WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Series Editor: Leon O. Chua Series A Vol. 66 DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS Jean-Marc Ginoux Université du Sud, France World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI
Contents I Dynamical Systems 18 1 Introduction 19 1.1 Galileo s pendulum......................... 19 1.2 D Alembert transformation..................... 21 1.3 From differential equations to dynamical systems......... 22 2 Dynamical Systems 23 2.1 State space phase space..................... 23 2.2 Definition.............................. 24 2.3 Existence and uniqueness...................... 24 2.4 Flow, fixed points and null-clines.................. 25 2.5 Stability theorems.......................... 29 2.5.1 Linearized system..................... 29 2.5.2 Hartman-Grobman linearization theorem......... 29 2.5.3 Liapounoff stability theorem................ 29 2.6 Phase portraits of dynamical systems................ 30 2.6.1 Two-dimensional systems................. 30 2.6.2 Three-dimensional systems................. 34 2.7 Various types of dynamical systems................ 37 2.7.1 Linear and nonlinear dynamical systems.......... 37 2.7.2 Homogeneous dynamical systems............. 38 2.7.3 Polynomial dynamical systems............... 38 2.7.4 Singularly perturbed systems................ 39 2.7.5 Slow-Fast dynamical systems............... 39 2.8 Two-dimensional dynamical systems................ 40 2.8.1 Poincaré index....................... 40 2.8.2 Poincaré contact theory................... 42 2.8.3 Poincaré limit cycle..................... 42 2.8.4 Poincaré-Bendixson Theorem............... 44 2.9 High-dimensional dynamical systems............... 46 2.9.1 Attractors.......................... 46
CONTENTS 2 2.9.2 Strange attractors...................... 47 2.9.3 First integrals and Lie derivative.............. 49 2.10 Hamiltonian and integrable systems................ 49 2.10.1 Hamiltonian dynamical systems.............. 49 2.10.2 Integrable system...................... 50 2.10.3 K.A.M. Theorem...................... 52 3 Invariant Sets 54 3.1 Manifold.............................. 54 3.1.1 Definition.......................... 54 3.1.2 Existence.......................... 55 3.2 Invariant sets............................ 55 3.2.1 Global invariance...................... 55 3.2.2 Local invariance...................... 56 4 Local Bifurcations 59 4.1 Introduction............................. 59 4.2 Center Manifold Theorem..................... 59 4.2.1 Center manifold theorem for flows............. 60 4.2.2 Center manifold approximation.............. 61 4.2.3 Center manifold depending upon a parameter....... 64 4.3 Normal Form Theorem....................... 65 4.4 Local Bifurcations of Codimension 1................ 70 4.4.1 Saddle-node bifurcation.................. 72 4.4.2 Transcritical bifurcation.................. 73 4.4.3 Pitchfork bifurcation.................... 75 4.4.4 Hopf bifurcation...................... 77 5 Slow-Fast Dynamical Systems 78 5.1 Introduction............................. 78 5.2 Geometric Singular Perturbation Theory.............. 81 5.2.1 Assumptions........................ 81 5.2.2 Invariance.......................... 82 5.2.3 Slow invariant manifold.................. 83 5.3 Slow-fast dynamical systems Singularly perturbed systems... 89 5.3.1 Singularly perturbed systems................ 89 5.3.2 Slow-fast autonomous dynamical systems......... 90 6 Integrability 92 6.1 Integrability conditions, integrating factor and multiplier..... 92 6.1.1 Two-dimensional dynamical systems........... 93
