Stochastic stability of invariant measures: The 2D Euler equation

In finite-dimensional dissipative dynamical systems, stochastic stability provides the selection of the physically relevant measures. That this might also apply to systems defined by partial differential equations, both dissipative and conservative, is the inspiration for this work. As an example, the 2D Euler equation is studied. Among other results this study suggests that the coherent structures observed in 2D hydrodynamics are associated with configurations that maximize stochastically stable measures uniquely determined by the boundary conditions in dynamical space.

The onset of chaos in nonautonomous dissipative dynamical systems: a low-order ocean-model case study

A four-dimensional nonlinear spectral ocean model is used to study the transition to chaos induced by periodic forcing in systems that are nonchaotic in the autonomous limit. The analysis relies on the construction of the system's pullback attractors (PBAs) through ensemble simulations, based on a large number of initial states in the remote past. A preliminary analysis of the autonomous system is carried out by investigating its bifurcation diagram, as well as by calculating a metric that measures the mean distance between two initially nearby trajectories, along with the system's entropy. We find that nonchaotic attractors can still exhibit sensitive dependence on initial data over some time interval; this apparent paradox is resolved by noting that the dependence only concerns the phase of the periodic trajectories, and that it disappears once the latter have converged onto the attractor. The periodically forced system, analyzed by the same methods, yields periodic or chaotic PBAs depending on the periodic forcing's amplitude ε. A new diagnostic method – based on the cross-correlation between two initially nearby trajectories – is proposed to characterize the transition between the two types of behavior. Transition to chaos is found to occur abruptly at a critical value εc and begins with the intermittent emergence of periodic oscillations with distinct phases. The same diagnostic method is finally shown to be a useful tool for autonomous and aperiodically forced systems as well.

Time Quasilattices in Dissipative Dynamical Systems

We establish the existence of ‘time quasilattices’ as stable trajectories in dissipative dynamical systems. These tilings of the time axis, with two unit cells of different durations, can be generated as cuts through a periodic lattice spanned by two orthogonal directions of time. We show that there are precisely two admissible time quasilattices, which we term the infinite Pell and Clapeyron words, reached by a generalization of the period-doubling cascade. Finite Pell and Clapeyron words of increasing length provide systematic periodic approximations to time quasilattices which can be verified experimentally. The results apply to all systems featuring the universal sequence of periodic windows. We provide examples of discrete-time maps, and periodically-driven continuous-time dynamical systems. We identify quantum many-body systems in which time quasilattices develop rigidity via the interaction of many degrees of freedom, thus constituting dissipative discrete ‘time quasicrystals’.