University of Southern Queensland

Computational Engineering and Science Research Centre,

Faculty of Health, Engineering and Sciences

Dynamics of active systems with nonlinear

excitation of the phase

A thesis submitted by

Mayada G. Mohammed Al-Badri

B. Sc., M. Sc., Iraq

in fulfilment of the requirements of

Doctor of Philosophy

2015

Dedication

Thee my Lord to be satisfiedAnd then

To my Mum who passed away inthe examination period of this dissertation

Abstract

Many physical and chemical systems exhibit self-oscillatory dynamics, for exam-ple systems involving the Belousov-Zhabotinsky reaction and systems used formaterial synthesis by solid-phase combustion, known as self-propagating high-temperature synthesis. Phase of oscillators crucially depend on diffusion (orthermal conductivity), which is reflected in the partial differential equation gov-erning the phase of oscillations. At first sight, the role of diffusion is to equalisethe phase in space. However, more complex situations are possible; for exam-ple the phase equation may involve self-excitation such as anti-diffusion in the(Kuramoto-Sivashinsky equation). In this research we investigate a version of thephase equation based on a nonlinear self-excitation. Previously it was shown thatnonlinear self-excitation can arise in chemical systems with non-local interaction.

In the present research, we analyse this kind of system in order to determinethe validity range of the nonlinearly excited phase equation in the parametricspace. Specifically, we numerically evaluate the values of the parameters thatguarantee the assumptions of slow variations of the phase in space and time and,simultaneously, the key role of the nonlinear self-excitation.

We also numerically solve the phase equation with nonlinear self-excitationin two spatial dimensions by finite-difference discretization in space and subse-quent numerical integration of a system of ordinary differential equation in time.Irregular dynamics intermitting with periods of slow evolution are revealed anddiscussed.

As a separate task, we derive a forced variant of the phase equation andpresent selected exact solutions – stationary and oscillatory. They are also usedto verify the numerical code. In the numerical experiments, we use a range ofsizes of spatial domain.

Lastly, different forms of the nonlinearly excited phase equation are investi-gated based on different types of dynamical balance.

Keywords

Active dissipative systems, reaction-diffusion systems, nonlinear partial differ-ential equation, nonlinear excitation, Kuramoto-Sivashinsky (KS) equation , fi-nite difference, irregular dynamics, Complex Ginzburg-Landau equation (CGLE),parametric space.

Publications by the candidate

The following publications were produced during the period of candidature:

• D.V. Strunin and M.G. Mohammed, Parametric space for nonlinearly ex-cited phase equation, ANZIAM Journal (E) [Oxford University Press] Vol.53(2012) pp. C236-C248.

• D.V. Strunin and M.G. Mohammed, Validity and dynamics in the nonlin-early excited 6th-order phase equation, Discrete and Continuous DynamicalSystems - Supplement [American Institute of Mathematical Sciences] ISBN-10: 1-60133-016-2; ISBN-13: 978-1-60133-016-1, 2013, pp. 719-728.

• M.G. Mohammed and D.V. Strunin. Finite-difference approach for a 6th-order nonlinear phase equation with self-excitation, JPS Conference Pro-ceedings [The Physical Society of Japan]. Proc.1, 2014, 016003.

• D.V. Strunin and M.G. Mohammed, Range of validity and intermittentdynamics of the phase of oscillators with nonlinear self-excitation, submit-ted to the journal Communications in Nonlinear Science and NumericalSimulation [Elsevier] in January 2015.

Certification of Dissertation

I certify that the ideas, experimental work, results, analyses and conclusionsset out in this dissertation are entirely my own effort, except where otherwiseindicated and acknowledged.I further certify that the work is original and has not been previously submittedfor assessment in any other course or institution, except where specifically stated.

Mayada Gassab Mohammed Date

ENDORSEMENT

A/Prof. Dmitry V. Strunin, Principal Supervisor Date [6ex]

Prof. Thanh Tran-Cong, Co-Supervisor Date

Acknowledgments

It is my great pleasure to take this opportunity to express my sincere gratitudeto several people who directly or indirectly played a key role in the successfulcompletion of this work. Their persistent support and encouragement was aconsiderable helping to me in many ways.