CONTENTS 3 6.1.2 Three-dimensional dynamical systems........... 96 6.2 First integrals Jacobi s last multiplier theorem.......... 101 6.2.1 Jacobi s last multiplier theorem.............. 102 6.3 Darboux theory of integrability................... 103 6.3.1 Algebraic particular integral General integral...... 103 6.3.2 General integral....................... 105 6.3.3 Multiplier.......................... 107 6.3.4 Algebraic particular integral and fixed points....... 108 6.3.5 Homogeneous polynomial dynamical system of degree m 109 6.3.6 Homogeneous polynomial dynamical system of degree two 115 6.3.7 Planar polynomial dynamical systems........... 121 II Differential Geometry 126 7 Differential Geometry 127 7.1 Concept of curves Kinematics vector functions......... 128 7.1.1 Trajectory curve...................... 128 7.1.2 Instantaneous velocity vector................ 128 7.1.3 Instantaneous acceleration vector............. 129 7.2 Gram-Schmidt process Generalized Frénet moving frame.... 129 7.2.1 Gram-Schmidt process................... 130 7.2.2 Generalized Frénet moving frame.............. 130 7.3 Curvatures of trajectory curves Osculating planes........ 131 7.4 Curvatures and osculating plane of space curves.......... 133 7.4.1 Frénet trihedron Serret-Frénet formulae......... 133 7.4.2 Osculating plane...................... 134 7.4.3 Curvatures of space curves................. 135 7.5 Flow curvature method....................... 136 7.5.1 Flow curvature manifold.................. 136 7.5.2 Flow curvature method................... 137 8 Dynamical Systems 138 8.1 Phase portraits of dynamical systems................ 138 8.1.1 Fixed points......................... 138 8.1.2 Stability theorems..................... 140 9 Invariant Sets 148 9.1 Invariant manifolds......................... 148 9.1.1 Global invariance...................... 149 9.1.2 Local invariance...................... 149
CONTENTS 4 9.2 Linear invariant manifolds..................... 151 9.3 Nonlinear invariant manifolds................... 157 10 Local Bifurcations 160 10.1 Center Manifold........................... 160 10.1.1 Center manifold approximation.............. 160 10.1.2 Center manifold depending upon a parameter....... 167 10.2 Normal Form Theorem....................... 174 10.3 Local bifurcations of codimension 1................ 179 11 Slow-Fast Dynamical Systems 180 11.1 Slow manifold of n-dimensional slow-fast dynamical systems... 181 11.2 Invariance.............................. 184 11.3 Flow Curvature Method Singular Perturbation Method..... 185 11.3.1 Darboux invariance Fenichel s invariance........ 187 11.3.2 Slow invariant manifold.................. 188 11.4 Non-singularly perturbed systems................. 197 12 Integrability 199 12.1 First integral............................. 199 12.1.1 Global first integral..................... 199 12.1.2 Local first integral..................... 200 12.2 Linear invariant manifolds as first integral............. 201 12.3 Darboux theory of integrability................... 205 12.3.1 General integral Multiplier................ 