LNCS 1790, pp. 451-464, 2000.

M. Heymann, F. Lin, G. Meyer, S. Resmerita, Analysis of Zeno behaviors in a class of hybrid systems. IEEE Trans. Automat. Control 50 (2005), no. 3, 376–383.

A. Lamperski, A.D. Ames, Lyapunov theory for Zeno stability. IEEE Trans. Automat. Control 58 (2013), no. 1, 100–112.

Reduction to Zero Dynamics / Invariant manifolds

Arkadi Ponossov Norwegian University of Life Sciences

Jim Schmiedeler University of Notre Dame, USA

Reading:

A.D. Ames, K. Galloway, K. Sreenath, J.W. Grizzle, Rapidly exponentially stabilizing control Lyapunov functions and hybrid zero dynamics. IEEE

Trans. Automat. Control 59 (2014), no. 4, 876–891.

J.W. Grizzle, G. Abba, F. Plestan, Asymptotically stable walking for biped robots: analysis via systems with impulse effects. IEEE Trans. Automat.

Control 46 (2001), no. 1, 51–64.

E.R. Westervelt, J.W. Grizzle, D.E. Koditschek, Hybrid zero dynamics of planar biped walkers. IEEE Trans. Automat. Control 48 (2003) 42–56.

E. Litsyn, Y.V. Nepomnyashchikh, A. Ponosov, Stabilization of linear differential systems via hybrid feedback controls. SIAM J. Control Optim. 38

(2000), no. 5, 1468–1480.

A.E. Martin, D.C. Post, J.P. Schmiedeler, Design and experimental implementation of a hybrid zero dynamics-based controller for planar bipeds

with curved feet. International Journal of Robostics Research 33 (2014), no. 7, 988-1005.

K.A., Hamed N. Sadati, W.A. Gruver, Stabilization of Periodic Orbits for Planar Walking With Noninstantaneous Double-Support Phase. IEEE

Transactions on Systems Man and Cybernetics part A-Systems and and Humans 42 (2012), no. 3, 685-706.

I., Poulakakis J.W. Grizzle, The spring loaded inverted pendulum as the hybrid zero dynamics of an asymmetric hopper. IEEE Trans. Automat.

Control 54 (2009), no. 8, 1779–1793.

R. Szalai, H. M. Osinga, Invariant polygons in systems with grazing-sliding. Chaos 18 (2008), no. 2, 023121, 11 pp.

D. Weiss; T. Küpper; H. A. Hosham, Invariant manifolds for nonsmooth systems. Phys. D 241 (2012), no. 22, 1895–1902.

E. Litsyn, Y. Nepomnyashchikh, A.Ponosov. Classification of linear dynamical systems in the plane admitting a stabilizing hybrid feedback control.

Journal on Dynamical and Control Systems, v. 6, no. 4 (2000), pp. 477-501.

Computing/Designing

the Poincare map

Luis Aguilar Instituto Politécnico Nacional Mexico

Reading:

L.T. Aguilar, I.M. Boiko, L.M. Fridman, L.B. Freidovich,. Generating oscillations in

inertia wheel pendulum via two-relay controller. Internat. J. Robust Nonlinear

Control 22 (2012), no. 3, 318–330.

C. Lin, Wang, Q.G. Lee, H. Tong. Local stability of limit cycles for MIMO relay

feedback systems. J. Math. Anal. Appl. 288 (2003), no. 1, 112–123.

K.J. Åström, Oscillations in systems with relay feedback. Adaptive control,

filtering, and signal processing. IMA Vol. Math. Appl. 74, Springer, New York,

1995.

M.J. Coleman, A. Chatterjee, A. Ruina, Motions of a rimless spoked wheel: a

simple three-dimensional system with impacts. Dynam. Stability Systems 12

(1997), no. 3, 139–159.

J.M. Gonçalves, A. Megretski, M.A. Dahleh, Global analysis of piecewise linear

systems using impact maps and surface Lyapunov functions. IEEE Trans.

Automat. Control 48 (2003), no. 12, 2089–2106.

Stick-slip

oscillations

Reading:

N. Begun, S. Kryzhevich, One-dimensional chaos in a system

with dry friction: analytical approach. Meccanica 50 (2015), no.

8, 1935–1948.

E.I. Butikov, Spring pendulum with dry and viscous damping.

Communications in Nonlinear Science and Numerical

Simulation 20 (2015), no. 1, 298-315.

Q. Li, Y. Chen, Z. Qin, Existence of Stick-Slip Periodic

Solutions in a Dry Friction Oscillator. Chinese Physics Letters

28 (2011), no. 3, 030502.

M.R. Jeffrey, Hidden dynamics in models of discontinuity and

switching. Phys. D 273/274 (2014) 34-45.

G.Licskó, G. Csernák, On the chaotic behaviour of a simple

dry-friction oscillator. Math. Comput. Simulation 95 (2014), 55–

62.

Pontryagin maximum principle

Nonlinear switching manifolds

Carolina Biolo

SISSA, Italy Reading:

P. Mason, M. Broucke, B. Piccoli, Time optimal swing-up of the planar pendulum. IEEE Trans. Automat.

Control 53 (2008), no. 8, 1876–1886.

S.A. Reshmin, F.L. Chernousko, Properties of the time-optimal feedback control for a pendulum-like system. J.

Optim. Theory Appl. 163 (2014), no. 1, 230–252.

Y. Horen, B.Z. Kaplan, Improved switching mode oscillators employing generalized switching lines.

International Journal of Circuit Theory and Applications 28 (2000), no. 1, 51-67.Sinha

C. Biolo, A. Agrachev, Switching in time-optimal problem the 3-D case with 2-D control, preprint.

Earthquake

fault

Reading:

J.M. Carlson, J.S. Langer, Mechanical model of an

earthquake fault. Phys. Rev. A (3) 40 (1989), no.

