Prevalent Behavior of Smooth Strongly Monotone Discrete-Time Dynamical Systems

For C1-smooth strongly monotone discrete-time dynamical systems, it is shown that ``convergence to linearly stable cycles" is a prevalent asymptotic behavior in the measuretheoretic sense. The results are then applied to classes of time-periodic parabolic equations and give new results on prevalence of convergence to periodic solutions. In particular, for equations with Neumann boundary conditions on convex domains, we show the prevalence of the set of initial conditions corresponding to the solutions that converge to spatiallyhomogeneous periodic solutions. While, for equations on radially symmetric domains, we obtain the prevalence of the set of initial values corresponding to solutions that are asymptotic to radially symmetric periodic solutions.

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