205 12.3.2 Homogeneous polynomial dynamical system of degree two 207 12.3.3 Planar polynomial dynamical systems........... 208 13 Inverse Problem 210 13.1 Flow curvature manifold of polynomial dynamical systems.... 210 13.1.1 Two-dimensional polynomial dynamical systems..... 210 13.1.2 Three-dimensional polynomial dynamical systems.... 212 13.2 Inverse problem for polynomial dynamical systems........ 214 13.2.1 Two-dimensional polynomial dynamical systems..... 214 13.2.2 Three-dimensional polynomial dynamical systems.... 215 III Applications 217 14 Dynamical Systems 218 14.1 FitzHugh-Nagumo model...................... 218 14.2 Pikovskii-Rabinovich-Trakhtengerts model............ 219
CONTENTS 5 15 Invariant sets - Integrability 220 15.1 Pikovskii-Rabinovich-Trakhtengerts model............ 220 15.2 Rikitake model........................... 223 15.3 Chua s model............................ 224 15.4 Lorenz model............................ 225 16 Local bifurcations 227 16.1 Chua s model............................ 227 16.2 Lorenz model............................ 229 17 Slow-Fast Dynamical Systems 231 17.1 Piecewise Linear Models 2D & 3D................. 231 17.1.1 Van der Pol piecewise linear model............ 231 17.1.2 Chua s piecewise linear model............... 233 17.2 Singularly Perturbed Systems 2D & 3D.............. 235 17.2.1 FitzHugh-Nagumo model................. 235 17.2.2 Chua s model........................ 236 17.3 Slow Fast Dynamical Systems 2D & 3D.............. 237 17.3.1 Brusselator model..................... 237 17.3.2 Pikovskii-Rabinovich-Trakhtengerts model........ 238 17.3.3 Rikitake model....................... 239 17.4 Piecewise Linear Models 4D & 5D................. 240 17.4.1 Chua s fourth-order piecewise linear model........ 240 17.4.2 Chua s fifth-order piecewise linear model......... 242 17.5 Singularly Perturbed Systems 4D & 5D.............. 244 17.5.1 Chua s fourth-order cubic model.............. 244 17.5.2 Chua s fifth-order cubic model............... 245 17.6 Slow Fast Dynamical Systems 4D & 5D.............. 246 17.6.1 Homopolar dynamo model................. 246 17.6.2 Mofatt model........................ 248 17.6.3 Magnetoconvection model................. 249 17.7 Slow manifold gallery........................ 251 17.8 Forced Van der Pol model...................... 252 Discussion 254 18 Appendix 257 18.1 Lie derivative............................ 257 18.2 Hessian............................... 257 18.3 Jordan form............................. 258 18.4 Connected region.......................... 259
CONTENTS 6 18.5 Fractal dimension.......................... 259 18.5.1 Kolmogorov or capacity dimension............ 260 18.5.2 Liapounoff exponents Wolf, Swinney, Vastano algorithm 261 18.5.3 Liapounoff dimension and Kaplan-Yorke conjecture... 262 18.5.4 Liapounoff dimension and Chlouverakis-Sprott conjecture 263 18.6 Identities.............................. 263 18.6.1 Concept of curves..................... 264 18.6.2 Gram-Schmidt process and Frénet moving frame..... 264 18.6.3 Frénet trihedron and curvatures of space curves...... 267 18.6.4 First identity........................ 268 18.6.5 Second identity....................... 269 18.6.6 Third identity........................ 270 18.7 Homeomorphism and diffeomorphism............... 270 18.7.1 Homeomorphism...................... 270 18.7.2 Diffeomorphism...................... 270 18.8 Differential equations....................... 271 18.8.1 Two-dimensional dynamical systems........... 271 18.8.2 Three-dimensional dynamical systems........... 271 18.9 Generalized Tangent Linear System Approximation........ 272 18.9.1 Assumptions........................ 272 18.9.2 Corollaries......................... 273 Mathematica Files 277 Bibliography 282 Index 297
