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Introduction to Dynamic Bifurcation Theory

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Bifurcation Theory of Functional Differential Equations

Part of the book series: Applied Mathematical Sciences ((AMS,volume 184))

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Abstract

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: \(\Lambda \times U \rightarrow {\mathbb{R}}^{n}\):

$$\displaystyle{ \dot{x} = f(\mu,x),\quad \mu \in \Lambda \subseteq {\mathbb{R}}^{m},\quad x \in U \subseteq {\mathbb{R}}^{n}, }$$
(1.1)

where U and Λ are open sets, x is the state variable, and μ is the (bifurcation) parameter.

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Notes

  1. 1.

    Two smooth manifolds M, \(N \in {\mathbb{R}}^{n}\) intersect transversally if there exist n linearly independent vectors that are tangent to at least one of these manifolds at every intersection point.

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Authors and Affiliations

  1. College of Mathematics and Econometrics, Hunan University, Changsha, Hunan, China, People’s Republic

    Shangjiang Guo

  2. Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada

    Jianhong Wu

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  1. Shangjiang Guo
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Guo, S., Wu, J. (2013). Introduction to Dynamic Bifurcation Theory. In: Bifurcation Theory of Functional Differential Equations. Applied Mathematical Sciences, vol 184. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6992-6_1

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