DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS

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1 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

2 Contents I Dynamical Systems 18 1 Introduction Galileo s pendulum D Alembert transformation From differential equations to dynamical systems Dynamical Systems State space phase space Definition Existence and uniqueness Flow, fixed points and null-clines Stability theorems Linearized system Hartman-Grobman linearization theorem Liapounoff stability theorem Phase portraits of dynamical systems Two-dimensional systems Three-dimensional systems Various types of dynamical systems Linear and nonlinear dynamical systems Homogeneous dynamical systems Polynomial dynamical systems Singularly perturbed systems Slow-Fast dynamical systems Two-dimensional dynamical systems Poincaré index Poincaré contact theory Poincaré limit cycle Poincaré-Bendixson Theorem High-dimensional dynamical systems Attractors

3 CONTENTS Strange attractors First integrals and Lie derivative Hamiltonian and integrable systems Hamiltonian dynamical systems Integrable system K.A.M. Theorem Invariant Sets Manifold Definition Existence Invariant sets Global invariance Local invariance Local Bifurcations Introduction Center Manifold Theorem Center manifold theorem for flows Center manifold approximation Center manifold depending upon a parameter Normal Form Theorem Local Bifurcations of Codimension Saddle-node bifurcation Transcritical bifurcation Pitchfork bifurcation Hopf bifurcation Slow-Fast Dynamical Systems Introduction Geometric Singular Perturbation Theory Assumptions Invariance Slow invariant manifold Slow-fast dynamical systems Singularly perturbed systems Singularly perturbed systems Slow-fast autonomous dynamical systems Integrability Integrability conditions, integrating factor and multiplier Two-dimensional dynamical systems

4 CONTENTS Three-dimensional dynamical systems First integrals Jacobi s last multiplier theorem Jacobi s last multiplier theorem Darboux theory of integrability Algebraic particular integral General integral General integral Multiplier Algebraic particular integral and fixed points Homogeneous polynomial dynamical system of degree m Homogeneous polynomial dynamical system of degree two Planar polynomial dynamical systems II Differential Geometry Differential Geometry Concept of curves Kinematics vector functions Trajectory curve Instantaneous velocity vector Instantaneous acceleration vector Gram-Schmidt process Generalized Frénet moving frame Gram-Schmidt process Generalized Frénet moving frame Curvatures of trajectory curves Osculating planes Curvatures and osculating plane of space curves Frénet trihedron Serret-Frénet formulae Osculating plane Curvatures of space curves Flow curvature method Flow curvature manifold Flow curvature method Dynamical Systems Phase portraits of dynamical systems Fixed points Stability theorems Invariant Sets Invariant manifolds Global invariance Local invariance

5 CONTENTS Linear invariant manifolds Nonlinear invariant manifolds Local Bifurcations Center Manifold Center manifold approximation Center manifold depending upon a parameter Normal Form Theorem Local bifurcations of codimension Slow-Fast Dynamical Systems Slow manifold of n-dimensional slow-fast dynamical systems Invariance Flow Curvature Method Singular Perturbation Method Darboux invariance Fenichel s invariance Slow invariant manifold Non-singularly perturbed systems Integrability First integral Global first integral Local first integral Linear invariant manifolds as first integral Darboux theory of integrability General integral Multiplier Homogeneous polynomial dynamical system of degree two Planar polynomial dynamical systems Inverse Problem Flow curvature manifold of polynomial dynamical systems Two-dimensional polynomial dynamical systems Three-dimensional polynomial dynamical systems Inverse problem for polynomial dynamical systems Two-dimensional polynomial dynamical systems Three-dimensional polynomial dynamical systems III Applications Dynamical Systems FitzHugh-Nagumo model Pikovskii-Rabinovich-Trakhtengerts model

6 CONTENTS 5 15 Invariant sets - Integrability Pikovskii-Rabinovich-Trakhtengerts model Rikitake model Chua s model Lorenz model Local bifurcations Chua s model Lorenz model Slow-Fast Dynamical Systems Piecewise Linear Models 2D & 3D Van der Pol piecewise linear model Chua s piecewise linear model Singularly Perturbed Systems 2D & 3D FitzHugh-Nagumo model Chua s model Slow Fast Dynamical Systems 2D & 3D Brusselator model Pikovskii-Rabinovich-Trakhtengerts model Rikitake model Piecewise Linear Models 4D & 5D Chua s fourth-order piecewise linear model Chua s fifth-order piecewise linear model Singularly Perturbed Systems 4D & 5D Chua s fourth-order cubic model Chua s fifth-order cubic model Slow Fast Dynamical Systems 4D & 5D Homopolar dynamo model Mofatt model Magnetoconvection model Slow manifold gallery Forced Van der Pol model Discussion Appendix Lie derivative Hessian Jordan form Connected region

7 CONTENTS Fractal dimension Kolmogorov or capacity dimension Liapounoff exponents Wolf, Swinney, Vastano algorithm Liapounoff dimension and Kaplan-Yorke conjecture Liapounoff dimension and Chlouverakis-Sprott conjecture Identities Concept of curves Gram-Schmidt process and Frénet moving frame Frénet trihedron and curvatures of space curves First identity Second identity Third identity Homeomorphism and diffeomorphism Homeomorphism Diffeomorphism Differential equations Two-dimensional dynamical systems Three-dimensional dynamical systems Generalized Tangent Linear System Approximation Assumptions Corollaries Mathematica Files 277 Bibliography 282 Index 297

