A remark on attractor bifurcation

In this paper we present some local dynamic bifurcation results in terms of invariant sets of nonlinear evolution equations. We show that if the trivial solution is an isolated invariant set of the system at the critical value $\lambda = \lambda_0$, then either there exists a one-sided neighborhood $I^{-}$ of $\lambda_0$ such that for each $\lambda \in I^{-}$, the system bifurcates from the trivial solution to an isolated nonempty compact invariant set $K_\lambda$ with $0 \notin K_\lambda$, or there is a one-sided neighborhood $I^{+}$ of $\lambda_0$ such that the system undergoes an attractor bifurcation for $\lambda \in I^{+}$ from $(0, \lambda_0)$. Then we give a modified version of the attractor bifurcation theorem. Finally, we consider the classical Swift–Hohenberg equation and illustrate how to apply our results to a concrete evolution equation.

Keywords

invariant-set bifurcation, attractor bifurcation, nonlinear evolution equation

2010 Mathematics Subject Classification

35B32, 37B30, 37G35

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Received 30 September 2020

Published 10 May 2021