Attractors of Local Semiflows on Topological Spaces

In this paper we introduce a notion of an attractor for local semiflows on topological spaces, which in some cases seems to be more suitable than the existing ones in the literature. Based on this notion we develop a basic attractor theory on topological spaces under appropriate separation axioms. First, we discuss fundamental properties of attractors such as maximality and stability and establish some existence results. Then, we give a converse Lyapunov theorem. Finally, the Morse decomposition of attractors is also addressed.Comment: 22 page

Equilibrium Index of Invariant Sets and Global Static Bifurcation for Nonlinear Evolution Equations

We introduce the notion of equilibrium index for statically isolated invariant sets of the system $u_t+A u=f_\lambda(u)$ on Banach space $X$ (where $A$ is a sectorial operator with compact resolvent) and present a reduction theorem and an index formula for bifurcating invariant sets near equilibrium points. Then we prove a new global static bifurcation theorem where the crossing number $\mathfrak{m}$ may be even. In particular, in case $\mathfrak{m}=2$, we show that the system undergoes either an attractor/repeller bifurcation, or a global static bifurcation. An illustrating example is also given by considering the bifurcations of the periodic boundary value problem of second-order differential equations.Comment: 51 page

Linking Theorems of Local Semiflows on Complete Metric Spaces

In this paper we prove some linking theorems and mountain pass type results for dynamical systems in terms of local semiflows on complete metric spaces. Our results provide an alternative approach to detect the existence of compact invariant sets without using the Conley index theory. They can also be applied to variational problems of elliptic equations without verifying the classical P.S. Condition. As an example, we study the resonant problem of the nonautonomous parabolic equation $u_t-\Delta u-\mu u=f(u)+g(x,t)$ on a bounded domain. The existence of a recurrent solution is proved under some Landesman-Laser type conditions by using an appropriate linking theorem of semiflows. Another example is the elliptic equation $-\Delta u+a(x)u=f(x,u)$ on $R^n$. We prove the existence of positive solutions by applying a mountain pass lemma of semiflows to the parabolic flow of the problem.Comment: We have removed the condition of separability on the phase $X$ and some other restrictions (such as nonsingularity of the flow) in this new versio