Nonlinear Dynamics A hands-on introductory survey Nonlinear Dynamics A hands-on introductory survey Marc R Roussel Alberta RNA Research and Training Institute, Department of Chemistry and Biochemistry, University of Lethbridge, Canada Morgan & Claypool Publishers Copyrightª2019Morgan&ClaypoolPublishers Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem ortransmittedinanyformorbyanymeans,electronic,mechanical,photocopying,recording orotherwise,withoutthepriorpermissionofthepublisher,orasexpresslypermittedbylawor undertermsagreedwiththeappropriaterightsorganization.Multiplecopyingispermittedin accordancewiththetermsoflicencesissuedbytheCopyrightLicensingAgency,theCopyright ClearanceCentreandotherreproductionrightsorganizations. Rights&Permissions Toobtainpermissiontore-usecopyrightedmaterialfromMorgan&ClaypoolPublishers,please [email protected]. ISBN 978-1-64327-464-5(ebook) ISBN 978-1-64327-461-4(print) ISBN 978-1-64327-462-1(mobi) DOI 10.1088/2053-2571/ab0281 Version:20190401 IOPConcisePhysics ISSN2053-2571(online) ISSN2054-7307(print) AMorgan&ClaypoolpublicationaspartofIOPConcisePhysics PublishedbyMorgan&ClaypoolPublishers,1210FifthAvenue,Suite250,SanRafael,CA, 94901,USA IOPPublishing,TempleCircus,TempleWay,BristolBS16HG,UK This book is dedicated to Simon J Fraser, my first teacher in nonlinear dynamics. Contents Preface x Author biography xii 1 Introduction 1-1 1.1 What is a dynamical system? 1-1 1.2 The law of mass action 1-3 1.3 Software 1-4 Reference 1-5 2 Phase-plane analysis 2-1 2.1 Introduction 2-1 2.2 The Lindemann mechanism 2-1 2.3 Dimensionless equations 2-3 2.4 The vector field 2-4 2.5 Exercises 2-8 3 Stability analysis for ODEs 3-1 3.1 Linear stability analysis 3-1 3.2 Lyapunov functions 3-8 3.3 Exercises 3-10 Reference 3-12 4 Introduction to bifurcations 4-1 4.1 Introduction 4-1 4.2 Saddle-node bifurcation 4-1 4.3 Transcritical bifurcation 4-3 4.4 Andronov–Hopf bifurcations 4-4 4.5 Dynamics in three dimensions 4-10 4.6 Exercises 4-17 References 4-18 5 Bifurcation analysis with AUTO 5-1 5.1 Bifurcation diagram of a gene expression model 5-1 5.2 The phase diagram of Griffith’s model 5-5 vii NonlinearDynamics 5.3 Bifurcation diagram of the autocatalator 5-7 5.4 Getting out of trouble in AUTO 5-11 5.5 Exercises 5-13 Reference 5-13 6 Invariant manifolds 6-1 6.1 Introduction 6-1 6.2 Flow dynamics away from the equilibrium point 6-2 6.3 Special eigenspaces of equilibrium points 6-5 6.4 From eigenspaces to invariant manifolds 6-7 6.5 Applications of invariant manifolds 6-9 6.5.1 The Lindemann mechanism revisited 6-9 6.5.2 A simple HIV model 6-12 6.6 Exercises 6-19 References 6-20 7 Singular perturbation theory 7-1 7.1 Introduction 7-1 7.2 Scaling and balancing 7-1 7.3 The outer solution 7-7 7.4 The inner solution 7-8 7.5 Matching the inner and outer solutions 7-9 7.6 Geometric singular perturbation theory and the outer solution 7-10 7.7 Exercises 7-12 References 7-14 8 Hamiltonian systems 8-1 8.1 Introduction 8-1 8.2 Integrable systems 8-3 8.3 Numerical integration 8-6 8.4 Exercises 8-10 9 Nonautonomous systems 9-1 9.1 Introduction 9-1 9.2 A driven Brusselator 9-5 9.3 Automated bifurcation analysis 9-8 9.4 Exercises 9-13 Reference 9-14 viii NonlinearDynamics 10 Maps and differential equations 10-1 10.1 Numerical methods as maps 10-1 10.2 Solution maps of differential equations 10-3 10.3 Poincaré maps of systems with periodic nonautonomous terms 10-4 10.4 Poincaré sections and maps in autonomous systems 10-6 10.5 Next-amplitude maps 10-7 10.6 Concluding comments 10-9 10.7 Exercises 10-9 References 10-10 11 Maps: stability and bifurcation analysis 11-1 11.1 Linear stability analysis of fixed points 11-1 11.2 Stability of periodic orbits 11-3 11.3 Lyapunov exponents 11-9 11.4 Exercises 11-13 References 11-15 12 Delay-differential equations 12-1 12.1 Introduction to infinite-dimensional dynamical systems 12-1 12.2 Introduction to delay-differential equations 12-2 12.3 Linearized stability analysis 12-6 12.4 Exercises 12-14 Reference 12-14 13 Reaction–diffusion equations 13-1 13.1 Introduction 13-1 13.1.1 The rate of diffusion 13-1 13.1.2 Reaction–diffusion equations 13-3 13.2 Stability analysis of reaction–diffusion equations 13-8 13.3 Exercises 13-15 Appendix A: Software installation A-1 ix