11, 6470–6484.

J. Nussbaum, A. Ruina, A two degree-of-freedom

earthquake model with static/dynamic friction, Pure

and Applied Geophysics 125 (1987), no. 4, 629-

656.

Neuroscience

Wilten Nicola Imperial College London, UK

Kyle Wedgwood University of Nottingham, UK

Reading:

A. Tonnelier, The McKean's caricature of the FitzHugh-Nagumo model. I. The

space-clamped system. SIAM J. Appl. Math. 63 (2002), no. 2, 459–484.

E. Shlizerman, P. Holmes, Neural dynamics, bifurcations, and firing rates in a

quadratic integrate-and-fire model with a recovery variable. I: Deterministic

behavior. Neural Comput. 24 (2012), no. 8, 2078–2118.

J.P. Keener, F.C. Hoppensteadt, J. Rinzel, Integrate-and-fire models of nerve

membrane response to oscillatory input. SIAM J. Appl. Math. 41 (1981), no. 3,

503–517.

A.A. Khajeh, Mode locking in a periodically forced resonate-and-fire neuron

model. E, Statistical, nonlinear, and soft matter physics 80 (2009), no. 5,

051922.

S. Coombes, R. Thul, K.C.A. Wedgwood, Non-smooth dynamics in spiking

neuron models. Phys. D 241 (2012), no. 22, 2042–2057.

W. Nicola, S.A. Campbell, Non-smooth Bifurcations of Mean Field Systems of

Two-Dimensional Integrate and Fire Neurons.

D. Zhou, Y. Sun, A.V. Rangan, D. Cai, Spectrum of Lyapunov exponents of

non-smooth dynamical systems of integrate-and-fire type. J. Comput.

Neurosci. 28 (2010), no. 2, 229–245.

Microscopy

Xiaopeng Zhao University of Tennessee

USA

Reading:

S. Misra, H. Dankowicz, M.R. Paul, Degenerate discontinuity-

induced bifurcations in tapping-mode atomic-force

microscopy. Phys. D 239 (2010), no. 1-2, 33–43.

H. Dankowicz, X. Zhao, S. Misra, Near-grazing dynamics in

tapping-mode atomic-force microscopy. International Jornal of

non-linear Mechanics 42 (2007), no. 4, 697-709.

Population dynamics

Irakli Loladze Arizona State University, USA

Amit Bhaya Universidade Federal do Rio de Janeiro, Brazil

Reading:

S. Rodrigues, J. Gonçalves, J.R. Terry, Existence and stability of limit cycles in a macroscopic

neuronal population model. Phys. D 233 (2007), no. 1, 39–65.

H. Wang, Y. Kuang, I. Loladze, Dynamics of a mechanistically derived stoichiometric producer-

grazer model. J. Biol. Dyn. 2 (2008), no. 3, 286–296.

M. Mendoza, E. Magno, A. Bhaya, Realistic threshold policy with hysteresis to control predator-

prey continuous dynamics, Theory in Biosciences 128 (2009), no. 2, 139-149.

M.E.M. Meza, A. Bhaya, E. Kaszkurewicz, M. I. da Silveira Costa, On-off policy and hysteresis

on-off policy control of the herbivore-vegetation dynamics in a semi-arid grazing system.

Ecological Engineering 28 (2006), no. 2, 114-123.

F. Dercole, S. Maggi, Detection and continuation of a border collision bifurcation in a forest fire

model. Appl. Math. Comput. 168 (2005), no. 1, 623–635.

Materials

science

Reading:

V.A. Kovtunenko, K. Kunisch, W. Ring, Propagation

and bifurcation of cracks based on implicit surfaces

and discontinuous velocities. Comput. Vis. Sci. 12

(2009), no. 8, 397–408.

C. R. Farrar, K. Worden, M. D. Todd, G. Park, J.

Nichols, D. E. Adams, M. T. Bement, K. Farinholt,

Nonlinear System Identification for Damage

Detection, LA-14353 report.

Cruise

Control

Reading:

R.A. DeCarlo, M.S. Branicky, S.

Pettersson, Perspectives and results on

the stability and stabilizability of hybrid

systems. Proceedings of the IEEE, 88

(2000), no. 7, 1069-1082.

A. Jacquemard, M.A. Teixeira, Periodic

solutions of a class of non-autonomous

second order differential equations with

discontinuous right-hand side. Phys. D

241 (2012), no. 22, 2003–2009.

Orthogonal

cutting

Zoltan

Dombovari Budapest University of Technology

and Economics, Hungary

Reading:

Z. Diombovar, A.W.B. David, R.E. Wilson, S.

Gabor, On the global dynamics of chatter in the

orthogonal cutting model, International Journal

of Non-Linear Mechanics 46 (2011) 330–338.

Cardiac

alternans

Alena

Talkachova University of Minnesota, USA

Reading:

M.A. Hassouneh, E.H. Abed, Border Collision Bifurcation

Control of Cardiac Alternans, Proc. American Control

Conference, Denver, Colorado June 4-6.2003, 459-464.

E.G. Tolkacheva, X. Zhao, Nonlinear dynamics of

periodically paced cardiac tissue. Nonlinear dynamics 68

(2012), no. 3, 347-363.

Pressure

relief

valve

Reading:

C. Bazsó, A.R. Champneys, C.J. Hös, Bifurcation

Analysis of a Simplified Model of a Pressure Relief

Valve Attached to a Pipe, SIAM J. Appl. Dyn. Syst. 13

(2014), no. 2, 704–721.

Drillstring

dynamics

Reading:

R.I. Leine, D.H. van Campen, Stick-slip whirl interaction in

drillstring dynamics. Journal of Vibration and Acoustics-

Transactions 124 (2002), no. 2, 209-220.