First of all, I would like to bow down and offer thanks and praises to theAlmighty God for providing me with the good health, wisdom and courage re-quired to pursue this research.

My foremost gratitude goes to my supervisor, Associate Professor DmitryStrunin. I would like to express my deepest and heartfelt appreciation to him.He offered me lots of invaluable suggestions, guidance, encouragement and ines-timable help. He has been a great example of an individual who has achievedexcellence in both scientific research and as a human being. His creativity anddeep respect for each of his students has empowered many bright minds andhelped them realize their talents. Over the years I have gained from Assoc ProfDmitry invaluable experience in conducting cutting edge research with the high-est standards of exposition and rigor. I will always remember our discussions onvarious subjects which helped me shape my opinion on many different aspects ofmy research area. His kindness and patience are greatly appreciated.

I would like to thank my co-supervisor Prof. Thanh Tran-Cong for his supportduring my PhD study journey.

I am indebted to the most important person in my life, my uncle “Abu Mo-hammed”, who compensated me for tenderness Father. Without his sacrifice andincredible support, love and patience through my many years in my life, I cannot achieve this level of education.

I would like to dedicate this PhD research to the soul of my mother, who leftus too soon, just in the examination stage of this dissertation. You may be movedto another world but you will still always in my heart and never be forgotten. Iwill always love you and keep your memories in my mind. I know you waited thisday from long time. I hope that this work makes you proud and happy.

I would like to address my deepest gratitude, love, and respect to the soul ofmy father (ALLAH bless upon him), that I wish he is alive to see what I haveachieved and to share my happiness for completing this research.

My special thanks go to my brothers and my sisters especially my lovelysister “Nehad” and her family. Their support and encouragement are greatlyappreciated. I would like to give a heartfelt thanks to my wonderful family, myhusband and my lovely children, “Zahraa and Mohammed Al-Mustafaa” who

viii ACKNOWLEDGMENTS

made all this possible to complete my study. I should not forget my relatives aswell as my friends here in Australia and there in Iraq.

Finally, my warmest regards go to my Iraqi government who supported mefinancially by offering the PhD scholarship. Their support is gratefully acknowl-edged.

Mayada G. Mohammed Al-Badri

University of Southern QueenslandFeburary 2015

Contents

Dedication i

Abstract i

Keywords ii

Publications by the candidate iii

Acknowledgments vii

List of Figures xiii

List of Tables xvii

Acronyms & Abbreviations xix

Chapter 1 Introduction 11.1 Overview and motivation . . . . . . . . . . . . . . . . . . . . . . . 11.2 Active nonlinear systems and oscillations . . . . . . . . . . . . . 21.3 Systems of coupled oscillators . . . . . . . . . . . . . . . . . . . . 3

1.3.1 Locally interacting oscillators and reaction-diffusion systems 41.3.2 Globally interacting oscillators . . . . . . . . . . . . . . . . 51.3.3 Nonlocally coupled oscillators . . . . . . . . . . . . . . . . 6

1.4 Phase equations and their hierarchy . . . . . . . . . . . . . . . . . 71.5 Scaling for the phase equations . . . . . . . . . . . . . . . . . . . 101.6 Focus and scope of the dissertation . . . . . . . . . . . . . . . . . 111.7 Outline of the dissertation . . . . . . . . . . . . . . . . . . . . . . 11

Chapter 2 Literature Review 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 General review of dynamical system concepts . . . . . . . . . . . 13

2.2.1 Dynamical systems . . . . . . . . . . . . . . . . . . . . . . 132.2.2 Equilibrium points and their stability . . . . . . . . . . . . 152.2.3 Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.4 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.5 Chaos and strange attractors . . . . . . . . . . . . . . . . 252.2.6 Conservative and dissipative systems . . . . . . . . . . . . 27

x CONTENTS

2.3 Complex Ginzburg-Landau equation (CGLE) . . . . . . . . . . . 312.4 Linear excitation in the phase equation . . . . . . . . . . . . . . . 322.5 Nonlinear excitation in the phase equation . . . . . . . . . . . . . 352.6 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Chapter 3 Parametric space for the nonlinearly excited phase nepequation 413.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Nonlocal CGLE . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Computational results . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3.1 Validity in the parametric space . . . . . . . . . . . . . . . 483.3.2 Comparison of validity areas between the nep equation and

the KS equation . . . . . . . . . . . . . . . . . . . . . . . . 493.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Chapter 4 Test of the numerical code. Forced nep equation 634.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2 Derivation of the nep equation . . . . . . . . . . . . . . . . . . . 644.3 Derivation of the forced nep equation . . . . . . . . . . . . . . . 654.4 Exact solutions of the forced equation.