List of Figures 1 Synopsis............................... 17 1.1 Galileo s pendulum......................... 20 2.1 Free fall............................... 26 2.2 Volterra-Lotka predator-prey model................ 27 2.3 Phase plane stability diagram.................... 32 2.4 Inverted pendulum......................... 33 2.5 stability diagram.......................... 36 2.6 Saddle-focus............................ 37 2.7 Poincaré limit cycle......................... 43 2.8 Duffing oscillator.......................... 45 2.9 Lorenz butterfly........................... 48 2.10 Spherical pendulum......................... 51 2.11 Hénon-Heiles Hamiltonian..................... 53 2.12 Transversal Poincaré section (p 2, q 2 ) of Hénon-Heiles Hamiltonian 53 3.1 Stable W S and unstable W U manifolds.............. 58 4.1 Part of the center manifold in green................ 61 6.1 General integral........................... 102 7.1 Osculating plane.......................... 134 8.1 Duffing oscillator.......................... 144 9.1 Local invariance........................... 150 10.1 Center manifold........................... 164 11.1 Van der Pol slow manifold..................... 192 11.2 Chua s slow invariant manifold in (xz)-plane........... 196 11.3 Lorenz slow manifold........................ 198
LIST OF FIGURES 8 12.1 Local first integral of Van der Pol model.............. 201 12.2 Volterra-Lotka s first integral.................... 204 12.3 First integral of quadratic system.................. 206 17.1 Van der Pol piecewise linear model slow invariant manifold.... 233 17.2 Chua s piecewise linear model slow invariant manifold...... 234 17.3 FitzHugh-Nagumo model slow invariant manifold......... 235 17.4 Chua s cubic model slow invariant manifold............ 236 17.5 Brusselator s model slow invariant manifold............ 237 17.6 (PRT) model slow invariant manifold............... 238 17.7 Rikitake model slow invariant manifold.............. 239 17.8 Chua s model invariant hyperplanes in (x 1, x 2, x 3 ) phase space.. 242 17.9 Chua s model invariant hyperplanes in (x 1, x 2, x 3 ) phase space.. 243 17.10 Chua s model slow invariant manifold............... 245 17.11 Chua s slow invariant manifold.................. 246 17.12 Dynamo model slow invariant manifold.............. 247 17.13 Mofatt model slow invariant manifold............... 248 17.14 Magnetoconvection slow invariant manifold............ 250 17.15 Chemical kinetics model - Neuronal bursting model....... 251 17.16 Forced Van der Pol model slow invariant manifold........ 253 17.17 Chua s cubic model attractor structure............... 256
List of Tables 15.1 Invariant manifolds of the (PRT) model.............. 222 15.2 Invariant manifolds of the Rikitake model............. 223 15.3 Invariant manifolds of the Lorenz model.............. 226