B. Besselink, N. van de Wouw, H. Nijmeijer, A Semi-Analytical

Study of Stick-Slip Oscillations in Drilling Systems.

Computational and nonlinear dynamics 6 (2011), no. 2, 021006

Q.J.Cao, M. Wiercigroch, E. Pavlovskaia, S.P. Yang,

Bifurcations and the penetrating rate analysis of a model for

percussive drilling. Acta Mech. Sin. 26 (2010), no. 3, 467–475.

School/Workshop on Applicable Theory of Switched Systems June 6-10, 2016

diagram of Topics Speakers Reading

Venue: UT Dallas, USA Information: www.utdallas.edu/sw16 , Organizer: Oleg Makarenkov

Common/Multiple

Lyapunov functions

Sue Ann Campbell University of Waterloo, Canada

Reading:

M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and

hybrid systems. IEEE Trans. Automat. Control 43 (1998), no. 4, 475–482.

A.A. Agrachev, D.Liberzon, Lie-algebraic stability criteria for switched systems.

SIAM J. Control Optim. 40 (2001), no. 1, 253–269.

R. Shorten, F. Wirth, O. Mason, K. Wulff, C. King, Stability criteria for switched and

hybrid systems. SIAM Rev. 49 (2007), no. 4, 545–592.

S. Kim, S. A. Campbell, X. Liu, Stability of a class of linear switching systems with

time delay. IEEE Trans. Circuits Syst. I Regul. Pap. 53 (2006), no. 2, 384-393.

Nonsmooth

Lyapunov

functions

Reading:

S.P. Bhat, D.S. Bernstein, Finite-time stability of continuous autonomous systems. SIAM J. Control

Optim. 38 (2000), no. 3, 751–766.

S.P. Bhat, D.S. Bernstein, Continuous finite-time stabilization of the translational and rotational double

integrators. IEEE Trans. Automat. Control 43 (1998), no. 5, 678–682.

E. Moulay, W. Perruquetti, Finite time stability of differential inclusions. IMA J. Math. Control Inform. 22

(2005), no. 4, 465–475.

R. Santiesteban, T. Floquet, Y. Orlov, S. Riachy, J.P. Richard, Second-order sliding mode control of

underactuated mechanical systems. II. Orbital stabilization of an inverted pendulum with application to

swing up/balancing control. J. Robust Nonlinear Control 18 (2008) 544–556.

R. Santiesteban, L. Fridman, J.A. Moreno, Finite-time convergence analysis for “Twisting” controller via

a strict Lyapunov function.

A. Polyakov, A. Poznyak, Unified Lyapunov function for a finite-time stability analysis of relay second-

order sliding mode control systems. IMA J. Math. Control Inform. 29 (2012), no. 4, 529–550.

J.A. Moreno, M. Osorio, Strict Lyapunov functions for the super-twisting algorithm. IEEE Trans.

Automat. Control 57 (2012) 1035–1040.

D. Shevitz, B. Paden, Lyapunov stability theory of nonsmooth systems. IEEE Trans. Automat. Control

39 (1994), no. 9, 1910–1914.

A. Levant, Principles of 2-sliding mode design. Automatica J. IFAC 43 (2007), no. 4, 576–586.

Maps

Xiaopeng Zhao University of Tennessee, USA

Francisco Torres

University of Seville, Spain

Reading:

L. Benadero, E. Freire, E. Ponce, F. Torres, Resonances in an area preserving continuous

piecewise linear map, slides at NPDDS 2014.

H.E. Nusse, E. Ott, J.A. Yorke, Border-collision bifurcations: an explanation for observed bifurcation

phenomena. Phys. Rev. E (3) 49 (1994), no. 2, 1073–1076.

V. Avrutin, P.S. Dutta, M. Schanz, S. Banerjee, Influence of a square-root singularity on the

behaviour of piecewise smooth maps. Nonlinearity 23 (2010), no. 2, 445–463.

P.Glendinning, C.H. Wong, Border collision bifurcations, snap-back repellers, and chaos. Phys.

Rev. E (3) 79 (2009), no. 2, 025202, 4 pp

L. Gardini, F. Tramontana, Snap-back repellers in non-smooth functions. Regul. Chaotic Dyn. 15

(2010), no. 2-3, 237–245.

I. Sushko, A. Agliari, L. Gardini, Bistability and border-collision bifurcations for a family of unimodal

piecewise smooth maps. Discrete Contin. Dyn. Syst. Ser. B 5 (2005), no. 3, 881–897.

P. Kowalczyk, Robust chaos and border-collision bifurcations in non-invertible piecewise-linear

maps. Nonlinearity 18 (2005), no. 2, 485–504.

X. Zhao, Discontinuity mapping for near-grazing dynamics in vibro-impact oscillators, R.A. Ibrahim

(Ed.), et al., Vibro-Impact Dynamics of Ocean Systems and Related Problems (2009) 275–285.

Hysteresis

Dmitrii Rachinskii (University of Texas at Dallas, USA)

Nikita Begun (Free University of Berlin, Germany)

Dinesh Ekanayake (Western Illinois University, USA)

Tamas Kalmar-Nagy (Budapest University of Technology and Economics, Hungary)

Reading:

S. McCarthy, D. Rachinskii, Dynamics of systems with Preisach memory near equilibria. Math. Bohem. 139 (2014), no. 1, 39–73.

T. Kalmár-Nagy, P. Wahi, A. Halder, Dynamics of a hysteretic relay oscillator with periodic forcing. SIAM J. Appl. Dyn. Syst. 10 (2011), no. 2, 403–422.

T. Kalmar-Nagy, R. Csikja, T. A. Elgohary, Nonlinear analysis of a 2-DOF piecewise linear aeroelastic system, Nonlinear Dynamics, 2016, online first..