Testing the numerical code . . . . . . . . . . . . . . . . . . . . . . 684.5 Numerical experiments with forced nep equation . . . . . . . . . 72

4.5.1 L=0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.5.2 L=0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.5.3 L=0.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.5.4 L=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Chapter 5 Irregular dynamics in the (unforced) nep equation in2D spatial domain 915.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.2 Numerical solution of the nep equation . . . . . . . . . . . . . . . 92

5.2.1 Trivial and transitional regimes at small–to–large sizes L . 925.2.2 Numerical experiment with L=2 . . . . . . . . . . . . . . . 925.2.3 Numerical experiment with L=2.5 . . . . . . . . . . . . . . 95

5.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Chapter 6 Other possible forms of the nep equation 1116.1 Form A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.2 Form B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.3 Form C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.4 Form D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

CONTENTS xi

Chapter 7 Conclusions 1177.1 Research outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . 1177.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

References 121

Appendix A Reduce program to compute the parametric space forthe nep equation 129

Appendix B Matlab program 135B.1 Matlab program to set the initial conditions

c© Tony Roberts . . . . . . . . . . . . . . . . . . . . . . . . . . . 135B.2 Matlab program to solve the forced nep equation c© Tony Roberts 136B.3 Matlab program to plot the solution of the forced nep equation c©

Tony Roberts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150B.4 Matlab program dae2.m for integration in time

c© Tony Roberts . . . . . . . . . . . . . . . . . . . . . . . . . . . 151B.5 Matlab program to solve the unforced nep equation c© Tony Roberts153

List of Figures

1.1 Trajectories in the phase space for (a) a dissipative system (Vander Pol oscillator) and (b) a Hamiltonian system (harmonic pen-dulum) (Kuramoto, 1984a). . . . . . . . . . . . . . . . . . . . . . 3

1.2 Three examples of possible nonlocal functions for system (1.13).(a) Gaussian: G(X) = exp−x

2; (b) Exponential: G(x) = exp−|x|;

(c) Non-monotonic: G(x) = 1a2

exp−x2

a2 − 1b2

exp−x2

b2 . Cited in (Casagrande,2006). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Real eigenvalues λ1 and λ2, (a) Stable Node, λ2 < λ1 <0, (b)Saddle, λ2 <0 <λ1 (Manneville, 2010). . . . . . . . . . . . . . . . 16

2.2 Complex eigenvalues λ = (σ ± iw), (a) Focus, (b) Center (σ=0) (Manneville, 2010). . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Double real eigenvalues, (a) Improper node, (b) Star (diagonaliz-able case) (Manneville, 2010). . . . . . . . . . . . . . . . . . . . . 17

2.4 Limit cycle in Van der Pol oscillator X − (r − gX2)X + X = 0.Left: Phase portraits showing the convergence toward a limit cycleindicated by a dashed line. Right: Limit cycles for increasingvalues of control parameter r (Manneville, 2010). . . . . . . . . . 18

2.5 Types of attractors (Shaw, 1981). . . . . . . . . . . . . . . . . . . 192.6 Attractors for Henon map fa,b = (a−x2 + by, x) (Manneville, 2010). 192.7 Basin of three coexisting attracting fixed points (Alligood et al.,

1997). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.8 A saddle-node bifurcation. The graph of the quadratic family

f(a, x) = a − x2 before, at, and following a saddle node bifur-cation is shown. (a) At a = −1, the graph does not intersect thedotted line y = x, (b) At a = −0.25, the graph and the line y = xintersect in one point, the point of tangency; for a > 0.25, theyintersect in two points, (c) At a = 1/2, f has a repelling fixedpoint and an attracting fixed point (Alligood et al., 1996). . . . . 21

