Overview of the advances in understanding chaos in low-dimensional dynamical systems subjected to parameter drift

Abstract

This paper offers a review while also studying yet unexplored features of the area of chaotic systems subjected to parameter drift of non-negligible rate, an area where the methods of traditional chaos theory are not applicable. Notably, periodic orbit expansion cannot be applied since no periodic orbits exist, nor do long-time limits, since for drifting physical processes the observational time can only be finite. This means that traditional Lyapunov-exponents are also ill-defined. Furthermore, such systems are non-ergodic, time and ensemble averages are different, the ensemble approach being superior to the single-trajectory view. In general, attractors and phase portraits are time-dependent in a non-periodic fashion. We describe the use of general methods which remain nevertheless applicable in such systems. In the phase space, the analysis is based on stable and unstable foliations, their intersections defining a Smale horseshoe, and the intersection points can be identified with the chaotic set governing the core of the drifting chaotic dynamics. Because of the drift, foliations and chaotic sets are also time-dependent, snapshot objects. We give a formal description for the time-dependent natural measure, illustrated by numerical examples. As a quantitative indicator for the strength of chaos, the so-called ensemble-averaged pairwise distance (EAPD) can be evaluated at any time instant. The derivative of this function can be considered the instantaneous (largest) Lyapunov exponent. We show that snapshot chaotic saddles, the central concept of transient chaos, can be identified in drifting systems as the intersections of the foliations, possessing a time-dependent escape rate in general. In dissipative systems, we find that the snapshot attractor coincides with the unstable foliation, and can consist of more than one component. These are a chaotic one, an extended snapshot chaotic saddle, and multiple regular time-dependent attractor points. When constructing the time-dependent basins of attraction of the attractor points, we find that the basin boundaries are time-dependent and fractal-like, containing the stable foliation, and that they can even exhibit Wada properties. In the Hamiltonian case, we study the phenomenon of the break-up of tori due to the drift in terms of both foliations and EAPD functions. We find that time-dependent versions of chaotic seas are not always fully chaotic, they can contain non-chaotic regions. Within such regions we identify time-dependent non-hyperbolic regions, the analogs of sticky zones of classical Hamiltonian phase spaces. We provide approximate formulas for the information dimension of snapshot objects, based on time-dependent Lyapunov exponents and escape rates. Besides these results, we also give possible applications of our methods e.g. in climate science and in the area of Lagrangian Coherent Structures.