Index attractor, 46, 47, 251, 255, 260, 262 attractor structure, 256 autonomous, 17, 23, 24, 28, 59, 249, 252, 255 Bendixson, 44 46 bifurcation, 59, 60, 64, 68, 72, 74, 76, 77, 127, 137, 160, 174, 179, 229, 230, 278 280 Brusselator, 16, 219, 237, 281 Cauchy, 13, 24, 25, 40, 46, 59, 89, 93, 96, 99, 127, 128, 138, 160, 167, 181, 210, 212 centre manifold, 15, 16, 54, 58, 60 65, 68, 70 72, 82, 127, 137, 160, 161, 163, 164, 166 171, 174, 179, 227, 229, 230, 254, 255, 278, 279 chaotic attractor, 238 Chua, 14, 16, 38, 80, 88, 89, 195, 219, 224, 225, 227, 233, 234, 236, 240 246, 256, 278 281 Chua s invariant hyperplanes, 242, 243 Chua s slow invariant manifold, 196, 236 codimension, 64, 70, 137, 160, 179, 230 complex dynamics, 80 conservative, 47, 50, 101, 262 curvature, 13, 14, 127, 129, 131 136, 154 156, 181, 192, 197, 202, 254, 255, 263, 264, 266 268 curvature of the flow, 13 curve, 13, 27, 40 42, 44, 72, 93, 99, 101, 102, 128, 131 136, 139, 154, 263, 264, 266, 267, 271 Darboux, 14, 16, 94, 99, 101, 103 118, 121, 123, 124, 148, 149, 159, 183, 184, 187, 199, 204, 205, 207 209, 221, 255 Darboux invariance theorem, 15, 16, 103, 104, 148, 149, 181, 183, 187, 232, 234 239, 241, 243, 244, 246, 247, 249, 250, 252, 254, 255 diffeomorphism, 29, 270 differential equation, 13, 19 24, 27, 41, 42, 44, 46, 59, 62, 64, 70, 92, 93, 95 101, 127, 138, 160, 167, 181, 219, 223, 224, 240, 242, 244, 245, 271, 272 differential geometry, 13, 14, 17, 127, 131, 154, 181, 254, 263, 278 dissipative, 46 48, 101, 260, 262 divergence, 47, 93, 97 Duffing, 45, 144, 169, 170, 277 279 dynamical system, 13 16, 18, 22 30, 33, 34, 37 47, 49, 50, 54 66, 68 70, 72 74, 76 78, 89 92, 95 97, 99 101, 103 115, 118, 119, 127, 128, 131 138, 140 142, 144 150, 155 157, 159 161, 163, 164, 166 169, 171, 174, 175, 178, 180 183, 185, 188, 192, 193, 197, 199 202, 205, 207, 208, 210, 212 216, 228, 229, 254, 255, 257, 264, 266, 272 275, 277, 278, 280 Fenichel, 15, 78, 79, 81, 82, 85, 88, 91, 180, 185, 187, 254 first integral, 14 16, 49 51, 100 102, 104 106, 108, 110, 111, 114 119, 122 125, 148, 149, 156, 199 209, 221, 224, 225, 254, 255, 278, 279 FitzHugh-Nagumo, 16, 218, 235, 280, 281
INDEX 300 non-autonomous, 231, 252 non-singularly perturbed systems, 16, 90, 91, 197, 255, 279 nonlinear invariant manifold, 157, 221, 254 normal forms, 15, 65, 66, 68 70, 127, 137, 160, 174, 175, 178, 179, 220, 254, 258 null-clines, 25, 26, 28 orbit, 29, 45, 51, 55, 78, 180, 261 osculating plane, 14, 131 136, 140 144, 146, 202, 203, 234 pendulum, 19, 20, 154 phase, 23, 24, 26, 27, 30, 32, 33, 44, 45, 47, 49, 59, 90, 101, 138, 232, 234, 242, 243, 245 248, 250, 253, 256, 260 piecewise linear model, 16, 80, 231 234, 240 243, 280, 281 Pikovskii, 16, 158, 219, 220, 238, 280, 281 pitchfork bifurcation, 14, 64, 72, 75, 76, 137, 160, 174, 179, 229, 230 Poincaré, 15, 43, 44, 46, 92, 128, 144 Poincaré index, 14 Poincaré section, 53, 260 quadratic system, 206 relaxation oscillations, 27, 39, 231 relaxation oscillator, 79 Rikitake, 16, 223, 239, 246, 255, 280, 281 saddle, 31, 33, 35, 37, 40, 45, 73, 137, 142, 144, 145, 160, 179 saddle points, 52 saddle-focus, 145 saddle-node, 145 saddle-node bifurcation, 72, 73 singular approximation, 39, 85, 88, 90, 190, 192, 195, 200, 201, 235 237 Singular Perturbation Method, 81, 185, 192, 196, 236, 278 singularly perturbed systems, 14 16, 28, 39, 58, 78 80, 82 84, 87, 89 91, 137, 148, 180, 181, 186, 188, 189, 193, 194, 197, 200, 231, 235, 236, 238, 239, 244, 247 249, 254, 