Asymptotically Autonomous Maps: Heteroclinic connections, Splitting and Phase-induced tipping

Abstract

We consider discrete-time dynamical systems generated by the iteration of asymptotically autonomous maps, whose asymptotic behavior may be either conservative or dissipative. These systems are of particular interest in the modeling of complex phenomena with time-dependent parameter variation. With an appropriate time compactification, the past and future infinity become normally hyperbolic invariant hyperplanes, and thus the plausible asymptotic states of the system can be determined by considering the appropriate fibers of the stable and unstable foliations or suitable sections of the asymptotic manifolds to their compact invariant subsets. In this context, we explore how tipping induced by a change of phase--namely, how a phase shift affects the evolution of the system--is encoded in the unstable manifolds of compact invariant subsets of the past equation. We further connect the tipping phenomenon to the splitting of invariant manifolds and identify certain system reversibilities able to prevent tipping. The theoretical framework we use clarifies the relationships among the various notions of tipping induced by phase-change introduced in the literature.