Bifurcation Theory and Methods of Dynamical Systems

The following sections are included:

• Preface

• Contents

The following sections are included:

• Hyperbolicity and Transversality. λ-Lemma
  • Linearization. Hyperbolicity of critical point. Local invariant manifolds
  • Hyperbolicity of periodic solutions. Global stable and unstable manifolds
  • Transversal intersection. λ-lemma

• Structural Stability and Bifurcation
  • Structural stability
  • Ω-Stability
  • Bifurcation

• Codimension and Unfoldings
  • Definition and codimension-1 examples
  • Bogdanov–Takens bifurcation

• Center Manifold Theory
  • Main theorems on center manifolds
  • Properties of center manifolds
  • Center manifolds depending on parameters

The following sections are included:

• Generalized Hopf Bifurcation
  • Focal values and main theorem
  • Examples
  • Applications to Hilbert's 16th problem

• Bifurcation of Multiple Limit Cycles
  • Main theorems
  • Island problem

• Homoclinic and Heteroclinic Bifurcations
  • Nondegenerate case
  • Degenerate case
  • Highly degenerate case

• Poincaré Bifurcation
  • Main theorems
  • Weakened Hilbert's 16th problem
  • Toral trigonometrical polynomial systems

The following sections are included:

• Boundedness of Solutions and Existence of Limit Cycles
  • Concepts of critical point system
  • Boundedness of solutions
  • Existence of limit cycles
  • Non-existence criteria of closed orbits

• Criteria for Deciding the Number of Limit Cycles
  • Preliminary lemmas
  • Generalization of some classical theorems
  • Several new results

• Global Bifurcation of Cubic Liénard Systems
  • General remarks
  • Integrable cases
  • One-critical point case
  • Two-critical point case
  • Three-critical point case

• Global Bifurcation in Some Applied Models
  • FitzHugh's nerve conduction equation
  • A self-excited system
  • Bogdanov-Takens system (continued)
  • Uniqueness of limit cycles for quadratic system (B)_ao2=0

The following sections are included:

• Bifurcation of Periodic Solutions
  • Poincaré maps and uniqueness of periodic solutions
  • The Liapunov-Schmidt method
  • An elementary method

• Method of Averaging and Integral Manifolds
  • Method of averaging
  • Integral manifolds

• Periodic Perturbed Systems on a Plane
  • Saddle-node bifurcation
  • Hyperbolic and semistable limit cycles
  • Generic Hopf bifurcation and local subharmonic solutions

• Hopf Bifurcation of Invariant Torus

• Poincaré Bifurcation of Invariant Torus
  • Case A
  • Case B
  • Case C (General case)

The following sections are included:

• Methods of Bifurcation Functions of Periodic Orbits
  • Liapunov-Schmidt reduction and Hopf bifurcation
  • An elementary bifurcation method
  • Bifurcation at non-semisimple eigenvalues

• Zero and Pure Imaginary Eigenvalues
  • Bifurcation analysis for limit cycles
  • Qualitative results for full system

• Two Pairs of Pure Imaginary Eigenvalues
  • Periodic orbits of reduced planar system
  • Existence of an invariant two-torus
  • Bifurcations of multiple periodic orbits

• Global Bifurcations of Large Periodic Orbits
  • Bifurcations of periodic orbits
  • Autonomous perturbations of linear system

• Uniqueness and Stability of Bifurcated Periodic Orbits

• Chaotic Dynamics Bifurcated by Symmetric Homoclinic Loops
  • Symbolic dynamics and Smale horseshoe
  • Conley-Moser conditions
  • Chaotic dynamics near double symmetric homoclinic loops

• Saddle-Focus Homoclinic Bifurcation. Chaos
  • Normal form and Poincaré map
  • Verification of Conley-Moser conditions
  • Complicated behavior with Hopf bifurcation

• Period-Doubling Bifurcation in Weak Sil'nikov Phenomenon
  • Existence and stability of bifurcated periodic orbits
  • Bifurcation diagram and asymptotic ratio of bifurcation values
  • Subsidiary homoclinic orbits

The following sections are included:

• Exponential Trichotomies

• Melnikov Vector in Higher Dimensions
  • A method using orthogonality condition and transversality theory
  • A more geometrical method

• Orbits Heteroclinic to Invariant Manifolds
  • Fenichel's invariant manifold theory
  • Existence and transversality of singular orbits

• Heteroclinic Orbits in Singular Perturbation Problems
  • Geometric singular perturbation theory
  • Transversal heteroclinic orbits
  • An example

• Heteroclinic to Nonhyperbolic Equilibria
  • Admissible variables and principal normal coordinates
  • Bifurcation equations
  • An example

The following sections are included:

• References

• Index