255, 281 slow invariant manifold, 14 17, 78, 82, 83, 85, 88, 90, 91, 137, 180, 182, 185 189, 191, 193, 194, 196, 220, 232, 233, 236 240, 243 251, 253 255 slow-fast dynamical systems, 14, 39, 78, 89, 90, 180, 181, 184, 185, 192, 231, 237, 255, 278 280 spherical pendulum, 50, 51, 200, 277 stability, 29, 30, 32, 36, 59, 73 75, 127, 137, 138, 140, 142, 144, 147, 148, 180, 218 strange attractor, 14, 47, 50, 260 263 tangent linear system approximation, 17, 181 183, 272, 274 276 torsion, 131, 134 136, 154, 181, 197, 266 268 trajectory curve, 13, 14, 24 26, 42, 44, 46, 47, 49 53, 101, 127 132, 136, 137, 155, 156, 181, 201, 202, 254 256, 260, 261, 263, 264, 266 268 transcritical, 137, 160, 170, 179 transcritical bifurcation, 72 74 Van der Pol, 14, 16, 27, 39, 41, 43, 44, 190, 192, 200, 201, 211, 215, 218, 231, 232, 235, 252, 253, 278 280 Volterra, 27, 28, 37, 118, 153, 156, 204, 279
INDEX 299 fixed point, 15, 16, 25, 26, 28 31, 33 35, 37, 40 46, 55 57, 60, 61, 70, 72 77, 108, 114, 127, 128, 132, 134, 137 147, 185, 214, 216, 218, 219, 234, 241, 243, 256 fixed point stability, 15, 254 flow curvature manifold, 13, 15, 16, 127, 136 144, 146, 148, 151 154, 156, 157, 159 161, 163, 164, 166 168, 170, 171, 174, 175, 179, 182, 184, 186, 188, 191 193, 196, 197, 201, 202, 204, 206, 207, 209 216, 218 221, 224, 225, 227, 232 241, 243, 244, 246 250, 252, 254 256 flow curvature method, 13, 15, 17, 90, 109, 127, 136, 137, 159, 185, 186, 190, 192, 195 197, 200, 252, 254 256, 276, 279 Forced Van der Pol, 17, 252, 255, 281 Galois, 147, 181 Geometric Singular Perturbation Theory, 14, 15, 58, 78, 79, 81, 82, 90, 91, 180, 185, 186, 188, 190 193, 195 197, 201, 254, 255, 278 Grobman, 29, 58 Groebner, 215, 216 Hénon-Heiles Hamiltonian, 52, 53, 277 Hamiltonian, 14, 49, 50, 52, 53, 199, 262 harmonic oscillator, 49 Hartman, 29, 58 homeomorphism, 29, 270 homopolar dynamo, 16, 246, 247 Hopf, 77, 137, 160 Hopf bifurcation, 76, 77 hyperbolic, 29, 40, 56, 57, 78, 81, 180 hyperbolic points, 52 implicit function theorem, 55, 82, 84, 85, 87, 88, 186 integrability, 54, 92 103, 127, 148, 159, 199, 205, 254, 255, 278 280 invariant manifold, 54, 56 58, 60, 71, 78, 104, 105, 107, 108, 110, 112 115, 127, 137, 148 152, 154 159, 180, 184, 199, 201, 224 226 invariant tori, 52 inverse problem, 16, 137, 210, 214, 255, 280 inverted pendulum, 33, 37 Jacobian, 29, 30, 33, 34, 37 39, 41, 54 57, 60, 61, 70, 71, 73 75, 77, 90, 91, 95, 129, 140 142, 145, 146, 161, 180 186, 234, 273 276 Jordan fom, 258, 259 Jordan form, 31, 35, 41, 258 K.A.M. theorem, 14, 52 K.A.M. tori, 52, 53 Kapteyn-Bautin, 125, 209, 278 LaSalle, 54 Liapounoff, 14, 29, 30 Liapounoff dimension, 260, 262, 263 Liapounoff exponents, 261 263 Lie derivative, 14, 15, 42, 49, 94, 119, 140 142, 149, 150, 153, 154, 157, 158, 184, 187, 192, 197, 200, 201, 221, 226, 232, 234, 237 239, 241, 244, 246, 247, 249, 250, 252, 255, 257 limit cycle, 14, 42 44, 46, 77, 277 linear, 29, 37, 38, 56, 65, 67, 134, 137, 148, 151, 152, 154 156, 159, 175, 180 183, 201, 233, 240 242, 244, 245, 258, 272 276 linear invariant manifold, 15, 153, 156, 202 205, 207, 208, 220, 254 Liouville, 47, 92, 199, 262 local bifurcations, 254 Lorenz, 16, 48, 90, 91, 147, 197, 198, 212, 216, 225, 226, 229, 230, 249, 255, 263, 278 280 Lorenz butterfly, 48, 277 Lorenz slow manifold, 198 magnetoconvection, 16, 249, 250, 281 Mofatt, 16, 246, 248, 249, 281