Climate change in mechanical systems: the snapshot view of parallel dynamical evolutions

Abstract

We argue that typical mechanical systems subjected to a monotonous parameter drift whose timescale is comparable to that of the internal dynamics can be considered to undergo their own climate change. Because of their chaotic dynamics, there are many permitted states at any instant, and their time dependence can be followed—in analogy with the real climate—by monitoring parallel dynamical evolutions originating from different initial conditions. To this end an ensemble view is needed, enabling one to compute ensemble averages characterizing the instantaneous state of the system. We illustrate this on the examples of (i) driven dissipative and (ii) Hamiltonian systems and of (iii) non-driven dissipative ones. We show that in order to find the most transparent view, attention should be paid to the choice of the initial ensemble. While the choice of this ensemble is arbitrary in the case of driven dissipative systems (i), in the Hamiltonian case (ii) either KAM tori or chaotic seas should be taken, and in the third class (iii) the best choice is the KAM tori of the dissipation-free limit. In all cases, the time evolution of the chosen ensemble on snapshots illustrates nicely the geometrical changes occurring in the phase space, including the strengthening, weakening or disappearance of chaos. Furthermore, we show that a Smale horseshoe (a chaotic saddle) that is changing in time is present in all cases. Its disappearance is a geometrical sign of the vanishing of chaos. The so-called ensemble-averaged pairwise distance is found to provide an easily accessible quantitative measure for the strength of chaos in the ensemble. Its slope can be considered as an instantaneous Lyapunov exponent whose zero value signals the vanishing of chaos. Paradigmatic low-dimensional bistable systems are used as illustrative examples whose driving in (i, ii) is chosen to decay in time in order to maintain an analogy with case (iii) where the total energy decreases all the time.

An ensemble based approach for the effect of climate change on the dynamics of extremes

“…When one of the parameters of the system is explicitly time-dependent (the equivalent of a changing climate), not even the long-term behavior of different single trajectories leads to the same outcome, meaning that this method does not yield representative results. An ensemble approach, however, proves to be useful in determining the snapshot attractor, a set whose shape (and probability distribution) is changing in time in a non-periodic manner (Serquina et al, 2008;Ku et al, 2015;Jánosi et al, 2021). It then follows, that in dissipative systems with explicit time-dependence, like the changing climate, the ensemble method is superior to the single-trajectory one (Tél et al, 2020).…”

Permutation entropy of indexed ensembles: quantifying thermalization dynamics

“…The alternate program to characterize entropy dynamics via formal ρ dynamics [1-3] arising from interest in the quantum limit, or via an ensemble ρ constructed from individual trajectories [4-8] established a connection between thermodynamic entropy growth and the dynamical loss of information about trajectories. Recent discussions use ensemble dynamics to help understand systems with parameter drift as well as to as snapshot techniques to capture the shape of invariant distributions [9,10]. However, progress is hampered since calculating ρ dynamics, either using many individual trajectories OR by propagating partial differential equations, prove computationally challenging.…”

Permutation entropy of indexed ensembles: Quantifying thermalization dynamics

“…The alternate program to characterize entropy dynamics via formal ρ dynamics [1][2][3] arising from interest in the quantum limit, or via an ensemble ρ constructed from individual trajectories [4][5][6][7][8] established a connection between thermodynamic entropy growth and the dynamical loss of information about trajectories. Recent discussions use ensemble dynamics to help understand systems with parameter drift as well as to as snapshot techniques to capture the shape of invariant distributions [9,10]. However, progress is hampered since calculating ρ dynamics, either using many individual trajectories OR by propagating partial differential equations, prove computationally challenging.…”