D. B. Ekanayake, R. V. Iyer, Proportional Derivative Control of Hysteretic Systems. SIAM Journal on Control and Optimization 51, no. 5, 3415–3433.

M. Zeitz, P. Gurevich, H. Stark, Feedback control of flow vorticity at low Reynolds numbers. European Physical Journal E 38 (2015), no. 3, 22.

output

input

Extending the concept

of the derivative

Extending

beyond Fillippov’s concept

Reading:

S.Adly, D. Goeleven, A stability theory for second-order nonsmooth dynamical systems

with application to friction problems. J. Math. Pures Appl. (9) 83 (2004) 17–51..

R.I. Leine, T.F. Heimsch, Global uniform symptotic attractive stability of the non-

autonomous bouncing ball system. Phys. D 241 (2012), no. 22, 2029–2041.

D.E. Stewart, Rigid-body dynamics with friction and impact. SIAM Rev. 42 (2000) 3–39.

R. Dzonou, M.D.P. Monteiro Marques, L. Paoli, A convergence result for a vibro-impact

problem with a general inertia operator. Nonlinear Dynam. 58 (2009) 361–384.

L. Han, J. Pang, Non-Zenoness of a class of differential quasi-variational inequalities.

Math. Program. 121 (2010), no. 1, Ser. A, 171–199.

J. Bastien, F. Bernardin, C.-H. Lamarque, Non Smooth Deterministic or Stochastic

Discrete Dynamical Systems: Applications to Models with Friction or Impact, Wiley

2013, 512 pp..

B. Brogliato, Nonsmooth mechanics, Springer, 2016, 629 pp.

Robot locomotion

Yildirim Hurmuzlu (Southern Methodist University, USA)

Jae-Sung Moon (UNIST University, Korea)

Andrew Lamperski (University of Minnesota, USA)

Mark Spong (University of Texas at Dallas, USA)

Robert Gregg (University of Texas at Dallas, USA)

Safya Belghith (National Engineering School of Tunis)

Hamid Reza Fahham (Marvasht Islamic Azad University, Iran)

Reading:

E.A. Yazdi, A. Alasty, Stabilization of Biped Walking Robot Using the Energy Shaping Method. Journal of Computational and Nonlinear Dynamics 3 (2008), no. 4, 041013.

M.W. Spong, G. Bhatia, Further results on control of the compass gait biped. Proceeding of the 2003 IEEE/RSJ International Conference on Intelligent Robots and Systems, vols 1-4(2003), 1933-1938.

H.R. Fahham, M. Farid, Minimum-time trajectory planning of spatial cable-suspended robots along a specified path considering both tension and velocity constraints. Eng. Optim. 42 (2010), no. 4, 387–402.

P.T. Piiroinen, H.J. Dankowicz, A.B. Nordmark, On a normal-form analysis for a class of passive bipedal walkers. International Journal of Bifurcation and Chaos 11 (2001), no. 9, 2411-2425.

S. Aoi, K. Tsuchiya, Stability analysis of a simple walking model driven by a nonlinear oscillator. IEEE International Conference on Systems. Man & Cybernetics, vols 1-7 (2004), 4450-4455.

P.L. Varkonyi, D. Gontier, J.W. Burdick, On the Lyapunov stability of quasistatic planar biped robots. IEEE International Conference on Robotics and Automation (2012), 63-70.

S.L. Das, A. Chatterjee, An alternative stability analysis technique for the simplest walker. Non linear Dynamics 28 (2002), no. 3-4, 273-284.

F. Asano, Stability analysis of underactuated compass gait based on linearization of motion. Multibody Syst. Dyn. 33 (2015), no. 1, 93–111.

J.A. Norris, A.P. Marsh, K.P. Granata, S.D. Ross, Revisiting the stability of 2D passive biped walking: local behavior. Phys. D 237 (2008), no. 23, 3038–3045.

L.M. Liu, Y.T. Tian, P.J. Zhang, An Analysis of Stability of Systems with Impulse Effects: Application to Biped Robots. IEEE Conference on Robotics, Automation and Mechatronics vols 1-2, (2008), 517-522.

L.B. Freidovich, W. Mettin, A.S. Shiriaev, A Passive 2-DOF Walker: Hunting for Gaits Using Virtual Holonomic Constraints. IEEE Transactions on Robotics 25 (2009), no. 5, 1202-1208.;

D. Efimov, W. Perruquetti, A. Shiriaev, On existence of oscillations in hybrid systems. Nonlinear Anal. Hybrid Syst. 12 (2014), 104–116.

A. Goswami, B. Espiau, A. Keramane, Limit cycles in a passive compass gait biped and passivity-mimicking control laws. Autonomous Robots 4 (1997), no. 3, 273-286.

Y. Hurmuzlu, G.D. Moskowitza, The role of impact in the stability of bipedal locomotion, Dynamics and Stability of Systems 1 (1986), no. 3, 217-234.

I.R. Manchester, U. Mettin, F. Iida, Stable dynamic walking over uneven terrain. International Journal of Robotics Research 30 (2011), no. 3, 265-279.

M. Wisse, A.L. Schwab, R.Q. van der Linde, How to keep from falling forward: Elementary swing leg action for passive dynamic walkers. IEEE Transactions on Robotics 21 (2005), no. 3, 393-401.

D.G.E. Hobbelen, M. Wisse, A disturbance rejection measure for limit cycle walkers: The Gait Sensitivity Norm. IEEE Transactions on Robotics 23 (2007), no. 6, 1213-1224.

H. Gritli, N. Khraief, S. Belghith, Chaos control in passive walking dynamics of a compass-gait model. Commun. Nonlinear Sci. Numer. Simul. 18 (2013), no. 8, 2048–2065.

