Hopf Bifurcation for an Infinite Delay Functional Equation
Part of the NATO ASI Series book series (volume 37)
In this lecture we discusse a Hopf bifurcation problem for the functional equation
Here x is a vector in R n, the parameter α is an element of a finite dimensional real Banach space A, and F is a mapping from Α × BUC(R;R n) into R n. Moreover, F(α, 0) = 0, so x ≡ 0 is a solution of (1.1). In addition we suppose that the linearization of (1.1) has a one-parameter family of nontrivial periodic solutions at a critical value α 0 of the parameter. Our aim is to show that also the nonlinear equation has a oneparameter family of nontrivial periodic solutions for some values of α close to α 0.
$$ x(t) = F(\alpha ,x_t ),t \in R. $$
KeywordsPeriodic Solution Hopf Bifurcation Functional Differential Equation Smoothness Assumption Bifurcation Equation
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