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 \in \Theta $ for a generic strongly monotone discrete-time dynamical system $\{ {T_\theta ^n :X \to X;n \in \mathbb{Z}_ + } \}$ is studied. Here, X and $\Theta $ are strongly ordered spaces, and the mapping $(x,\theta ) \mapsto T_\theta x$ from $\mathcal{X}$ 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 $\mathcal{S}$. No hyperbolicity hypothesis is assumed. If $\mathcal{X}$ is an open subset of a strongly ordered separable Banach space $\mathcal{V}$, it is proved that (1) $\mu (\mathcal{X} \backslash \mathcal{S}) = 0$ for every Gaussian measure $\mu $ on $\mathcal{V}$; (2) $(x,\theta ) \in \mathcal{S}$ implies $\omega _\theta (x) \times \{ \theta \} \subset \mathcal{S}$, where $\omega _\theta (x)$ denotes the $\omega $-limit set of $x \in X$ under the semigroup $\{ {T_\theta ^n :...