J. Moon, M.W. Spong, Bifurcations and Chaos in Passive Walking of a Compass-Gait Biped with Asymmetries. IEEE International Conference on Robotics and Automation Book Series (2010), 1721-1726.

M.W. Spong, J.K. Holm, D. Lee, Passivity-based control of bipedal locomotion - Regulating walking by exploiting passive gaits in 2-D and 3-D bipeds. IEEE Robotics & Automation Magazine 14 (2007), no. 2, 30-40.

R.R. Burridge,A.A. Rizzi, D.E. Koditschek, Sequential composition of dynamically dexterous robot behaviors. International Journal of Ronorics Research 18 (1999) , no. 6, 534-555.

A. Lamperski, A.D. Ames, Lyapunov theory for Zeno stability. IEEE Trans. Automat. Control 58 (2013), no. 1, 100–112.

R.D. Gregg, A.K. Tilton, S. Candido, Control and Planning of 3-D Dynamic Walking With Asymptotically Stable Gait Primitives. IEEE Transactions on Robotics 28 (2012), no. 6, 1415-1423.

M. Garcia, A. Chatterjee, A. Ruina, The simplest walking model: Stability, complexity, and scaling. Journal of Biomechanical Engineering-Transactions of the ASME 120 (1998), no. 2, 281-288.

H. Park, K. Sreenath, A. Ramezani, Switching Control Design for Accommodating Large Step-down Disturbances in Bipedal Robot Walking. IEEE International Conference on Robotics and Automation (2012), 45-50.

Braking systems

Reading:

H. Jing, Z. Liu, H. Chen, A Switched Control Strategy for Antilock

Braking System With On/Off Valves. IEEE Transactions on

Vehicular Technology, 60 (2011), no. 4, 1470-1484.

C.F. Lee,C. Manzie, Near-time-optimal tracking controller design

for an automotive electromechanical brake. Institution of

Mechanical Engineers, 226 (2012), no. I4, 537-549.

E. de Bruijn, M. Gerard, M. Corno, On the performance increase

of wheel deceleration control through force sensing. 2010 IEEE

Multi-Conference on Systems and Control, IEEE International

Conference on Control Applications, (2010), 161-166.

M. Corno, M. Gerard, M. Verhaegen, Hybrid ABS Control Using

Force Measurement. IEEE Transactions on Control Systems

Technology 20 (2012), no. 5, 1223-1235.

E. Dincmen, B.A. Guvenc, T. Acarman, Extremum-Seeking

Control of ABS Braking in Road Vehicles With Lateral Force

Improvement. IEEE Transactions on Control Systems

Technology, 22 (2014), no. 1, 230-237.

S. Drakunov, U. Ozguner, P. Dix, ABS Control using optimum

search via sliding modes. IEEE Transactions on Control Systems

Technology, 3 (1995), no. 1, 79-85.

T.A. Johansen, I. Petersen, J. Kalkkuhl, Gain-scheduled wheel

slip control in automotive brake systems. IEEE Transactions on

Control Systems Technology, 11 (2003), no. 6, 799-811.

B.J. Olson, S.W. Shaw, G. Stépán, Stability and bifurcation of

longitudinal vehicle braking. Nonlinear Dynam. 40 (2005), no. 4,

339–365.

W. Pasillas, Hybrid modeling and limit cycle analysis for a class of

five-phase anti-lock brake algorithms. 7th International

Symposium on Advanced Vehicle Control, Vehicle System

Dynamics, 44 (2006), no. 2, 173-188.

M. Tanelli, G. Osorio, M. di Bernardo, S.M. Savaresi, A. Astolfi,

Existence, stability and robustness analysis of limit cycles in

hybrid anti-lock braking systems. Internat. J. Control 82 (2009),

no. 4, 659–678.

Internet

Protocol Priya

Ranjan Amity University, India

Reading:

A. Mukhopadhyay, P. Ranjan, Nonlinear Instabilities in

D2TCP-II, arXiv:1212.6907.

P. Ranjan, E.H. Abed, R.J. La, Nonlinear instabilities in TCP-

RED. IEEE-ACM Transactions on Networking 12 (2004), no.

6, 1079-1092.

Fuel

consumption

Reading:

X. Wei, L. Guzzella, V.I. Utkin,

Model-based fuel optimal control of

hybrid electric vehicle using variable

structure control systems. Journal of

Dynamic Systems Measurement and

Control 129 (2007), no. 1 13-19.

M.G. Wu, A.V. Sadovsky, Minimum-

Cost Aircraft Descent Trajectories

with a Constrained Altitude Profile,

NASA/TM-2015-218734 report

Networks (consensus, scheduling, etc)

Alexander Sadovsky (NASA Ames Research Center, USA)

Qing Hui (University of Nebraska – Lincoln, USA)

Nicholas Gans (University of Texas at Dallas, USA)

Reading:

J. Gebert, N. Radde, G.W. Weber, Modeling gene regulatory networks with piecewise linear differential equations. European J. Oper.

Res. 181 (2007), no. 3, 1148–1165.

R. Edwards, S. Kim, P. van den Driessche, Control design for sustained oscillation in a two-gene regulatory network. J. Math. Biol. 62

(2011), no. 4, 453–478.

H. de Jong, J. Geiselmann, C. Hernandez, Genetic Network Analyzer: qualitative simulation of genetic regulatory networks.

Bioinfomatics 19 (2013), no. 3, 336-344.

H. de Jong, J. Geiselmann, G. Batt, C. Hernandez, M. Page, Qualitative simulation of the initiation of sporulation in Bacillus subtilis.

Bull. Math. Biol. 66 (2004), no. 2, 261–299.

