A Remark on Attractor Bifurcation

Abstract

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 $0\not\in K_\lambda$, 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.