2.9 Simple bifurcations from a time independent state. The curvesshow how the solution evolves as a control parameter is varied.Solid lines denote stable solutions, dashed lines unstable solutions.The dotted line in (e) shows the continuous growth of the ampli-tude of the limit cycle (Cross and Hohenberg, 1993). . . . . . . . 23

xiv LIST OF FIGURES

2.10 A pair of complex conjugate eigenvalues crosses the imaginary axisat the Hopf bifurcation, (a) Before the bifurcation point (R <Rc), a steady stable state exist, (b) It looses its stability at thebifurcation (R = Rc), (c) The new stationary solution is a limitcycle whose amplitude increases as the square root of the distancefrom the bifurcation point. The frequency of the limit cycle equalsthe imaginary part of the eigenvalue (Imλ) (Lakshmanan andRajasekar, 2003). . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.11 Attractors of the Rossler system x = −y − z, y = x + ay, z =b+ (x− c)z as c is varied (Alligood et al., 1996). . . . . . . . . . 26

2.12 (a) The time series of the concentration in the Belousov-Zhabotinskyreaction (BZ), (b) Reconstructed attractor in the plane (x(t), x(t+τ)), (c) Poincare section of the attractor along the cut, (d) Firstreturn map of the Poincare section (Roux and Swinney, 1981). . . 28

2.13 Numerical solution of the convection equations. Graph of Y as afunction of time for the first 1000 iterations (upper curve), second1000 iterations (middle curve), and third 1000 iterations (lowercurve) (Lorenz, 1963). . . . . . . . . . . . . . . . . . . . . . . . . 29

2.14 Numerical solution of the convection equations. Projections on theX − Y plane and the Y − Z plane in phase space of the segmentof the trajectory extending from iteration 1400 to iteration 1900.Numerals 14, 15, etc., denote positions at iterations 1400, 1500, etc.States of steady convection are denoted by C and C ′ (Lorenz, 1963). 30

2.15 Solution of KS equation in the (x,t) plane. L = 64, n = 128,Ax = 0.5, with a time interval 4T = 1 between two successivesnapshots (Manneville, 2010). . . . . . . . . . . . . . . . . . . . . 33

2.16 The model of a viscoelastic medium with internal oscillators. σis the stress, EI + EIL, EII are the frozen and equilibrium elasticmoduli, respectively, and MI + MII , MIL are the correspondingmasses of oscillators (Nikolaevskii, 1989). . . . . . . . . . . . . . 33

2.17 Schematic presentation of the reaction front (Strunin, 1999). . . . 362.18 The settled solution at k = 0.12. Two periods are shown (Strunin,

1999). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1 The area of validity of the nep equation. . . . . . . . . . . . . . . 503.2 The area of validity of the nep equation from different angle. . . . 513.3 2-D slide of the parametric space of the nep equation. . . . . . . . 523.4 Valid and invalid points for the nep and KS equations along a

selected line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.5 A comparison along a selected line. . . . . . . . . . . . . . . . . . 543.6 A comparison along a selected line. . . . . . . . . . . . . . . . . . 543.7 A comparison along a selected line. . . . . . . . . . . . . . . . . . 553.8 A comparison along a selected line. . . . . . . . . . . . . . . . . . 553.9 Valid and invalid points for the nep equation along selected hori-

zontal lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

LIST OF FIGURES xv

3.10 Valid and invalid points for the KS equation along selected hori-zontal lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.11 Valid and invalid points for the nep equation along selected verticallines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.12 Valid and invalid points for the KS equation along selected verticallines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.1 Exact stationary solution (4.23) of equation (4.26). M = 1.3 ·1013,a = 10, b = c = 1, L = 0.4. The axes show grid points. . . . . . . 70

4.2 The force function (4.25). M = 1.3 · 1013, L = 0.4. The axes showgrid points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 Evolution of the numerical solution of (4.20) towards the stationaryregime. t = 1.1 · 10−11, 1.24 · 10−10, 1.735 · 10−10, 10.0 · 10−10,20.0 · 10−10. The equation coefficients A = 10, B = C = 1. Thedomain size L = 0.4. . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4 Settling of the maximum of u in the 60×60-grid experiment. . . . 724.5 The exact solution (4.23) and settled numerical solution (circles)

for 40×40, 60×60 and 90×90-grid experiments (u(x, y0) at y0 = 0.2(grid points 20, 30 and 45 respectively)). . . . . . . . . . . . . . . 73