K. Aihara, H. Suzuki, Theory of hybrid dynamical systems and its applications to biological and medical systems. Philoshophical

Transactions of The Royal Society A-Mathematical Physical and Engineering Sciences 368 (2010), no. 1930, 4893-4914.

H. Suzuki, J. Imura, Y. Horio, K. Aihara: Chaotic Boltzmann machines, Scientific Reports 3, 1610 (2013).

M. Forti, P. Nistri, Global convergence of neural networks with discontinuous neuron activations. IEEE Trans. Circuits Systems I Fund.

Theory Appl. 50 (2003), no. 11, 1421–1435.

W.L. Lu, T.P. Chen, Dynamical behaviors of Cohen-Grossberg neural networks with discontinuous activation functions. Neural

Networks 18 (2005), no. 3, 231-242.

J. Wang, L. Huang, Z. Guo, Global asymptotic stability of neural networks with discontinuous activations, Neural Networks 22 (2009)

931-937.

A. Machina, R. Edwards, P. van den Driessche, Singular dynamics in gene network models. SIAM J. Appl. Dyn. Syst. 12 (2013), no. 1,

95–125.

A.V. Sadovsky, D. Davis, D.R. Isaacson, Efficient Computation of Separation-Compliant Speed Advisories for Air Traffic Arriving in

Terminal Airspace. Journal of Dynamic Systems Measurement and Control-Transactions of The ASME 136 (2014), no. 4, 041027.

Q. Hui, W.M. Haddad, P.S. Bhat, Semi-stability theory for differential inclusions with applications to consensus problems in dynamical

networks with switching topology. American Control Conference 2008, VOLS 1-12 (2008), 3981-3986.

C. Qian, W. Lin, A continuous feedback approach to global strong stabilization of nonlinear systems. IEEE Trans. Automat. Control 46

(2001), no. 7, 1061–1079.

H. Sayyaadi, M.R. Doostmohammadian, Finite-time consensus in directed switching network topologies and time-delayed

communications. Scientia Iranica 18 (2011), no. 1, 75-85.

Y. Kim, S. Wee, N. Gans, Decentralized cooperative mean approach to collision avoidance for nonholonomic mobile robots. 2015

IEEE International Conference on Robotics and Automation (ICRA), (2015), 35-41.

Y. Kim, S. Wee, N. Gans, Consensus based attractive vector approach for formation control of nonholonomic mobile robots. 2015

IEEE International Conference on Advanced Intelligent Mechatronics (AIM), (2015),977 - 983.

Switching therapy/mutation rate

Gouhei Tanaka (The University of Tokyo, Japan)

Cynthia Sanchez Tapia (University of California, USA)

Reading:

A. Wang, Y. Xiao, R.A. Cheke, Global dynamics of a piece-wise epidemic model with switching vaccination strategy. Discrete

Contin. Dyn. Syst. Ser. B 19 (2014), no. 9, 2915–2940.

G. Tanaka, K. Tsumoto, S.Tsuji, K. Aihara, Bifurcation analysis on a hybrid systems model of intermittent hormonal therapy for

prostate cancer. Phys. D 237 (2008), no. 20, 2616–2627.

C. Sanchez Tapia, F.Y.M. Wan, Fastest time to cancer by loss of tumor suppressor genes. Bull. Math. Biol. 76 (2014), no. 11,

2737–2784.

F.Y.M. Wan, A.V. Sadovsky, N.L. Komarova, Genetic instability in cancer: an optimal control problem. Stud. Appl. Math. 125

(2010), no. 1, 1–38.

N.L. Komarova, V.A. Sadovsky, Y.M.F. Wan, Selective pressures for and against genetic instability in cancer: a time-dependent

problem. Journal of the royal society interface 5(2008), no. 18, 105-121.

Robot manipulator

Mark Spong University of Texas at Dallas, USA

Reading:

R. Santiesteban, Time Convergence Estimation of a Perturbed Double Integrator: Family of Continuous

Sliding Mode Based Output Feedback Synthesis. European Control Conference (2013), 3764-3769.

Q. Wei, W.P. Dayawansa, W.S. Levine, Nonlinear controller for an inverted pendulum having restricted travel.

Automatica J. IFAC 31 (1995), no. 6, 841–850.

J. Zhao, W.M. Spong, Hybrid control for global stabilization of the cart-pendulum system. Automatica J. IFAC

37 (2001), no. 12, 1941–1951.

P. Mason, M. Broucke, B. Piccoli, Time optimal swing-up of the planar pendulum. IEEE Trans. Automat.

Control 53 (2008), no. 8, 1876–1886.

B.E. Paden, S.S. Sastry, A calculus for computing Filippov's differential inclusion with application to the

variable structure control of robot manipulators. IEEE Trans. Circuits and Systems 34 (1987), no. 1, 73–82.

Power converters

Enrique Ponce (University of Seville, Spain) Valentina Sessa (Rio de Janeiro State University, Brazil)

Amit Patra (Indian Institute of Technology Kharagpur) Bengt Lennartson (Chalmers University of Technology, Sweden)

Reading:

Y. Lu, X. Huang, B. Zhang, Hybrid Feedback Switching Control in a Buck Converter. IEEE International Conference on Automation and Logistics, vol. 1-6 (2008), 207-210.

S.K. Mazumder, K. Acharya, Multiple Lyapunov function based reaching condition for orbital existence of switching power converters. IEEE Transactions of Power Electronics 23

(2008), no. 3, 1449-1471.

T. Hu, A Nonlinear-System Approach to Analysis and Design of Power-Electronic Converters With Saturation and Bilinear Terms. IEEE Transactions on Power Electronics 26

(2011), no. 2, 399-410.

V. Stramosk, L. Benadero, D.J. Pagano, E. Ponce, Sliding Mode Control of Interconnected Power Electronic Converters in DC Microgrids. 39th Annual Conference of the IEEE

Industrial-Electronics-Society, IEEE Industrial Electronics Society (2013), 8385-8390.

