Mitigating long transient time in deterministic systems by resetting (original) (raw)

Timing of transients: quantifying reaching times and transient behavior in complex systems

New Journal of Physics

In dynamical systems, one may ask how long it takes for a trajectory to reach the attractor, i.e. how long it spends in the transient phase. Although for a single trajectory the mathematically precise answer may be infinity, it still makes sense to compare different trajectories and quantify which of them approaches the attractor earlier. In this article, we categorize several problems of quantifying such transient times. To treat them, we propose two metrics, area under distance curve and regularized reaching time, that capture two complementary aspects of transient dynamics. The first, area under distance curve, is the distance of the trajectory to the attractor integrated over time. It measures which trajectories are 'reluctant', i.e. stay distant from the attractor for long, or 'eager' to approach it right away. Regularized reaching time, on the other hand, quantifies the additional time (positive or negative) that a trajectory starting at a chosen initial condition needs to approach the attractor as compared to some reference trajectory. A positive or negative value means that it approaches the attractor by this much 'earlier' or 'later' than the reference, respectively. We demonstrated their substantial potential for application with multiple paradigmatic examples uncovering new features.

Slowing Down of So-called Chaotic States: “Freezing” the Initial State

The interdisciplinary journal of Discontinuity, Nonlinearity, and Complexity

The so-called chaotic states that emerge on the model of XY interacting on regular critical range networks are analyzed. Typical time scales are extracted from the time series analysis of the global magnetization. The large spectrum confirms the chaotic nature of the observable, anyhow different peaks in the spectrum allows for typical characteristic timescales to emerge. We find that these time scales τ(N) display a critical slowing down, i.e they diverge as N → ∞. The scaling law is analyzed for different energy densities and the behavior τ(N) ∼ √ N is exhibited. This behavior is furthermore explained analytically using the formalism of thermodynamic-equations of the motion and analyzing the eigenvalues of the adjacency matrix.

Routes to extreme events in dynamical systems: Dynamical and statistical characteristics

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2020

Intermittent large amplitude events are seen in the temporal evolution of a state variable of many dynamical systems. Such intermittent large events suddenly start appearing in dynamical systems at a critical value of a system parameter and continues for a range of parameter values. Three important processes of instabilities, namely, interior crisis, Pomeau-Manneville intermittency and the breakdown of quasiperiodic motion, are most common as observed in many systems that lead to such occasional and rare transitions to large amplitude spiking events. We characterize these occasional large events as extreme events if they are larger than a statistically defined significant height. We present two exemplary systems, a single system and a coupled system to illustrate how the instabilites work to originate as extreme events and they manifest as non-trivial dynamical events. We illustrate the dynamical and statistical properties of such events.

Natural time analysis of critical phenomena

Proceedings of the National Academy of Sciences, 2011

A quantity exists by which one can identify the approach of a dynamical system to the state of criticality, which is hard to identify otherwise. This quantity is the variance κ 1 ð≡hχ 2 i − hχ i 2 Þ of natural time χ , where hfðχ Þi ¼ ∑ p k f ðχ k Þ and p k is the normalized energy released during the kth event of which the natural time is defined as χ k ¼ k∕N and N stands for the total number of events. Then we show that κ 1 becomes equal to 0.070 at the critical state for a variety of dynamical systems. This holds for criticality models such as 2D Ising and the Bak-Tang-Wiesenfeld sandpile, which is the standard example of self-organized criticality. This condition of κ 1 ¼ 0.070 holds for experimental results of critical phenomena such as growth of rice piles, seismic electric signals, and the subsequent seismicity before the associated main shock.

How, Why and When Tsallis Statistical Mechanics Provides Precise Descriptions of Natural Phenomena

Entropy

The limit of validity of ordinary statistical mechanics and the pertinence of Tsallis statistics beyond it is explained considering the most probable evolution of complex systems processes. To this purpose we employ a dissipative Landau–Ginzburg kinetic equation that becomes a generic one-dimensional nonlinear iteration map for discrete time. We focus on the Renormalization Group (RG) fixed-point maps for the three routes to chaos. We show that all fixed-point maps and their trajectories have analytic closed-form expressions, not only (as known) for the intermittency route to chaos but also for the period-doubling and the quasiperiodic routes. These expressions have the form of q-exponentials, while the kinetic equation’s Lyapunov function becomes the Tsallis entropy. That is, all processes described by the evolution of the fixed-point trajectories are accompanied by the monotonic progress of the Tsallis entropy. In all cases the action of the fixed-point map attractor imposes a sev...

Critical fluctuations, intermittent dynamics and Tsallis statistics

Physica A: Statistical Mechanics and its Applications, 2004

It is pointed out that the dynamics of the order parameter at a thermal critical point obeys the precepts of the nonextensive Tsallis statistics. We arrive at this conclusion by putting together two welldefined statistical-mechanical developments. The first is that critical fluctuations are correctly described by the dynamics of an intermittent nonlinear map. The second is that intermittency in the neighborhood of a tangent bifurcation in such map rigorously obeys nonextensive statistics. We comment on the implications of this result.

physics pp. 353{370 Statistical methods in nonlinear dynamics

2015

Abstract. Sensitivity to initial conditions in nonlinear dynamical systems leads to ex-ponential divergence of trajectories that are initially arbitrarily close, and hence to un-predictability. Statistical methods have been found to be helpful in extracting useful information about such systems. In this paper, we review brie°y some statistical methods employed in the study of deterministic and stochastic dynamical systems. These include power spectral analysis and aliasing, extreme value statistics and order statistics, recur-rence time statistics, the characterization of intermittency in the Sinai disorder problem, random walk analysis of di®usion in the chaotic pendulum, and long-range correlations in stochastic sequences of symbols.

Critical dynamic approach to stationary states in complex systems

The European Physical Journal Special Topics, 2007

A dynamic scaling Ansatz for the approach to stationary states in complex systems is proposed and tested by means of extensive simulations applied to both the Bak-Sneppen (BS) model, which exhibits robust Self-Organised Critical (SOC) behaviour, and the Game of Life (GOL) of J. Conway, whose critical behaviour is under debate. Considering the dynamic scaling behaviour of the density of sites (ρ(t)), it is shown that i) by starting the dynamic measurements with configurations such that ρ(t = 0) → 0, one observes an initial increase of the density with exponents θ = 0.12(2) and θ = 0.11(2) for the BS and GOL models, respectively; ii) by using initial configurations with ρ(t = 0) → 1, the density decays with exponents δ = 0.47(2) and δ = 0.28(2) for the BS and GOL models, respectively. It is also shown that the temporal autocorrelation decays with exponents C a = 0.35(2) (Ca = 0.35(5)) for the BS (GOL) model. By using these dynamically determined critical exponents and suitable scaling relationships, we also obtain the dynamic exponents z = 2.10(5) (z = 2.10(5)) for the BS (GOL) model. Based on this evidence we conclude that the dynamic approach to stationary states of the investigated models can be described by suitable power-law functions of time with well-defined exponents. −z/x 0 0 , where m0 → 0 is the initial magnetisation [4]

Long Chaotic Transients in Complex Networks

Physical Review Letters, 2004

We show that long chaotic transients dominate the dynamics of randomly diluted networks of pulsecoupled oscillators. This contrasts with the rapid convergence towards limit cycle attractors found in networks of globally coupled units. The lengths of the transients strongly depend on the network connectivity and vary by several orders of magnitude, with maximum transient lengths at intermediate connectivities. The dynamics of the transients exhibit a novel form of robust synchronization. An approximation to the largest Lyapunov exponent characterizing the chaotic nature of the transient dynamics is calculated analytically.