DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS
|
|
- Willis Tyler
- 6 years ago
- Views:
Transcription
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
Dynamical Systems Analysis Using Differential Geometry
Dynamical Systems Analysis Using Differential Geometry Jean-Marc GINOUX and Bruno ROSSETTO Laboratoire P.R.O.T.E.E., équipe EBMA / ISO Université du Sud Toulon-Var, B.P. 3, 83957, La Garde, ginoux@univ-tln.fr,
More informationIntroduction to Applied Nonlinear Dynamical Systems and Chaos
Stephen Wiggins Introduction to Applied Nonlinear Dynamical Systems and Chaos Second Edition With 250 Figures 4jj Springer I Series Preface v L I Preface to the Second Edition vii Introduction 1 1 Equilibrium
More informationDYNAMICAL SYSTEMS. I Clark: Robinson. Stability, Symbolic Dynamics, and Chaos. CRC Press Boca Raton Ann Arbor London Tokyo
DYNAMICAL SYSTEMS Stability, Symbolic Dynamics, and Chaos I Clark: Robinson CRC Press Boca Raton Ann Arbor London Tokyo Contents Chapter I. Introduction 1 1.1 Population Growth Models, One Population 2
More informationEE222 - Spring 16 - Lecture 2 Notes 1
EE222 - Spring 16 - Lecture 2 Notes 1 Murat Arcak January 21 2016 1 Licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Essentially Nonlinear Phenomena Continued
More information8.1 Bifurcations of Equilibria
1 81 Bifurcations of Equilibria Bifurcation theory studies qualitative changes in solutions as a parameter varies In general one could study the bifurcation theory of ODEs PDEs integro-differential equations
More informationDIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS
DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS Morris W. Hirsch University of California, Berkeley Stephen Smale University of California, Berkeley Robert L. Devaney Boston University
More informationNonlinear Autonomous Dynamical systems of two dimensions. Part A
Nonlinear Autonomous Dynamical systems of two dimensions Part A Nonlinear Autonomous Dynamical systems of two dimensions x f ( x, y), x(0) x vector field y g( xy, ), y(0) y F ( f, g) 0 0 f, g are continuous
More informationI NONLINEAR EWORKBOOK
I NONLINEAR EWORKBOOK Chaos, Fractals, Cellular Automata, Neural Networks, Genetic Algorithms, Gene Expression Programming, Wavelets, Fuzzy Logic with C++, Java and SymbolicC++ Programs Willi-Hans Steeb
More informationLecture 1: A Preliminary to Nonlinear Dynamics and Chaos
Lecture 1: A Preliminary to Nonlinear Dynamics and Chaos Autonomous Systems A set of coupled autonomous 1st-order ODEs. Here "autonomous" means that the right hand side of the equations does not explicitly
More informationLecture 20: ODE V - Examples in Physics
Lecture 20: ODE V - Examples in Physics Helmholtz oscillator The system. A particle of mass is moving in a potential field. Set up the equation of motion. (1.1) (1.2) (1.4) (1.5) Fixed points Linear stability
More informationDIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS
DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS Morris W. Hirsch University of California, Berkeley Stephen Smale University of California, Berkeley Robert L. Devaney Boston University
More informationOrdinary Differential Equations and Smooth Dynamical Systems
D.V. Anosov S.Kh. Aranson V.l. Arnold I.U. Bronshtein V.Z. Grines Yu.S. Il'yashenko Ordinary Differential Equations and Smooth Dynamical Systems With 25 Figures Springer I. Ordinary Differential Equations
More informationBIFURCATION PHENOMENA Lecture 1: Qualitative theory of planar ODEs
BIFURCATION PHENOMENA Lecture 1: Qualitative theory of planar ODEs Yuri A. Kuznetsov August, 2010 Contents 1. Solutions and orbits. 2. Equilibria. 3. Periodic orbits and limit cycles. 4. Homoclinic orbits.
