Weak tracking in nonautonomous chaotic systems

TL;DR

This paper investigates a new phenomenon called weak tracking in nonautonomous chaotic systems, where pullback attractors limit to subsets of future attractors, demonstrated in a nonautonomous Rössler system with multiple critical rates.

Contribution

It introduces the concept of weak tracking in nonautonomous chaotic systems and demonstrates its occurrence in a Rössler system, highlighting the complexity of rate-induced transitions.

Findings

Weak tracking occurs in nonautonomous chaotic systems.

Infinitely many critical rates exist where weak tracking happens.

Pullback attractors can limit to subsets of future attractors.

Abstract

Previous studies have shown that rate-induced transitions can occur in pullback attractors of systems subject to "parameter shifts" between two asymptotically steady values of a system parameter. For cases where the attractors limit to equilibrium or periodic orbit in past and future limits of such an nonautonomous systems, these can occur as the parameter change passes through a critical rate. Such rate-induced transitions for attractors that limit to chaotic attractors in past or future limits has been less examined. In this paper, we identify a new phenomenon is associated with more complex attractors in the future limit: weak tracking, where a pullback attractor of the system limits to a proper subset of an attractor of the future limit system. We demonstrate weak tracking in a nonautonomous R\"ossler system, and argue there are infinitely many critical rates at each of which the pullback attracting solution of the system tracks an embedded unstable periodic orbit of the future chaotic attractor. We also state some necessary conditions that are needed for weak tracking.