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 $u_t+Au=f_{\lambda }(u)$ 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.