Chaos in Hamiltonian systems subjected to parameter drift

Abstract

Based on the example of a paradigmatic low-dimensional Hamiltonian system subjected to different scenarios of parameter drifts of non-negligible rates, we show that the dynamics of such systems can best be understood by following ensembles of initial conditions corresponding to tori of the initial system. When such ensembles are followed, toruslike objects called snapshot tori are obtained, which change their location and shape. In their center, one finds a time-dependent, snapshot elliptic orbit. After some time, many of the tori break up and spread over large regions of the phase space; however, one may find some smaller tori, which remain as closed curves throughout the whole scenario. We also show that the cause of torus breakup is the collision with a snapshot hyperbolic orbit and the surrounding chaotic sea, which forces the ensemble to adopt chaotic properties. Within this chaotic sea, we demonstrate the existence of a snapshot horseshoe structure and a snapshot saddle. An easily visualizable condition for torus breakup is found in relation to a specific snapshot stable manifold. The average distance of nearby pairs of points initiated on an original torus at first hardly changes in time but crosses over into an exponential growth when the snapshot torus breaks up. This new phase can be characterized by a novel type of a finite-time Lyapunov exponent, which depends both on the torus and on the scenario followed. Tori not broken up are shown to be the analogs of coherent vortices in fluid flows of arbitrary time dependence, and the condition for breakup can also be demonstrated by the so-called polar rotation angle method.

Climate Change in Mechanical Systems: The Snapshot View of Parallel Dynamical Evolutions

“…One can nevertheless imagine that a snasphot attractor contains the union of an infinite number of special aperiodic orbits which are analogs of the periodic ones in the sense that they keep their (in)stability practically unchanged over long stretches of time. Such points with hyperbolic character shall be called snapshot hyperbolic points (SHPs), as introduced in [29]. Note that analogous objects have been considered in the literature (see e.g.…”

“…Smale horseshoes (or chaotic saddles) are generally thought to be the most essential compenents of both permanent and transient chaos [53,62]. The literature on cases of arbitrary time dependence is sparse but such objects were identified in random maps [45], in time-continuous systems subjected to drivings of changing strength in both dissipative [36,37] and Hamiltonian systems [29]. A system where changing tangles of manifolds can be observed are fluid flows of aperiodic time dependence, as illustrated in a turbulence-related experiment in [47].…”

“…The main issue is that the intersection points of the manifolds must belong to the same instant, which for moving hyperbolic points (i.e. SHPs) implies that the points from which the manifolds are initiated must belong to different instances [45,36,37,29].…”

“…If a plausible picture is needed behind the snapshot view, we can speak of the theory of parallel dynamical evolutions of the systems, in analogy with parallel climate realizations. A recent effort in this direction is the application of the snapshot view to epidemics under changing environmental conditions [41], to aperiodically driven Hamiltonian systems [29,30], to tipping phenomena [38,4], and to advection in open flows of changing intensity [65].…”

“…As time goes on, these ensembles take up different shapes in every instant, thus we call them snapshot tori. Outside tori, time-depedent chaotic regions exist, called snapshot chaotic seas [29,30] and they can also be used as sub-ensembles.…”