8 List of Figures 1 Synopsis Galileo s pendulum Free fall Volterra-Lotka predator-prey model Phase plane stability diagram Inverted pendulum stability diagram Saddle-focus Poincaré limit cycle Duffing oscillator Lorenz butterfly Spherical pendulum Hénon-Heiles Hamiltonian Transversal Poincaré section (p 2, q 2 ) of Hénon-Heiles Hamiltonian Stable W S and unstable W U manifolds Part of the center manifold in green General integral Osculating plane Duffing oscillator Local invariance Center manifold Van der Pol slow manifold Chua s slow invariant manifold in (xz)-plane Lorenz slow manifold

9 LIST OF FIGURES Local first integral of Van der Pol model Volterra-Lotka s first integral First integral of quadratic system Van der Pol piecewise linear model slow invariant manifold Chua s piecewise linear model slow invariant manifold FitzHugh-Nagumo model slow invariant manifold Chua s cubic model slow invariant manifold Brusselator s model slow invariant manifold (PRT) model slow invariant manifold Rikitake model slow invariant manifold Chua s model invariant hyperplanes in (x 1, x 2, x 3 ) phase space Chua s model invariant hyperplanes in (x 1, x 2, x 3 ) phase space Chua s model slow invariant manifold Chua s slow invariant manifold Dynamo model slow invariant manifold Mofatt model slow invariant manifold Magnetoconvection slow invariant manifold Chemical kinetics model - Neuronal bursting model Forced Van der Pol model slow invariant manifold Chua s cubic model attractor structure

10 List of Tables 15.1 Invariant manifolds of the (PRT) model Invariant manifolds of the Rikitake model Invariant manifolds of the Lorenz model

11 Index attractor, 46, 47, 251, 255, 260, 262 attractor structure, 256 autonomous, 17, 23, 24, 28, 59, 249, 252, 255 Bendixson, bifurcation, 59, 60, 64, 68, 72, 74, 76, 77, 127, 137, 160, 174, 179, 229, 230, 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, , 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, , 256, 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, , , 181, 192, 197, 202, 254, 255, 263, 264, curvature of the flow, 13 curve, 13, 27, 40 42, 44, 72, 93, 99, 101, 102, 128, , 139, 154, 263, 264, 266, 267, 271 Darboux, 14, 16, 94, 99, 101, , 121, 123, 124, 148, 149, 159, 183, 184, 187, 199, 204, 205, , 221, 255 Darboux invariance theorem, 15, 16, 103, 104, 148, 149, 181, 183, 187, 232, , 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, , 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, 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, , , 118, 119, 127, 128, , , , , , 163, 164, , 171, 174, 175, 178, , 185, 188, 192, 193, 197, , 205, 207, 208, 210, , 228, 229, 254, 255, 257, 264, 266, , 277, 278, 280 Fenichel, 15, 78, 79, 81, 82, 85, 88, 91, 180, 185, 187, 254 first integral, 14 16, 49 51, , , 108, 110, 111, , , 148, 149, 156, , 221, 224, 225, 254, 255, 278, 279 FitzHugh-Nagumo, 16, 218, 235, 280, 281

12 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, , , 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, , 250, 253, 256, 260 piecewise linear model, 16, 80, , , 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, 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, , 254, 255, 281 slow invariant manifold, 14 17, 78, 82, 83, 85, 88, 90, 91, 137, 180, 182, , 191, 193, 194, 196, 220, 232, 233, , , slow-fast dynamical systems, 14, 39, 78, 89, 90, 180, 181, 184, 185, 192, 231, 237, 255, 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, tangent linear system approximation, 17, , 272, torsion, 131, , 154, 181, 197, trajectory curve, 13, 14, 24 26, 42, 44, 46, 47, 49 53, 101, , 136, 137, 155, 156, 181, 201, 202, , 260, 261, 263, 264, transcritical, 137, 160, 170, 179 transcritical bifurcation, Van der Pol, 14, 16, 27, 39, 41, 43, 44, 190, 192, 200, 201, 211, 215, 218, 231, 232, 235, 252, 253, Volterra, 27, 28, 37, 118, 153, 156, 204, 279

13 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, , 185, 214, 216, 218, 219, 234, 241, 243, 256 fixed point stability, 15, 254 flow curvature manifold, 13, 15, 16, 127, , 146, 148, , 156, 157, , 163, 164, , 170, 171, 174, 175, 179, 182, 184, 186, 188, , 196, 197, 201, 202, 204, 206, 207, , , 224, 225, 227, , 243, 244, , 252, flow curvature method, 13, 15, 17, 90, 109, 127, 136, 137, 159, 185, 186, 190, 192, , 200, 252, , 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, , , 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, , 127, 148, 159, 199, 205, 254, 255, invariant manifold, 54, 56 58, 60, 71, 78, 104, 105, 107, 108, 110, , 127, 137, , , 180, 184, 199, 201, 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, , 145, 146, 161, , 234, 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, Lie derivative, 14, 15, 42, 49, 94, 119, , 149, 150, 153, 154, 157, 158, 184, 187, 192, 197, 200, 201, 221, 226, 232, 234, , 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, , 159, 175, , 201, 233, , 244, 245, 258, linear invariant manifold, 15, 153, 156, , 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, Lorenz butterfly, 48, 277 Lorenz slow manifold, 198 magnetoconvection, 16, 249, 250, 281 Mofatt, 16, 246, 248, 249, 281

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