Methods of Qualitative Theory in Nonlinear Dynamics

The following sections are included:

• INTRODUCTION TO PART II

• Contents

The following sections are included:

• Rough systems on a plane. Andronov–Pontryagin theorem

• The set of center motions

• General classification of center motions

• Remarks on roughness of high-order dynamical systems

• Morse–Smale systems

• Some properties of Morse–Smale systems

The following sections are included:

• Systems of first degree of non-roughness

• Remarks on bifurcations of multi-dimensional systems

• Structurally unstable homoclinic and heteroclinic orbits. Moduli of topological equivalence

• Bifurcations in finite-parameter families of systems. Andronov's setup

The following sections are included:

• The reduction theorems. The Lyapunov functions

• The first critical case

• The second critical case

The following sections are included:

• The reduction of the Poincaré map. Lyapunov functions

• The first critical case

• The second critical case

• The third critical case. Weak resonances

• Strong resonances

• Passage through strong resonance on stability boundary

• Additional remarks on resonances

The following sections are included:

• Bifurcation surface and transverse families

• Bifurcation of an equilibrium state with one zero exponent

• Bifurcations of periodic orbits with multiplier +1

• Bifurcations of periodic orbits with multiplier −1

• Andronov–Hopf bifurcation

• Birth of invariant torus

• Bifurcations of resonant periodic orbits accompanying the birth of invariant torus

The following sections are included:

• Bifurcations of a homoclinic loop to a saddle-node equilibrium state

• Creation of an invariant torus

• The formation of a Klein bottle

• The blue sky catastrophe

• On embedding into the flow

The following sections are included:

• Stability of a separatrix loop on the plane

• Bifurcation of a limit cycle from a separatrix loop of a saddle with non-zero saddle value

• Bifurcations of a separatrix loop with zero saddle value

• Birth of periodic orbits from a homoclinic loop (the case dim W^u=1)

• Behavior of trajectories near a homoclinic loop in the case dim W^u > 1

• Codimension-two bifurcations of homoclinic loops
  • The case A = 0 — inclination-flip
  • The case Γ ⊂ W^SS — orbit-flip
  • The case σ = 0

• Bifurcations of the homoclinic-8 and heteroclinic cycles

• Estimates of the behavior of trajectories near a saddle equilibrium state

The following sections are included:

• Main stability boundaries of equilibrium states and periodic orbits

• Classification of codimension-one boundaries of stability regions
  • Criteria for safe boundaries
  • Criteria for dangerous boundaries

• Dynamically definite and indefinite boundaries of stability regions

The following sections are included:

• Appendix C EXAMPLES, PROBLEMS AND EXERCISES
  • Qualitative integration
  • Rough equilibrium states and stability boundaries
    • Routh-Hurwitz criterion
    • 3D case
  • Periodically forced systems
  • Derivation of normal forms
  • Behavior on stability boundaries
  • Bifurcations of fixed points and periodic orbits
  • Homoclinic bifurcations

• Bibliography

• Index — Parts I & II