Connecting Orbit Structure of Monotone Solutions in the Shadow System

The shadow system \begin{align}u_t= & \varepsilon ?2u_{xx}+f(u)-\xi ,\\ \xi = & \int?{}_{I} g(u, \xi) dx,\end{align}\quad I=[0, 1] is a scalar reaction diffusion equation coupled with an ODE. The extra freedom coming from the ODE drastically influences the solution structure and dynamics as compared to that of a single scalar reaction diffusion system. In fact, it causes secondary bifurcations and coexistence of multiple stable equilibria. Our long term goal is a complete description of the global dynamics on its global attractor as a function of ,f, andg. Since this is still far beyond our capabilities, we focus on describing the dynamics of solutions to the shadow system which are monotone inx, and classify the global connecting orbit structures in the monotone solution space up to the semi-conjugacy. The maximum principle and hence the lap number arguments, which have played a central role in the analysis of one dimensional scalar reaction diffusion equations, cannot be directly applied to the shadow system, although there is a Lyapunov function in an appropriate parameter regime. In order to overcome this difficulty, we resort to the Conley index theory. This method is topological in nature, and allows us to reduce the connection problem to a series of algebraic computations. The semi-conjugacy property can be obtained once the connection problem is solved. The shadow system turns out to exhibit minimal dynamics which displays the mechanism of basic pattern formation, namely it explains the dynamic relation among the trivial rest states (constant solutions) and the event patterns (large amplitude inhomogeneous solutions).