Static and Dynamic Bifurcations

Synonyms

Center manifold; Hopf bifurcation; Normal form; Singularity; Stability

Definitions

Steady solutions of differential equations may lose their stability under parameter variations, and new solution types, e.g., periodic solutions, may emerge. To explore the dynamics close to the loss of stability, the originally high-dimensional system is reduced to a low-dimensional set of bifurcation equations by center manifold theory. The reduced system can be simplified further by normal form theory. These methods are demonstrated for the Hopf bifurcation, when a pair of complex eigenvalues crosses the imaginary axis and a family of periodic solutions branches off from the static equilibrium.

Introduction

We consider systems of ordinary differential equations (ODEs)

$$\displaystyle \begin{aligned} \dot{\boldsymbol{x}}=\boldsymbol{f}(t, \boldsymbol{x}; \boldsymbol{\lambda}), \end{aligned} $$

(1)

where x(t) ∈Rⁿ denotes the state vector, the time t is the independent variable, and λ ∈R^m...