An infinite-dimensional Morse-Smale map

Abstract

A semi-implicit discretization on time of the one-dimensional parabolic equationu _t=u_xx+f(u(x)), (x, t) ∈ (0, 1)×R⁺, with Dirichlet boundary conditions, gives rise to an infinite dimensional map Φ _h (u): starting from$\frac{{u_n + 1 - u_n }}{h} = \Delta u_{n + 1} + f(u_n ),\Delta = \frac{{d^2 }}{{dx^2 }},h > 0$, we define a discrete flow on the Hilbert spaceH ¹₀ (0, 1), given by Φ _h (u) = (I -hΔ)^-1(u+h f o u). The corresponding flow has gradient structure, compact attractor, lap number and Morse-Smale properties and structural stability with respect to the attractor.