V. Sessa, L. Iannelli, F. Vasca, A Complementarity Model for Closed-Loop Power Converters. IEEE Transactions on Power Electronics 29 (2014), no. 12, 6821-6835.

C. Fang, E.H. Abed, Robust feedback stabilization of limit cycles in PWM DC-DC converters. Nonlinear Dynam. 27 (2002), no. 3, 295–309.

A. Patra, S. Banerjee, A Current-Controlled Tristate Boost Converter With Improved Performance Through RHP Zero Elimination. IEEE Transactions on Power Electronics 24

(2009), no. 3, 776 - 786.

C. Sreekumar, V. Agarwal, A hybrid control algorithm for voltage regulation in dc-dc boost converter. IEEE Transactions on Industrial Electronics 55 (2008), no. 6, 2530-2538.

V. Utkin, Sliding Mode Control of DC/DC Multiphase Power Converters. 13TH International Power Electronics and Motion Control Conference, vol 1-5 (2008), 512-514.

S. Almér, U. Jönsson, C. Kao, J. Mari, Stability analysis of a class of PWM systems. IEEE Trans. Automat. Control 52 (2007), no. 6, 1072–1078.

S.Almér, U.T. Jönsson, Dynamic phasor analysis of pulse-modulated systems. SIAM J. Control Optim. 50 (2012), no. 3, 1110–1138.

M. Rubensson, B. Lennartson, Global convergence analysis for piecewise linear systems applied to limit cycles in a DC/DC converter. Proceedings of The 2002 American Control

Conference, vols 1-6 (2002), 1272-1277.

W. Xiao, B. Zhang, D. Qiu, Dongyan Control strategy based on discrete-time Lyapunov theory for DC-DC converters. 33rd Annual Conference of The IEEE Industrial Electronics

Society vols 1-3 (2007), 1501-1505.

A. Schild, J. Lunze, J. Krupar, Design of Generalized Hysteresis Controllers for DC-DC Switching Power Converters. IEEE Transactions on Power Electronics 24 (2009), no. 1-2,

138-146.

T. Saito, H. Torikai, W. Schwarz, Switched dynamical systems with double periodic inputs: An analysis tool and its application to the buck-boost converter. IEEE Transactions on

Circuits and Systems I-Fundamental Theory and Applications 47 (2000), no. 7, 1038-1046.

I.A. Hiskens, J.W. Park, V. Donde, Dynamic embedded optimization and shooting methods for power system performance assessment. Applied Mathematics For Restructured

Electric Power Systems: Optimization, Control and Computational Intelligence (2005), 179-199.

C.K. Tse, Y.M. Lai, H.H.C. Iu, Hopf bifurcation and chaos in a free-running current-controlled Cuk switching regulator. IEEE Transactions on Circuits and Systems I-Fundamental

Theory and Applications 47 (2000), no. 4, 448-457.

S. Banerjee, P. Ranjan, C. Grebogi, Bifurcations in two-dimensional piecewise smooth maps - Theory and applications in switching circuits. IEEE Transactions on Circuits and

Systems I-Regular Papers 47 (2000), on.5, 633-643.

Hard ball gas

Reading:

D. Turaev, V. Rom-Kedar, Elliptic islands appearing in

near-ergodic flows. Nonlinearity 11 (1998), no. 3, 575–

600.

A. Kaplan, N. Friedman, M. Andersen, Observation of

islands of stability in soft wall atom-optics billiards.

Physical Reciew Letters 87 (2001), no. 27, 274101.

N. Chernov, A. Korepanov, N. Simányi, Stable regimes

for hard disks in a channel with twisting walls. Chaos

22 (2012), no. 2, 026105, 13 pp.

Climate change

Kaitlin Hill Northwestern University, USA

Esther Widiasih University of Hawaii, USA

Reading:

J. Walsh, E. Widiasih, J. Hahn, R. McGehee, Periodic Orbits for a Discontinuous

Vector Field Arising from a Conceptual Model of Glacial Cycles. (2015)

S.F. Abe-Ouchi, K.Kawamura, Insolation-driven 100,000-year glacial cycles and

hysteresis of ice-sheet volume. Nature 500 (2013), no. 7461, 190-+.

D. Paillard, F. Parrenin, The Antarctic ice sheet and the triggering of deglaciations.

Earth and Planetary Science Letters 227 (2004), no. 3-4, 263-271

P. Welander, A Simple heat salt oscillator, Dynamics of Atomspheres and Oceans

6 (1982), no. 4, 233-242.

K. Hill, D.S. Abbot, M. Silber, Analysis of an Arctic sea ice loss model in the limit of

a discontinuous albedo, arXiv:1509.00059.

Nonlinear pendulum

Alexander Ivanov (Moscow Institute of Physics and Technology)

Tassilo Kuepper (University of Cologne, Germany)

Petri Piiroinen (National University of Ireland)

Michele Bonnin (Politecnico di Torino, Italy)

Reading:

A. Belendez, C. Pascual, D.I. Mendez, T. Belendez, C. Neipp, Exact solution for the nonlinear pendulum, Rev. Bras. Ensino Fís. 29 (2007), 645-648.

M. Sabatini, On the period function of x″+f(x)x' 2+g(x)=0, J. Differential Equations 196 (2004) 151–168.

M. Han, J. Yang, P. Yu, Hopf bifurcations for near-Hamiltonian systems. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 19 (2009), no. 12, 4117–4130.

M. Bonnin , F. Corinto, V. Lanza, A mathematical framework for amplitude and phase noise analysis of coupled oscillators. The European Physical

Journal Special Topics 225 (2016), no. 1, 171-186.

