Contents -- Preface -- Acknowledgments -- List of Figures -- List of Examples -- Dynamical Systems -- 1. Differential Equations -- 1.1 Galileo's pendulum -- 1.2 D'Alembert transformation -- 1.3 From differential equations to dynamical systems -- 2. Dynamical Systems -- 2.1 State space - phase space -- 2.2 Definition -- 2.3 Existence and uniqueness -- 2.4 Flow, fixed points and null-clines -- 2.5 Stability theorems -- 2.5.1 Linearized system -- 2.5.2 Hartman-Grobman linearization theorem -- 2.5.3 Liapouno. stability theorem -- 2.6 Phase portraits of dynamical systems -- 2.6.1 Two-dimensional systems -- 2.6.2 Three-dimensional systems -- 2.7 Various types of dynamical systems -- 2.7.1 Linear and nonlinear dynamical systems -- 2.7.2 Homogeneous dynamical systems -- 2.7.3 Polynomial dynamical systems -- 2.7.4 Singularly perturbed systems -- 2.7.5 Slow-Fast dynamical systems -- 2.8 Two-dimensional dynamical systems -- 2.8.1 Poincare index -- 2.8.2 Poincare contact theory -- 2.8.3 Poincare limit cycle -- 2.8.4 Poincare-Bendixson Theorem -- 2.9 High-dimensional dynamical systems -- 2.9.1 Attractors -- 2.9.2 Strange attractors -- 2.9.3 First integrals and Lie derivative -- 2.10 Hamiltonian and integrable systems -- 2.10.1 Hamiltonian dynamical systems -- 2.10.2 Integrable system -- 2.10.3 K.A.M. Theorem -- 3. Invariant Sets -- 3.1 Manifold -- 3.1.1 Definition -- 3.1.2 Existence -- 3.2 Invariant sets -- 3.2.1 Global invariance -- 3.2.2 Local invariance -- 4. Local Bifurcations -- 4.1 CenterManifold Theorem -- 4.1.1 Center manifold theorem for flows -- 4.1.2 Center manifold approximation -- 4.1.3 Center manifold depending upon a parameter -- 4.2 Normal FormTheorem. -- 4.3 Local Bifurcations of Codimension 1 -- 4.3.1 Saddle-node bifurcation -- 4.3.2 Transcritical bifurcation -- 4.3.3 Pitchfork bifurcation -- 4.3.4 Hopf bifurcation.

5. Slow-Fast Dynamical Systems -- 5.1 Introduction -- 5.2 Geometric Singular Perturbation Theory -- 5.2.1 Assumptions -- 5.2.2 Invariance -- 5.2.3 Slow invariant manifold -- 5.3 Slow-fast dynamical systems - Singularly perturbed systems -- 5.3.1 Singularly perturbed systems -- 5.3.2 Slow-fast autonomous dynamical systems -- 6. Integrability -- 6.1 Integrability conditions, integrating factor, multiplier -- 6.1.1 Two-dimensional dynamical systems -- 6.1.2 Three-dimensional dynamical systems -- 6.2 First integrals - Jacobi's last multiplier theorem -- 6.2.1 First integrals -- 6.2.2 Jacobi's last multiplier theorem -- 6.3 Darboux theory of integrability -- 6.3.1 Algebraic particular integral - General integral -- 6.3.2 General integral -- 6.3.3 Multiplier -- 6.3.4 Algebraic particular integral and fixed points -- 6.3.5 Homogeneous polynomial dynamical systems of degree m -- 6.3.6 Homogeneous polynomial dynamical systems of degree two -- 6.3.7 Planar polynomial dynamical systems -- Differential Geometry -- 7. Differential Geometry -- 7.1 Concept of curves - Kinematics vector functions -- 7.1.1 Trajectory curve -- 7.1.2 Instantaneous velocity vector -- 7.1.3 Instantaneous acceleration vector -- 7.2 Gram-Schmidt process - Generalized Frénet moving frame -- 7.2.1 Gram-Schmidt process -- 7.2.2 Generalized Frénetmoving frame -- 7.3 Curvatures of trajectory curves - Osculating planes -- 7.4 Curvatures and osculating plane of space curves -- 7.4.1 Frenet trihedron - Serret-Frenet formulae -- 7.4.2 Osculating plane -- 7.4.3 Curvatures of space curves -- 7.5 Flow curvaturemethod -- 7.5.1 Flow curvature manifold -- 7.5.2 Flow curvaturemethod -- 8. Dynamical Systems -- 8.1 Phase portraits of dynamical systems -- 8 .1.1 Fixed points -- 8.1.2 Stability theorems -- 9. Invariant Sets -- 9.1 Invariantmanifolds -- 9.1.1 Global invariance -- 9.1.2 Local invariance.

