Paper detail

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 $λ=λ_0$, then either there exists a one-sided neighborhood $I^-$ of $λ_0$ such that for each $λ\in I^-$, the system bifurcates from the trivial solution to an isolated nonempty compact invariant set $K_λ$ with $0\not\in K_λ$, or there is a one-sided neighborhood $I^+$ of $λ_0$ such that the system undergoes an attractor bifurcation for $λ\in I^+$ from $(0,λ_0)$. 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.