A Remark on Attractor Bifurcation

TL;DR

This paper investigates local bifurcation phenomena in nonlinear evolution equations, establishing conditions for bifurcation types, modifying existing theorems, and applying results to the Swift-Hohenberg equation.

Contribution

It provides new bifurcation criteria based on invariant sets, modifies the attractor bifurcation theorem, and demonstrates applications to specific evolution equations.

Findings

Bifurcation from trivial solutions depends on invariant set properties.

Modified attractor bifurcation theorem extends classical results.

Application to Swift-Hohenberg equation illustrates practical relevance.

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.