More informationSlow Manifold of a Neuronal Bursting Model
Slow Manifold of a Neuronal Bursting Model Jean-Marc Ginoux 1 and Bruno Rossetto 2 1 PROTEE Laboratory, Université du Sud, B.P. 2132, 83957, La Garde Cedex, France, ginoux@univ-tln.fr 2 PROTEE Laboratory,
More informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 5. Global Bifurcations, Homoclinic chaos, Melnikov s method Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Motivation 1.1 The problem 1.2 A
More informationMODELING BY NONLINEAR DIFFERENTIAL EQUATIONS
MODELING BY NONLINEAR DIFFERENTIAL EQUATIONS Dissipative and Conservative Processes WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua University of California, Berkeley Series A. Volume
More informationCHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: January 08, 2018, at 08 30 12 30 Johanneberg Kristian
More informationvii Contents 7.5 Mathematica Commands in Text Format 7.6 Exercises
Preface 0. A Tutorial Introduction to Mathematica 0.1 A Quick Tour of Mathematica 0.2 Tutorial 1: The Basics (One Hour) 0.3 Tutorial 2: Plots and Differential Equations (One Hour) 0.4 Mathematica Programs
More informationAlligood, K. T., Sauer, T. D. & Yorke, J. A. [1997], Chaos: An Introduction to Dynamical Systems, Springer-Verlag, New York.
Bibliography Alligood, K. T., Sauer, T. D. & Yorke, J. A. [1997], Chaos: An Introduction to Dynamical Systems, Springer-Verlag, New York. Amann, H. [1990], Ordinary Differential Equations: An Introduction
More informationChaos and Time-Series Analysis
Chaos and Time-Series Analysis Julien Clinton Sprott Department of Physics Universitv of Wisconsin Madison OXTORD UNIVERSITY PRESS Contents Introduction 1.1 Examples of dynamical systems 1.1.1 Astronomical
More informationDynamical Systems. Pierre N.V. Tu. An Introduction with Applications in Economics and Biology Second Revised and Enlarged Edition.
Pierre N.V. Tu Dynamical Systems An Introduction with Applications in Economics and Biology Second Revised and Enlarged Edition With 105 Figures Springer-Verlag Berlin Heidelberg New York London Paris
More information10 Back to planar nonlinear systems
10 Back to planar nonlinear sstems 10.1 Near the equilibria Recall that I started talking about the Lotka Volterra model as a motivation to stud sstems of two first order autonomous equations of the form
More informationTWO DIMENSIONAL FLOWS. Lecture 5: Limit Cycles and Bifurcations
TWO DIMENSIONAL FLOWS Lecture 5: Limit Cycles and Bifurcations 5. Limit cycles A limit cycle is an isolated closed trajectory [ isolated means that neighbouring trajectories are not closed] Fig. 5.1.1
More informationDIFFERENTIAL GEOMETRY APPLICATIONS TO NONLINEAR OSCILLATORS ANALYSIS
DIFFERENTIAL GEOMETR APPLICATIONS TO NONLINEAR OSCILLATORS ANALSIS NDES6, J.M. Ginoux and B. Rossetto Laboratoire P.R.O.T.E.E., I.U.T. de Toulon, Université du Sud, B.P., 8957, La Garde Cedex, France e-mail:
More informationA New Hyperchaotic Attractor with Complex Patterns
A New Hyperchaotic Attractor with Complex Patterns Safieddine Bouali University of Tunis, Management Institute, Department of Quantitative Methods & Economics, 41, rue de la Liberté, 2000, Le Bardo, Tunisia
More informationDifferential Geometry and Mechanics Applications to Chaotic Dynamical Systems
Differential Geometry and Mechanics Applications to Chaotic Dynamical Systems Jean-Marc Ginoux, Bruno Rossetto To cite this version: Jean-Marc Ginoux, Bruno Rossetto. Differential Geometry and Mechanics
More informationCHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: August 22, 2018, at 08 30 12 30 Johanneberg Jan Meibohm,
More informationarxiv: v1 [math.ds] 7 Aug 2014
arxiv:1408.1712v1 [math.ds] 7 Aug 2014 Slow Invariant Manifolds as Curvature of the Flow of Dynamical Systems August 11, 2014 JEAN-MARC GINOUX, BRUNO ROSSETTO, LEON O. CHUA Laboratoire PROTEE, I.U.T. of
More informationMathematical Foundations of Neuroscience - Lecture 7. Bifurcations II.
