The existence of multiple equilibrium points in a global attractor for some symmetric dynamical systems

In this paper, we are mainly concerned with some properties of the global attractor for some symmetric dynamical systems with a Lyapunov function in a Banach space. Under some suitable assumptions, we give an estimate of lower bound of  Z2Z2 index of the global attractor and get the existence of the multiple equilibrium points in the global attractor for the symmetric dynamical systems. As an application of these results, we consider the existence of multiple stationary solutions for some semilinear reaction–diffusion equations.