Introduction to Dynamic Bifurcation Theory

  • Shangjiang Guo
  • Jianhong Wu
Part of the Applied Mathematical Sciences book series (AMS, volume 184)


The change in the qualitative behavior of solutions as a control parameter (or control parameters) in a system is varied and is known as a bifurcation. When the solutions are restricted to neighborhoods of a given equilibrium, a bifurcation occurs often when the zero solution of the linearization of the system at the equilibrium changes its stability. To illustrate the basic concepts of bifurcation phenomena, we consider the following continuous dynamical system defined by the C r (r≥1) vector field f: \(\Lambda \times U \rightarrow {\mathbb{R}}^{n}\):
$$\displaystyle{ \dot{x} = f(\mu,x),\quad \mu \in \Lambda \subseteq {\mathbb{R}}^{m},\quad x \in U \subseteq {\mathbb{R}}^{n}, }$$
where U and Λ are open sets, x is the state variable, and μ is the (bifurcation) parameter.


Periodic Solution Periodic Orbit Hopf Bifurcation Unstable Manifold Homoclinic Orbit 
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© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Shangjiang Guo
    • 1
  • Jianhong Wu
    • 2
  1. 1.College of Mathematics and EconometricsHunan UniversityChangshaChina, People’s Republic
  2. 2.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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