Mathematical Foundations of Neuroscience - Lecture 7. Bifurcations II. Filip Piękniewski Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland Winter 2009/2010 Filip
More informationSubmanifolds of. Total Mean Curvature and. Finite Type. Bang-Yen Chen. Series in Pure Mathematics Volume. Second Edition.
le 27 AIPEI CHENNAI TAIPEI - Series in Pure Mathematics Volume 27 Total Mean Curvature and Submanifolds of Finite Type Second Edition Bang-Yen Chen Michigan State University, USA World Scientific NEW JERSEY
More informationDynamical Systems with Applications using Mathematica
Stephen Lynch Dynamical Systems with Applications using Mathematica Birkhäuser Boston Basel Berlin Contents Preface xi 0 A Tutorial Introduction to Mathematica 1 0.1 A Quick Tour of Mathematica 2 0.2 Tutorial
More informationOutline. Learning Objectives. References. Lecture 2: Second-order Systems
Outline Lecture 2: Second-order Systems! Techniques based on linear systems analysis! Phase-plane analysis! Example: Neanderthal / Early man competition! Hartman-Grobman theorem -- validity of linearizations!
More informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 4. Bifurcations Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Local bifurcations for vector fields 1.1 The problem 1.2 The extended centre
More informationIndex. Bertrand mate, 89 bijection, 48 bitangent, 69 Bolyai, 339 Bonnet s Formula, 283 bounded, 48
Index acceleration, 14, 76, 355 centripetal, 27 tangential, 27 algebraic geometry, vii analytic, 44 angle at a corner, 21 on a regular surface, 170 angle excess, 337 angle of parallelism, 344 angular velocity,
More informationSimple conservative, autonomous, second-order chaotic complex variable systems.
Simple conservative, autonomous, second-order chaotic complex variable systems. Delmar Marshall 1 (Physics Department, Amrita Vishwa Vidyapeetham, Clappana P.O., Kollam, Kerala 690-525, India) and J. C.
More informationFROM EQUILIBRIUM TO CHAOS
FROM EQUILIBRIUM TO CHAOS Practica! Bifurcation and Stability Analysis RÜDIGER SEYDEL Institut für Angewandte Mathematik und Statistik University of Würzburg Würzburg, Federal Republic of Germany ELSEVIER
More informationDynamical Systems with Applications
Stephen Lynch Dynamical Systems with Applications using MATLAB Birkhauser Boston Basel Berlin Preface xi 0 A Tutorial Introduction to MATLAB and the Symbolic Math Toolbox 1 0.1 Tutorial One: The Basics
More informationOne Dimensional Dynamical Systems
16 CHAPTER 2 One Dimensional Dynamical Systems We begin by analyzing some dynamical systems with one-dimensional phase spaces, and in particular their bifurcations. All equations in this Chapter are scalar
More informationTowards a Global Theory of Singularly Perturbed Dynamical Systems John Guckenheimer Cornell University
Towards a Global Theory of Singularly Perturbed Dynamical Systems John Guckenheimer Cornell University Dynamical systems with multiple time scales arise naturally in many domains. Models of neural systems
More informationUpon successful completion of MATH 220, the student will be able to:
MATH 220 Matrices Upon successful completion of MATH 220, the student will be able to: 1. Identify a system of linear equations (or linear system) and describe its solution set 2. Write down the coefficient
More informationBIFURCATION PHENOMENA Lecture 4: Bifurcations in n-dimensional ODEs
BIFURCATION PHENOMENA Lecture 4: Bifurcations in n-dimensional ODEs Yuri A. Kuznetsov August, 2010 Contents 1. Solutions and orbits: equilibria cycles connecting orbits compact invariant manifolds strange
More informationChapter 24 BIFURCATIONS
Chapter 24 BIFURCATIONS Abstract Keywords: Phase Portrait Fixed Point Saddle-Node Bifurcation Diagram Codimension-1 Hysteresis Hopf Bifurcation SNIC Page 1 24.1 Introduction In linear systems, responses