4.6 The exact and numerical (circles) solutions in the oscillatory regime. 744.7 The cross-section for the force function surface for different values

of L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.8 Evolution of the solution of the forced equation for L=0.3. . . . . 764.9 Evolution of the cross-section of the solution of the forced equation

for L=0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.10 Evolution of the maximum value of u for L=0.3. . . . . . . . . . . 784.11 Evolution of the solution of the forced equation for L=0.5. . . . . 804.12 Evolution of the solution of the forced equation for L=0.5. . . . . 814.13 Evolution of the cross-section of the solution of the forced equation

for L=0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.14 Evolution of the cross-section of the solution of the forced equation

for L=0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.15 Evolution of the maximum value of u for L=0.5. . . . . . . . . . . 844.16 Evolution of the solution of the forced equation for L=0.6. . . . . 854.17 Evolution of the cross-section of the solution of the forced equation

for L=0.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.18 Evolution of the maximum value of u for L=0.6. . . . . . . . . . . 874.19 Evolution of the solution of the forced equation for L=1. . . . . . 884.20 Evolution of the cross-section of the solution of the forced equation

for L=1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.21 Evolution of the maximum value of u for L=1. . . . . . . . . . . . 90

5.1 The evolution of u for L=0.9. t = t0, 15 · 10−9, 180 · 10−9. . . . . . 935.2 The evolution of u for L=1.8. t = t0, 20 · 10−9, 190 · 10−9. . . . . . 945.3 t = t0, 9 · 10−9, 150 · 10−9, 225 · 10−9. . . . . . . . . . . . . . . . . 965.4 t = 462 · 10−9, 510 · 10−9, 541.5 · 10−9, 552 · 10−9. . . . . . . . . . . 97

xvi LIST OF FIGURES

5.5 t = 558 · 10−9, 570 · 10−9, 588 · 10−9, 609 · 10−9. . . . . . . . . . . . 985.6 t = 651 · 10−9, 679.5 · 10−9, 738 · 10−9, 769.5 · 10−9. . . . . . . . . . 995.7 t = 822 · 10−9, 1281 · 10−9, 1554 · 10−9, 1804 · 10−9. . . . . . . . . . 1005.8 t = 2001 · 10−9, 2300 · 10−9, 2425 · 10−9 . . . . . . . . . . . . . . . 1015.9 Integral of the absolute value of the excitation term. . . . . . . . . 1025.10 The positions of the global maximum of the excitation term at

different moments through the period of time (0, 1804 · 10−9). . . 1025.11 The positions of the second-largest maximum of the excitation

term at different moments through the period of time (0, 1804 ·10−9).1035.12 t = 0, 3 · 10−9, 10 · 10−9, 45 · 10−9. . . . . . . . . . . . . . . . . . . 1045.13 t = 86 · 10−9, 209 · 10−9, 230 · 10−9, 430 · 10−9. . . . . . . . . . . . 1055.14 t = 461 · 10−9, 938 · 10−9, 960 · 10−9. . . . . . . . . . . . . . . . . . 1065.15 1220 · 10−9, 1265 · 10−9. . . . . . . . . . . . . . . . . . . . . . . . . 1075.16 Integral of the absolute excitation. . . . . . . . . . . . . . . . . . . 1085.17 The positions of the global maximum of the excitation at different

moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.18 The positions of the second-largest maximum of the excitation at

different moments. . . . . . . . . . . . . . . . . . . . . . . . . . . 109

List of Tables

1.1 Hierarchy of truncations of the phase equations (1.14) . . . . . . . 10

2.1 Stability of equilibrium points. . . . . . . . . . . . . . . . . . . . . 15

Acronyms & Abbreviations

CGLE Complex Ginzburg-Landau equationGL equation Ginzburg-Landau equationnep equation Nonlinearly excited phase equationKS equation Kuramoto-Sivashinsky equation