Crisis in time-dependent dynamical systems

TL;DR

The paper addresses crises in time-dependent dynamical systems, focusing on abrupt expansions of attractor basins when the boundary is crossed under bounded fluctuations. It combines a mean-field reduction of globally coupled Kuramoto rotors with inertia and simplified stochastic models (Hénon and logistic maps) to derive and validate a universal scaling for the escape probability, $G_M(\delta) \approx \exp[-\alpha(\ln\delta)^2+\beta\ln\delta]$ with $\alpha=1/(2\ln\mu)$. The results show that crises can occur in non-autonomous settings due to tunneling-like transitions across a fluctuating basin boundary, with the grey zone acting as the critical conduit, and fractal basin boundaries governing escape dynamics. This advances understanding of rare transitions under bounded forcing and has implications for systems such as climate dynamics and synchronization where external drives shape stability and transitions.

Abstract

Many dynamical systems operate in a fluctuating environment. However, even in low-dimensional setups, transitions and bifurcations have not yet been fully understood. In this Letter we focus on crises, a sudden flooding of the phase space due to the crossing of the boundary of the basin of attraction. We find that crises occur also in non-autonomous systems although the underlying mechanism is more complex. We show that in the vicinity of the transition, the escape probability scales as $\exp[-α(\ln δ)^2]$, where $δ$ is the distance from the critical point, while $α$ is a model-dependent parameter. This prediction is tested and verified in a few different systems, including the Kuramoto model with inertia, where the crisis controls the loss of stability of a chimera state.

Figures (3)

• Figure 1: (a) Snapshot of the Kuramoto model in the phase plane $(\dot{\vartheta},\vartheta)$. The green square denotes the cluster position, while the open circles denote the dust. ($f_{cl}=0.393$). Inset: time dependence of the dust-cluster distance in the presence of a migration event. The blue (red) curve identifies the maximal (average) Euclidean distance between the dust and the cluster. The black curve identifies the distance between the cluster and the nearest dust oscillator. Around t=100 the minimal distance virtually vanishes, indicating the occurrence of a migration event. (Initially, $f_{cl}=0.465.$). (b) Basin of attraction of the cluster. The greyscale in ${\bf P}=(\vartheta, \dot{\vartheta})$ identifies the Euclidean distance of a probe oscillator (initially in ${\bf P})$ from the cluster after a time $t_e=100$. Initial conditions are varied in a grid of size 0.01 in both directions. The cluster is initially in $(2.48160,3.12)$ and $f_{cl}= 0.5$.
• Figure 2: (a) Kuramoto model: escape probability towards the cluster as a function of the initial value of $f_d$. Each value is obtained by averaging over 10 different realisations each of length $t=2000000$. The dashed line is a fit with Eq. (\ref{['eq:theoryf']}). In the inset, the escape probability vs the distance from the critical point is reported for the Hénon map. (b) Critical noise amplitude vs the average value of the parameter $a$ for the logistic map. (c) Escape probability vs the logarithm of the distance from criticality for the linear stochastic model of the GZ dynamics. (d) Scaling behavior of the escape probability for the logistic and Hénon map. Black (red) dots represent simulation data for the logistic (Hénon) map, while the green (red) dashed line represent the corresponding scaling behavior estimated by following Eq. (7).
• Figure 3: (a) Snapshot of the Kuramoto model in the phase plane $(\dot{\vartheta},\vartheta)$. The red dot corresponds to the cluster position, while the black and green circles correspond to the dust. (initially, $f_{cl}= 0.393$). (b) Enlargement of panel (a) around the cluster position corresponding to the evolution at time $T_1$. Here are reported enlargements of the same dynamical evolution for successive times $T_2,..,T_5$ (identifiable by different symbols) to characterize the tunnel zone in the cluster growth problem.