More informationANALYTICAL MECHANICS. LOUIS N. HAND and JANET D. FINCH CAMBRIDGE UNIVERSITY PRESS
ANALYTICAL MECHANICS LOUIS N. HAND and JANET D. FINCH CAMBRIDGE UNIVERSITY PRESS Preface xi 1 LAGRANGIAN MECHANICS l 1.1 Example and Review of Newton's Mechanics: A Block Sliding on an Inclined Plane 1
More informationQualitative Analysis of Dynamical Systems and Models in Life Science
Ecole Polytechnique UPSay 2015-16 Master de Mécanique M2 Biomécanique Qualitative Analysis of Dynamical Systems and Models in Life Science Alexandre VIDAL Dernière modification : February 5, 2016 2 Contents
More informationOn Universality of Transition to Chaos Scenario in Nonlinear Systems of Ordinary Differential Equations of Shilnikov s Type
Journal of Applied Mathematics and Physics, 2016, 4, 871-880 Published Online May 2016 in SciRes. http://www.scirp.org/journal/jamp http://dx.doi.org/10.4236/jamp.2016.45095 On Universality of Transition
More informationAPPLIED SYMBOLIC DYNAMICS AND CHAOS
DIRECTIONS IN CHAOS VOL. 7 APPLIED SYMBOLIC DYNAMICS AND CHAOS Bai-Lin Hao Wei-Mou Zheng The Institute of Theoretical Physics Academia Sinica, China Vfö World Scientific wl Singapore Sinaaoore NewJersev
More informationUNIVERSIDADE DE SÃO PAULO
UNIVERSIDADE DE SÃO PAULO Instituto de Ciências Matemáticas e de Computação ISSN 010-577 GLOBAL DYNAMICAL ASPECTS OF A GENERALIZED SPROTT E DIFFERENTIAL SYSTEM REGILENE OLIVEIRA CLAUDIA VALLS N o 41 NOTAS
More information= F ( x; µ) (1) where x is a 2-dimensional vector, µ is a parameter, and F :
1 Bifurcations Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 A bifurcation is a qualitative change
More informationNonlinear Dynamics. Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna.
Nonlinear Dynamics Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna http://www.moreno.marzolla.name/ 2 Introduction: Dynamics of Simple Maps 3 Dynamical systems A dynamical
More informationHandout 2: Invariant Sets and Stability
Engineering Tripos Part IIB Nonlinear Systems and Control Module 4F2 1 Invariant Sets Handout 2: Invariant Sets and Stability Consider again the autonomous dynamical system ẋ = f(x), x() = x (1) with state
More informationAnalysis of Dynamical Systems
2 YFX1520 Nonlinear phase portrait Mathematical Modelling and Nonlinear Dynamics Coursework Assignments 1 ϕ (t) 0-1 -2-6 -4-2 0 2 4 6 ϕ(t) Dmitri Kartofelev, PhD 2018 Variant 1 Part 1: Liénard type equation
More informationCONTROLLING IN BETWEEN THE LORENZ AND THE CHEN SYSTEMS
International Journal of Bifurcation and Chaos, Vol. 12, No. 6 (22) 1417 1422 c World Scientific Publishing Company CONTROLLING IN BETWEEN THE LORENZ AND THE CHEN SYSTEMS JINHU LÜ Institute of Systems
More informationModelling biological oscillations
Modelling biological oscillations Shan He School for Computational Science University of Birmingham Module 06-23836: Computational Modelling with MATLAB Outline Outline of Topics Van der Pol equation Van
More informationAPPPHYS217 Tuesday 25 May 2010
APPPHYS7 Tuesday 5 May Our aim today is to take a brief tour of some topics in nonlinear dynamics. Some good references include: [Perko] Lawrence Perko Differential Equations and Dynamical Systems (Springer-Verlag
More informationDynamical Systems in Neuroscience: The Geometry of Excitability and Bursting
Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting Eugene M. Izhikevich The MIT Press Cambridge, Massachusetts London, England Contents Preface xv 1 Introduction 1 1.1 Neurons
More informationPHYSICS 110A : CLASSICAL MECHANICS
PHYSICS 110A : CLASSICAL MECHANICS 1. Introduction to Dynamics motion of a mechanical system equations of motion : Newton s second law ordinary differential equations (ODEs) dynamical systems simple 2.
