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We introduce the notion of equilibrium index for statically isolatedinvariant 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 reductiontheorem and an index formula for bifurcating invariant sets near equilibriumpoints. Then we prove a new global static bifurcation theorem where thecrossing number $\mathfrak{m}$ may be even. In particular, in case$\mathfrak{m}=2$, we show that the system undergoes either anattractor/repeller bifurcation, or a global static bifurcation. An illustratingexample is also given by considering the bifurcations of the periodic boundaryvalue problem of second-order differential equations.

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