Methods of Qualitative Theory in Nonlinear Dynamics

The following sections are included:

• Preface

• Contents

The following sections are included:

• Necessary background from the theory of ordinary differential equations

• Dynamical systems. Basic notions

• Qualitative integration of dynamical systems

The following sections are included:

• Notion of an equilibrium state. A linearized system

• Qualitative investigation of 2- and 3-dimensional linear systems

• High-dimensional linear systems. Invariant subspaces

• Behavior of trajectories of a linear system near saddle equilibrium states

• Topological classification of structurally stable equilibrium states

• Stable equilibrium states. Leading and non-leading manifolds

• Saddle equilibrium states. Invariant manifolds

• Solution near a saddle. The boundary-value problem

• Problem of smooth linearization. Resonances

The following sections are included:

• A Poincaré map. A fixed point. Multipliers

• Non-degenerate linear one- and two-dimensional maps

• Fixed points of high-dimensional linear maps

• Topological classification of fixed points

• Properties of nonlinear maps near a stable fixed point

• Saddle fixed points. Invariant manifolds

• The boundary-value problem near a saddle fixed point

• Behavior of linear maps near saddle fixed points. Examples

• Geometrical properties of nonlinear saddle maps

• Normal coordinates in a neighborhood of a periodic trajectory

• The variational equations

• Stability of periodic trajectories. Saddle periodic trajectories

• Smooth equivalence and resonances

• Autonomous normal forms

• The principle of contraction mappings. Saddle maps

The following sections are included:

• Non-autonomous systems

• Theorem on the existence of an invariant torus. The annulus principle

• Theorem on persistence of an invariant torus

• Basics of the theory of circle diffeomorphisms. Synchronization problems

The following sections are included:

• Reduction to the center manifold

• A boundary-value problem

• Theorem on invariant foliation

• Proof of theorems on center manifolds

The following sections are included:

• Center manifold theorem for a homoclinic loop

• The Poincaré map near a homoclinic loop

• Proof of the center manifold theorem near a homoclinic loop

• Center manifold theorem for heteroclinic cycles

The following sections are included:

• Appendix A: SPECIAL FORM OF SYSTEMS NEAR A SADDLE EQUILIBRIUM STATE

• Appendix B: FIRST ORDER ASYMPTOTIC FOR THE TRAJECTORIES NEAR A SADDLE FIXED POINT

• Bibliography

• Index