LARGE-TIME BEHAVIOR OF A TIME-PERIODIC COOPERATIVE SYSTEM OF REACTION-DIFFUSION EQUATIONS DEPENDING ON PARAMETERS

A kind of structural stability with respect to a parameter theta is-an-element-of THETA for a generic strongly monotone discrete-time dynamical system {T(theta)n: X --> X; n is-an-element-of Z+} is studied. Here, X and THETA are strongly ordered spaces, and the mapping (x, theta) bar-arrow-pointing-right T(theta)x from chi into X is assumed to be continuous, strongly monotone and satisfying a compactness hypothesis. A classification of structurally stable points in chi is introduced; the set of all such points is denoted by S. No hyperbolicity hypothesis is assumed. If chi is an open subset of a strongly ordered separable Banach space nu, it is proved that (1) mu(chi/S) = 0 for every Gaussian measure mu on nu; (2) (x, theta) is-an-element-of S implies omega(theta)(x) x {theta} subset-of S, where omega(theta)(x) denotes the omega-limit set of x is-an-element-of X under the semigroup {T(theta)n: n is-an-element-of Z+}; and (3) omega(theta)(x) is a "quasi cycle" for T(theta) whenever (x, theta) is-an-element-of S. These results are applied to a very general strictly cooperative time-periodic system of weakly coupled reaction-diffusion equations with (space- and/or time-dependent) parameters in both the reaction functions and Robin's boundary conditions. Here T(theta) is the period map.