Robust dynamics of non-autonomous systems with nested invariant cone structures

Abstract

We develop a perturbation theory for skew-product semiflows with nested invariant cone (NIC) structures over compact almost periodic minimal flows. The theory describes how the restrictions imposed by these structures on asymptotic dynamics persist under small $C^1$ perturbations. We first establish a trichotomy for differences of perturbed trajectories: finite-time coincidence, exponential contraction outside the terminal cone, or eventual separation within a fixed cone layer. On compact invariant sets near the reference set, the perturbed semiflow extends to a flow. Under a transversality condition, compact invariant sets whose orbit differences remain in fixed cone layers admit finite-dimensional fibre realizations over the original base flow. In the codimension-one case, every compact minimal set in the perturbative neighborhood is an almost $1$-cover of the base. For precompact positive orbits whose $\omega$-limit sets lie in this neighborhood, we prove that each $\omega$-limit set contains at most two minimal sets and classify the remaining orbits by their forward and backward limit sets. Applications include nonlocal and chemotaxis-type perturbations of scalar parabolic equations with homogeneous Neumann boundary conditions and perturbations of competitive--cooperative tridiagonal systems. Under dissipativity assumptions, these applications give global conclusions for almost periodically forced equations, including perturbations that destroy the original zero-number or sign-variation mechanism.

arXiv comment60pages