A.B. Nordmark, P.T. Piiroinen, Simulation and stability analysis of impacting systems with complete chattering. Non-linear Dynamics 58 (2009), no. 1-2,

85-106.

C.C. Chung, J. Hauser, Nonlinear control of a swinging pendulum. Automatica J. IFAC 31 (1995), no. 6, 851–862.

T. Witelski, L.N. Virgin, C. George, A driven system of impacting pendulums: Experiments and simulations. Journal of Sound and Vibration 333 (2014),

no. 6, 1734-1753.

T. Kupper, H.A. Hosham, K. Dudtschenko, The dynamics of bells as impacting system, Proceedings of the Institution of Mechanical Engineers, Part C:

Journal of Mechanical Engineering Science, 225 (2001) 2436-2443.

G. Luoa, J. Xieb, X. Zhuc, J. Zhanga, Periodic motions and bifurcations of a vibro-impact system, Chaos, Solitons and Fractals 36 (2008) 1340–1347.

A.X.C.N. Valente, N.H. McClamroch, I. Mezić, Hybrid dynamics of two coupled oscillators that can impact a fixed stop. Internat. J. Non-Linear Mech. 38

(2003), no. 5, 677–689.

D.J. Wagg, Periodic sticking motion in a two-degree-of-freedom impact oscillator. International Journal of Non-linear Mechanics 40 (2001), no. 8, 1076-

1087.

P. Thota, H. Dankowicz, Continuous and discontinuous grazing bifurcations in impacting oscillators. Phys. D 214 (2006), no. 2, 187–197.

F.Casas, W. Chin, C. Grebogi, E. Ott, Universal grazing bifurcations in impact oscillators. Phys. Rev. E (3) 53 (1996), no. 1, part A, 134–139.

C. Duan, R. Singh, Dynamic analysis of preload nonlinearity in a mechanical oscillator. Journal of Sound and Vibration 301 (2007), no. 3-5, 963-978.

V.Sh. Burd, Resonance vibrations of impact oscillator with biharmonic excitation. Phys. D 241 (2012), no. 22, 1956–1961.

M.H. Fredriksson, A.B. Nordmark, Bifurcations caused by grazing incidence in many degrees of freedom impact oscillators. Proc. Roy. Soc. London Ser.

A 453 (1997), no. 1961, 1261–1276.

X. Zhao, Discontinuity Mapping for Near-Grazing Dynamics in Vibro-Impact Oscillators. Vibro-Impact Dynamics of Ocean Systems and Related

Problems Book Series: Lecture Notes in Applied and Computational Mechanics 44 (2009), 275-285.

O.Janin, C.H. Lamarque, Stability of singular periodic motions in a vibro-impact oscillator. Nonlinear Dynam. 28 (2002), no. 3-4, 231–241.

A.P. Ivanov, Stabilization of an impact oscillator near grazing incidence owing to resonance. Journal of Sound and Vibration 162 (1993), no 3, 562-565.

Hybrid automata and optimization

Bengt Lennartson Chalmers University of Technology,

Sweden

Vadim Azhmyakov Universidad de Medellin, Colombia

Reading:

B. Lennartson, K. Bengtsson, O. Wigstrom, S. Riazi, Modeling and Optimization of Hybrid

Systems for the Tweeting Factory, IEEE Transactions on Automation Science and Engineering

13 (2016), no. 1, 191-205.

V. Azhmyakov, R. Galvan-Guerra, M. Egerstedt, Hybrid LQ-optimization using Dynamic

Programming, in Proceedings of the 2009 American Control Conference, St. Louis, USA, 2009,

pp. 3617 - 3623.

R. Galvan-Guerra, V. Azhmyakov, M. Egerstedt, On the LQ-based optimization techniques for

impulsive hybrid control systems, in Proceedings of the 2010 American Control Conference,

Baltimore, USA, 2010, pp. 129 - 135.

0 0 1

1

Bifurcation approach

David Simpson (Massey University, New Zealand)

Tassilo Kuepper (University of Cologne, Germany)

Oleg Makarenkov (University of Texas at Dallas)

Javier Ros (University of Seville, Spain)

Zalman Balanov (University of Texas at Dallas)

Reading:

M.Guardia, T. M. Seara, M. A. Teixeira, Generic bifurcations of low codimension of planar Filippov systems. J. Differential Equations 250 (2011), no. 4, 1967–2023.

D.J.W. Simpson, J.D. Meiss, Andronov-Hopf bifurcations in planar, piecewise-smooth, continuous flows. Phys. Lett. A 371 (2007), no. 3, 213–220.

M. R. Jeffrey, D. J. W. Simpson, Non-Filippov dynamics arising from the smoothing of nonsmooth systems, and its robustness to noise. Nonlinear Dynam. 76

(2014), no. 2, 1395–1410.

M. di Bernardo, C.J. Budd, A.R. Champneys, P. Kowalczyk, Piecewise-smooth dynamical systems. Theory and applications. Springer, 2008. 481 pp

T. Küpper; H.A. Hosham; D. Weiss, Bifurcation for non-smooth dynamical systems via reduction methods. Recent trends in dynamical systems, 79–105, Proc.

Math. Stat., 35, Springer, 2013.

O. Makarenkov, Bifurcation of limit cycles from a fold-fold singularity in planar switched systems, arXiv:1603.03117

E. Ponce, J. Ros, E. Vela, A unified approach to piecewise linear Hopf and Hopf-pitchfork bifurcations. Analysis, modelling, optimization, and numerical techniques,

173–184, Springer Proc. Math. Stat., 121, Springer, Cham, 2015.

E. Hooton, Z. Balanov, W. Krawcewicz, D. Rachinskii, Sliding Hopf bifurcation in interval systems, arXiv:1507.08596.

www.utdallas.edu/sw16