More information7 Two-dimensional bifurcations
7 Two-dimensional bifurcations As in one-dimensional systems: fixed points may be created, destroyed, or change stability as parameters are varied (change of topological equivalence ). In addition closed
More informationConstructing a chaotic system with any number of equilibria
Nonlinear Dyn (2013) 71:429 436 DOI 10.1007/s11071-012-0669-7 ORIGINAL PAPER Constructing a chaotic system with any number of equilibria Xiong Wang Guanrong Chen Received: 9 June 2012 / Accepted: 29 October
More informationChaotic Vibrations. An Introduction for Applied Scientists and Engineers
Chaotic Vibrations An Introduction for Applied Scientists and Engineers FRANCIS C. MOON Theoretical and Applied Mechanics Cornell University Ithaca, New York A WILEY-INTERSCIENCE PUBLICATION JOHN WILEY
More informationChaos. Lendert Gelens. KU Leuven - Vrije Universiteit Brussel Nonlinear dynamics course - VUB
Chaos Lendert Gelens KU Leuven - Vrije Universiteit Brussel www.gelenslab.org Nonlinear dynamics course - VUB Examples of chaotic systems: the double pendulum? θ 1 θ θ 2 Examples of chaotic systems: the
More informationMathematical Modeling I
Mathematical Modeling I Dr. Zachariah Sinkala Department of Mathematical Sciences Middle Tennessee State University Murfreesboro Tennessee 37132, USA November 5, 2011 1d systems To understand more complex
More informationEQUATIONS WITH APPLICATIONS
ORDINARY DIFFERENTIAL EQUATIONS WITH APPLICATIONS SERIES ON APPLIED MATHEMATICS Editor-in-Chief: Frank Hwang Associate Editors-in-Chief: Zhong-ci Shi and U Rothblum VOl. 1 VOl. 2 VOl. 3 VOl. 4 VOl. 5 Vol.
More informationAN ELECTRIC circuit containing a switch controlled by
878 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 46, NO. 7, JULY 1999 Bifurcation of Switched Nonlinear Dynamical Systems Takuji Kousaka, Member, IEEE, Tetsushi
More informationTHREE DIMENSIONAL SYSTEMS. Lecture 6: The Lorenz Equations
THREE DIMENSIONAL SYSTEMS Lecture 6: The Lorenz Equations 6. The Lorenz (1963) Equations The Lorenz equations were originally derived by Saltzman (1962) as a minimalist model of thermal convection in a
More informationElements of Applied Bifurcation Theory
Yuri A. Kuznetsov Elements of Applied Bifurcation Theory Second Edition With 251 Illustrations Springer Preface to the Second Edition Preface to the First Edition vii ix 1 Introduction to Dynamical Systems
More informationHomework 2 Modeling complex systems, Stability analysis, Discrete-time dynamical systems, Deterministic chaos
Homework 2 Modeling complex systems, Stability analysis, Discrete-time dynamical systems, Deterministic chaos (Max useful score: 100 - Available points: 125) 15-382: Collective Intelligence (Spring 2018)
More informationDynamics of Modified Leslie-Gower Predator-Prey Model with Predator Harvesting
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:05 55 Dynamics of Modified Leslie-Gower Predator-Prey Model with Predator Harvesting K. Saleh Department of Mathematics, King Fahd
More informationHamiltonian Chaos and the standard map
Hamiltonian Chaos and the standard map Outline: What happens for small perturbation? Questions of long time stability? Poincare section and twist maps. Area preserving mappings. Standard map as time sections
More informationState Regulation of Rikitake Two-Disk Dynamo Chaotic System via Adaptive Control Method
International Journal of ChemTech Research CODEN (USA): IJCRGG ISSN: 0974-4290 Vol.8, No.9 pp 374-386, 205 State Regulation of Rikitake Two-Disk Dynamo Chaotic System via Adaptive Control Method Sundarapandian
More informationADAPTIVE CONTROL AND SYNCHRONIZATION OF A GENERALIZED LOTKA-VOLTERRA SYSTEM
ADAPTIVE CONTROL AND SYNCHRONIZATION OF A GENERALIZED LOTKA-VOLTERRA SYSTEM Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University Avadi, Chennai-600
More information7 Planar systems of linear ODE
7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution
More information2 Qualitative theory of non-smooth dynamical systems
2 Qualitative theory of non-smooth dynamical systems In this chapter, we give an overview of the basic theory of both smooth and non-smooth dynamical systems, to be expanded upon in later chapters. In
More informationBifurcation Analysis, Chaos and Control in the Burgers Mapping
ISSN 1749-3889 print, 1749-3897 online International Journal of Nonlinear Science Vol.4007 No.3,pp.171-185 Bifurcation Analysis, Chaos and Control in the Burgers Mapping E. M. ELabbasy, H. N. Agiza, H.
