Weak tracking in nonautonomous chaotic systems
arXiv:2007.04675 · doi:10.1103/PhysRevE.102.052210
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 Rössler 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.
arXiv comment15 pages and 5 figures
References in corpus (9)
- Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system
- Parameter shifts for nonautonomous systems in low dimension: Bifurcation- and Rate-induced tipping
- Early-warning indicators for rate-induced tipping
- Basin bifurcations, oscillatory instability and rate-induced thresholds for AMOC in a global oceanic box model
- Rate-induced tipping from periodic attractors: partial tipping and connecting orbits
- Probability of noise- and rate-induced tipping
- Adapting to a Changing Environment: Non-obvious Thresholds in Multi-Scale Systems
- Detecting transient rate-tipping using Steklov averages and Lyapunov vectors
- Tipping Phenomena and Points of No Return in Ecosystems: Beyond Classical Bifurcations