Connecting orbit structure of monotone solutions in the shadow system (Q1377535)

The authors study the dynamical system given by the following system of reaction-diffusion equations \[ {\partial u\over\partial t}= \varepsilon^2 {\partial^2u\over\partial x^2}+ f(u)- \xi,\quad {d\xi\over dt}= \int_I g(u,\xi)dx \] for \((t, x)\in [0,\infty)\times I\), where \(u\) satisfies the Neumann boundary condition on \([0,\infty)\times \partial I\). Here \(I= (0,1)\). The structure of the attractor (consisting of functions monotone in \(x\)) is investigated. The stability of equilibrium solutions and the connection orbits between them are classified.