9.2 Linear invariantmanifolds -- 9.3 Nonlinear invariantmanifolds -- 10. Local Bifurcations -- 10.1 Center Manifold -- 10.1.1 Center manifold approximation -- 10.1.2 Center manifold depending upon a parameter -- 10.2 Normal Form Theorem. -- 10.3 Local bifurcations of codimension 1 -- 11. Slow-Fast Dynamical Systems -- 11.1 Slow manifold of n-dimensional slow-fast dynamical systems -- 11.2 Invariance -- 11.3 Flow Curvature Method - Singular Perturbation Method -- 11.3.1 Darboux invariance - Fenichel's invariance -- 11.3.2 Slow invariant manifold -- 11.4 Non-singularly perturbed systems -- 12. Integrability -- 12.1 First integral -- 12.1.1 Global first integral -- 12.1.2 Local first integral -- 12.2 Linear invariant manifolds as first integral -- 12.3 Darboux theory of integrability -- 12.3.1 General integral - Multiplier -- 12.3.2 Darboux homogeneous polynomial dynamical systems of degree two -- 12.3.3 Planar polynomial dynamical systems -- 13. Inverse Problem -- 13.1 Flow curvature manifold of polynomial dynamical systems -- 13.1.1 Two-dimensional polynomial dynamical systems -- 13.1.2 Three-dimensional polynomial dynamical systems -- 13.2 Flow curvature manifold symmetry (parity) -- 13.2.1 Two-dimensional polynomial dynamical systems -- 13.2.2 n-dimensional polynomial dynamical systems -- 13.3 Inverse problem for polynomial dynamical systems -- 13.3.1 Two-dimensional polynomial dynamical systems -- 13.3.2 Three-dimensional polynomial dynamical systems -- Applications -- 14. Dynamical Systems -- 14.1 FitzHugh-Nagumomodel -- 14.2 Pikovskii-Rabinovich-Trakhtengerts model -- 15. Invariant Sets - Integrability -- 15.1 Pikovskii-Rabinovich-Trakhtengerts model -- 15.2 Rikitakemodel -- 15.3 Chua'smodel -- 15.4 Lorenzmodel -- 16. Local Bifurcations -- 16.1 Chua'smodel -- 16.2 Lorenz model -- 17. Slow-Fast Dynamical Systems - Singularly Perturbed Systems.

17.1 Piecewise Linear Models 2D & 3D -- 17.1.1 Van der Pol piecewise linear model -- 17.1.2 Chua's piecewise linear model -- 17.2 Singularly Perturbed Systems 2D & 3D -- 17.2.1 FitzHugh-Nagumo model -- 17.2.2 Chua's model -- 17.3 Slow Fast Dynamical Systems 2D & 3D -- 17.3.1 Brusselator model -- 17.3.2 Pikovskii-Rabinovich-Trakhtengerts model -- 17.3.3 Rikitake model -- 17.4 Piecewise Linear Models 4D & 5D -- 17.4.1 Chua's fourth-order piecewise linear model -- 17.4.2 Chua's .fth-order piecewise linear model -- 17.5 Singularly Perturbed Systems 4D & 5D -- 17.5.1 Chua's fourth-order cubic model -- 17.5.2 Chua's fifth-order cubic model -- 17.6 Slow Fast Dynamical Systems 4D & 5D -- 17.6.1 Homopolar dynamo model -- 17.6.2 Mofatt model -- 17.6.3 Magnetoconvection model -- 17.7 Slow manifold gallery -- 17.8 Forced Van der Polmodel -- Discussion -- Appendix A -- A.1 Lie derivative -- A.2 Hessian -- A.3 Jordan form -- A.4 Connected region -- A.5 Fractal dimension -- A.5.1 Kolmogorov or capacity dimension -- A.5.2 Liapounoff exponents - Wolf, Swinney, Vastano algorithm -- A.5.3 Liapounoff dimension and Kaplan-Yorke conjecture -- A.5.4 Liapounoff dimension and Chlouverakis-Sprott conjecture -- A.6 Identities -- A.6.1 Concept of curves -- A.6.2 Gram-Schmidt process and Frénet moving frame -- A.6.3 Frénet trihedron and curvatures of space curves -- A.6.4 First identity -- A.6.5 Second identity -- A.6.6 Third identity -- A.7 Homeomorphismand diffeomorphism. -- A.7.1 Homeomorphism. -- A.7.2 Diffeomorphism -- A.8 Differential equations -- A.8.1 Two-dimensional dynamical systems -- A.8.2 Three-dimensional dynamical systems -- A.9 Generalized Tangent Linear System Approximation -- A.9.1 Assumptions -- A.9.2 Corollaries -- Mathematica Files -- Bibliography -- Index.