More informationMATHEMATICAL PHYSICS
Advanced Methods of MATHEMATICAL PHYSICS R.S. Kaushal D. Parashar Alpha Science International Ltd. Contents Preface Abbreviations, Notations and Symbols vii xi 1. General Introduction 1 2. Theory of Finite
More informationPart II. Dynamical Systems. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 34 Paper 1, Section II 30A Consider the dynamical system where β > 1 is a constant. ẋ = x + x 3 + βxy 2, ẏ = y + βx 2
More informationFundamentals of Dynamical Systems / Discrete-Time Models. Dr. Dylan McNamara people.uncw.edu/ mcnamarad
Fundamentals of Dynamical Systems / Discrete-Time Models Dr. Dylan McNamara people.uncw.edu/ mcnamarad Dynamical systems theory Considers how systems autonomously change along time Ranges from Newtonian
More information11 Chaos in Continuous Dynamical Systems.
11 CHAOS IN CONTINUOUS DYNAMICAL SYSTEMS. 47 11 Chaos in Continuous Dynamical Systems. Let s consider a system of differential equations given by where x(t) : R R and f : R R. ẋ = f(x), The linearization
More informationJournal of Differential Equations
J. Differential Equations 248 (2010 2841 2888 Contents lists available at ScienceDirect Journal of Differential Equations www.elsevier.com/locate/jde Local analysis near a folded saddle-node singularity
More informationStability and Bifurcation in the Hénon Map and its Generalizations
Chaotic Modeling and Simulation (CMSIM) 4: 529 538, 2013 Stability and Bifurcation in the Hénon Map and its Generalizations O. Ozgur Aybar 1, I. Kusbeyzi Aybar 2, and A. S. Hacinliyan 3 1 Gebze Institute
More informationAdaptive Control of Rikitake Two-Disk Dynamo System
International Journal of ChemTech Research CODEN (USA): IJCRGG ISSN: 0974-4290 Vol.8, No.8, pp 121-133, 2015 Adaptive Control of Rikitake Two-Disk Dynamo System Sundarapandian Vaidyanathan* R & D Centre,
More informationTWELVE LIMIT CYCLES IN A CUBIC ORDER PLANAR SYSTEM WITH Z 2 -SYMMETRY. P. Yu 1,2 and M. Han 1
COMMUNICATIONS ON Website: http://aimsciences.org PURE AND APPLIED ANALYSIS Volume 3, Number 3, September 2004 pp. 515 526 TWELVE LIMIT CYCLES IN A CUBIC ORDER PLANAR SYSTEM WITH Z 2 -SYMMETRY P. Yu 1,2
More informationCyclicity of common slow fast cycles
Available online at www.sciencedirect.com Indagationes Mathematicae 22 (2011) 165 206 www.elsevier.com/locate/indag Cyclicity of common slow fast cycles P. De Maesschalck a, F. Dumortier a,, R. Roussarie
More informationUnit Ten Summary Introduction to Dynamical Systems and Chaos
Unit Ten Summary Introduction to Dynamical Systems Dynamical Systems A dynamical system is a system that evolves in time according to a well-defined, unchanging rule. The study of dynamical systems is
More informationCHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: January 14, 2019, at 08 30 12 30 Johanneberg Kristian
More informationA Novel Three Dimension Autonomous Chaotic System with a Quadratic Exponential Nonlinear Term
ETASR - Engineering, Technology & Applied Science Research Vol., o.,, 9-5 9 A Novel Three Dimension Autonomous Chaotic System with a Quadratic Exponential Nonlinear Term Fei Yu College of Information Science
More informationAdaptive Synchronization of Rikitake Two-Disk Dynamo Chaotic Systems
International Journal of ChemTech Research CODEN (USA): IJCRGG ISSN: 0974-4290 Vol.8, No.8, pp 00-, 205 Adaptive Synchronization of Rikitake Two-Disk Dynamo Chaotic Systems Sundarapandian Vaidyanathan
More informationBifurcation of Fixed Points
Bifurcation of Fixed Points CDS140B Lecturer: Wang Sang Koon Winter, 2005 1 Introduction ẏ = g(y, λ). where y R n, λ R p. Suppose it has a fixed point at (y 0, λ 0 ), i.e., g(y 0, λ 0 ) = 0. Two Questions:
