Introduction / 1: |
References |
Preliminary Results / 2: |
Linear Functional Analysis / 2.1: |
Nonlinear Functional Analysis / 2.2: |
Banach Fixed Point Theorem / 2.2.1: |
Implicit Function Theorem / 2.2.2: |
Lyapunov-Schmidt Method / 2.2.3: |
Brouwer Degree / 2.2.4: |
Local Invertibility / 2.2.5: |
Global Invertibility / 2.2.6: |
Multivalued Mappings / 2.3: |
Differential Topology / 2.4: |
Differentiable Manifolds / 2.4.1: |
Vector Bundles / 2.4.2: |
Tubular Neighbourhoods / 2.4.3: |
Dynamical Systems / 2.5: |
Homogenous Linear Equations / 2.5.1: |
Chaos in Diffeomorphisms / 2.5.2: |
Periodic ODEs / 2.5.3: |
Vector Fields / 2.5.4: |
Global Center Manifolds / 2.5.5: |
Two-Dimensional Flows / 2.5.6: |
Averaging Method / 2.5.7: |
Carathéodory Type ODEs / 2.5.8: |
Singularities of Smooth Maps / 2.6: |
Jet Bundles / 2.6.1: |
Transversality / 2.6.2: |
Malgrange Preparation Theorem / 2.6.4: |
Complex Analysis / 2.6.5: |
Chaos in Discrete Dynamical Systems / 3: |
Transversal Bounded Solutions / 3.1: |
Difference Equations / 3.1.1: |
Variational Equation / 3.1.2: |
Perturbation Theory / 3.1.3: |
Bifurcation from a Manifold of Homoclinic Solutions / 3.1.4: |
Applications to Impulsive Differential Equations / 3.1.5: |
Transversal Homoclinic Orbits / 3.2: |
Higher Dimensional Difference Equations / 3.2.1: |
Bifurcation Result / 3.2.2: |
Applications to McMillan Type Mappings / 3.2.3: |
Planar Integrable Maps with Separatrices / 3.2.4: |
Singular Impulsive ODEs / 3.3: |
Singular ODEs with Impulses / 3.3.1: |
Linear Singular ODEs with Impulses / 3.3.2: |
Derivation of the Melnikov Function / 3.3.3: |
Examples of Singular Impulsive ODEs / 3.3.4: |
Singularly Perturbed Impulsive ODEs / 3.4: |
Singularly Perturbed ODEs with Impulses / 3.4.1: |
Melnikov Function / 3.4.2: |
Second Order Singularly Perturbed ODEs with Impulses / 3.4.3: |
Inflated Deterministic Chaos / 3.5: |
Inflated Dynamical Systems / 3.5.1: |
Inflated Chaos / 3.5.2: |
Chaos in Ordinary Differential Equations / 4: |
Higher Dimensional ODEs / 4.1: |
Parameterized Higher Dimensional ODEs / 4.1.1: |
Variational Equations / 4.1.2: |
Melnikov Mappings / 4.1.3: |
The Second Order Melnikov Function / 4.1.4: |
Application to Periodically Perturbed ODEs / 4.1.5: |
ODEs with Nonresonant Center Manifolds / 4.2: |
Parameterized Coupled Oscillators / 4.2.1: |
Chaotic Dynamics on the Hyperbolic Subspace / 4.2.2: |
Chaos in the Full Equation / 4.2.3: |
Applications to Nonlinear ODEs / 4.2.4: |
ODEs with Resonant Center Manifolds / 4.3: |
ODEs with Saddle-Center Parts / 4.3.1: |
Example of Coupled Oscillators at Resonance / 4.3.2: |
General Equations / 4.3.3: |
Singularly Perturbed and Forced ODEs / 4.3.4: |
Forced Singular ODEs / 4.4.1: |
Center Manifold Reduction / 4.4.2: |
ODEs with Normal and Slow Variables / 4.4.3: |
Homoclinic Hopf Bifurcation / 4.4.4: |
Bifurcation from Degenerate Homoclinics / 4.5: |
Periodically Forced ODEs with Degenerate Homoclinics / 4.5.1: |
Bifurcation Equation / 4.5.2: |
Bifurcation for 2-Parametric Systems / 4.5.3: |
Bifurcation for 4-Parametric Systems / 4.5.4: |
Autonomous Perturbations / 4.5.5: |
Inflated ODEs / 4.6: |
Inflated Carathéodory Type ODEs / 4.6.1: |
Inflated Periodic ODEs / 4.6.2: |
Inflated Autonomous ODEs / 4.6.3: |
Nonlinear Diatomic Lattices / 4.7: |
Forced and Coupled Nonlinear Lattices / 4.7.1: |
Spatially Localized Chaos / 4.7.2: |
Chaos in Partial Differential Equations / 5: |
Beams on Elastic Bearings / 5.1: |
Weakly Nonlinear Beam Equation / 5.1.1: |
Setting of the Problem / 5.1.2: |
Chaotic Solutions / 5.1.3: |
Useful Numerical Estimates / 5.1.5: |
Lipschitz Continuity / 5.1.6: |
Infinite Dimensional Non-Resonant Systems / 5.2: |
Buckled Elastic Beam / 5.2.1: |
Abstract Problem / 5.2.2: |
Chaos on the Hyperbolic Subspace / 5.2.3: |
Applications to Vibrating Elastic Beams / 5.2.4: |
Planer Motion with One Buckled Mode / 5.2.6: |
Nonplaner Symmetric Beams / 5.2.7: |
Nonplaner Nonsymmetric Beams / 5.2.8: |
Multiple Buckled Modes / 5.2.9: |
Periodically Forced Compressed Beam / 5.3: |
Resonant Compressed Equation / 5.3.1: |
Formulation of Weak Solutions / 5.3.2: |
Chaos in Discontinuous Differential Equations / 5.3.3: |
Transversal Homoclinic Bifurcation / 6.1: |
Discontinuous Differential Equations / 6.1.1: |
Geometric Interpretation of Nondegeneracy Condition / 6.1.2: |
Orbits Close to the Lower Homoclinic Branches / 6.1.4: |
Orbits Close to the Upper Homoclinic Branch / 6.1.5: |
Chaotic Behaviour / 6.1.6: |
Almost and Quasiperiodic Cases / 6.1.8: |
Periodic Case / 6.1.9: |
Piecewise Smooth Planar Systems / 6.1.10: |
3D Quasiperiodic Piecewise Linear Systems / 6.1.11: |
Multiple Transversal Crossings / 6.1.12: |
Sliding Homoclinic Bifurcation / 6.2: |
Higher Dimensional Sliding Homoclinics / 6.2.1: |
Planar Sliding Homoclinics / 6.2.2: |
Three-Dimensional Sliding Homoclinics / 6.2.3: |
Outlook / 6.3: |
Concluding Related Topics / 7: |
Notes on Melnikov Function / 7.1: |
Role of Melnikov Function / 7.1.1: |
Melnikov Function and Calculus of Residues / 7.1.2: |
Second Order ODEs / 7.1.3: |
Applications and Examples / 7.1.4: |
Transverse Heteroclinic Cycles / 7.2: |
Blue Sky Catastrophes / 7.3: |
Symmetric Systems with First Integrals / 7.3.1: |
D'Alembert and Penalized Equations / 7.3.2: |
Index |