More informationAVERAGING AND RECONSTRUCTION IN HAMILTONIAN SYSTEMS
AVERAGING AND RECONSTRUCTION IN HAMILTONIAN SYSTEMS Kenneth R. Meyer 1 Jesús F. Palacián 2 Patricia Yanguas 2 1 Department of Mathematical Sciences University of Cincinnati, Cincinnati, Ohio (USA) 2 Departamento
More informationHigh-Dimensional Dynamics in the Delayed Hénon Map
EJTP 3, No. 12 (2006) 19 35 Electronic Journal of Theoretical Physics High-Dimensional Dynamics in the Delayed Hénon Map J. C. Sprott Department of Physics, University of Wisconsin, Madison, WI 53706,
More information4 Second-Order Systems
4 Second-Order Systems Second-order autonomous systems occupy an important place in the study of nonlinear systems because solution trajectories can be represented in the plane. This allows for easy visualization
More informationHalf of Final Exam Name: Practice Problems October 28, 2014
Math 54. Treibergs Half of Final Exam Name: Practice Problems October 28, 24 Half of the final will be over material since the last midterm exam, such as the practice problems given here. The other half
More informationThermodynamics of Chaotic systems by C. Beck and F Schlögl (1993) LecturesonGeometryandDynamicalSystemsbyY.PesinandV.Clemenhaga
ME597B/MATH597G/PHYS597C Spring, 2015 Chapter 01: Dynamical Systems Source Thermodynamics of Chaotic systems by C. Beck and F Schlögl (1993) LecturesonGeometryandDynamicalSystemsbyY.PesinandV.Clemenhaga
More information6.2 Brief review of fundamental concepts about chaotic systems
6.2 Brief review of fundamental concepts about chaotic systems Lorenz (1963) introduced a 3-variable model that is a prototypical example of chaos theory. These equations were derived as a simplification
More informationDynamics at infinity and a Hopf bifurcation arising in a quadratic system with coexisting attractors
Pramana J. Phys. 8) 9: https://doi.org/.7/s43-7-55-x Indian Academy of Sciences Dynamics at infinity and a Hopf bifurcation arising in a quadratic system with coexisting attractors ZHEN WANG,,,IRENEMOROZ
More informationNonlinear dynamics & chaos BECS
Nonlinear dynamics & chaos BECS-114.7151 Phase portraits Focus: nonlinear systems in two dimensions General form of a vector field on the phase plane: Vector notation: Phase portraits Solution x(t) describes
More informationThe Forced van der Pol Equation I: The Slow Flow and Its Bifurcations
SIAM J. APPLIED DYNAMICAL SYSTEMS Vol., No., pp. 35 c 3 Society for Industrial and Applied Mathematics Downloaded 4/8/8 to 48.5.3.83. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
More informationStationary radial spots in a planar threecomponent reaction-diffusion system
Stationary radial spots in a planar threecomponent reaction-diffusion system Peter van Heijster http://www.dam.brown.edu/people/heijster SIAM Conference on Nonlinear Waves and Coherent Structures MS: Recent
More informationChaos & Recursive. Ehsan Tahami. (Properties, Dynamics, and Applications ) PHD student of biomedical engineering
Chaos & Recursive Equations (Properties, Dynamics, and Applications ) Ehsan Tahami PHD student of biomedical engineering Tahami@mshdiau.a.ir Index What is Chaos theory? History of Chaos Introduction of
More informationThe Higgins-Selkov oscillator
The Higgins-Selkov oscillator May 14, 2014 Here I analyse the long-time behaviour of the Higgins-Selkov oscillator. The system is ẋ = k 0 k 1 xy 2, (1 ẏ = k 1 xy 2 k 2 y. (2 The unknowns x and y, being
More informationSIMPLE CHAOTIC FLOWS WITH ONE STABLE EQUILIBRIUM
International Journal of Bifurcation and Chaos, Vol. 23, No. 11 (2013) 1350188 (7 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127413501885 SIMPLE CHAOTIC FLOWS WITH ONE STABLE EQUILIBRIUM
More information