Crisis in time-dependent dynamical systems (2503.13152v1)

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 $\exp[-\alpha (\ln \delta)^2]$, where $\delta$ is the distance from the critical point, while $\alpha$ 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.

Crisis in Time-Dependent Dynamical Systems

The paper "Crisis in Time-Dependent Dynamical Systems" by Olmi and Politi explores the complex phenomena concerning crises in non-autonomous dynamical systems. Crises, understood as sudden expansions of phase space due to the crossing of an attractor's boundary, have been prominently studied in autonomous systems. This paper extends the concept to non-autonomous systems, which pose a more complicated mechanism due to their time variability and dependence on fluctuating external conditions.

Key Findings and Methodological Approach

The authors identify crises in non-autonomous systems and develop a scaling law for escape probabilities in these contexts. The escape probability near a crisis transitions according to the scaling relation $\exp[-\alpha (\ln \delta)^2]$, where $\delta$ is the proximity to the critical crisis point, and $\alpha$ is a model-dependent constant. This scaling law is validated numerically across various dynamical models including the Kuramoto model with inertia and simpler systems such as the Hénon and logistic maps.

The paper begins by investigating the Kuramoto model, which is representative of a network of identically coupled rotators influenced by inertia, modifying the natural dynamics. The presence of both a synchronized cluster and an unsynchronized dust (asynchronous state) characterizes the chimera state, which becomes destabilized during crises. In this non-autonomous context, the crisis is driven by fluctuating parameters acting as a deterministic chaotic forcing, demonstrating the model’s ability to predict transition points that were traditionally examined in deterministic setups.

Analytical Development and Implications

The paper progresses through simpler examples, such as the modulated Hénon and logistic maps. These models reaffirm the scaling behavior observed in the Kuramoto scenario by introducing a control parameter susceptible to stochastic fluctuations. Crucially, the noise in these systems not only changes the stability landscape but modifies the critical transition threshold, presenting a lower boundary for instability than expected from deterministic perspectives.

Escape dynamics are dissected using a linear stochastic model within which phase points exhibit expansion and contraction influenced by underlying random saddle dynamics. This mechanism provides a theoretical frame to derive escape times and predict crisis transitions in fluctuating systems. Particularly, it is conjectured that the fractal dimension of attractors might interact with instabilities, a hypothesis necessitating further exploration.

Speculation on Future Directions

The implications of this paper extend theoretical understandings of chaotic dynamics beyond static configurations into domains informed by external perturbations and non-autonomy. The focus on scaling laws in crisis behavior prompts speculation about generalized methods in predicting systemic transitions under realistic conditions where environmental fluctuations are inherent. The analytical insights provided may guide future research in adaptive systems and those modeled after physiological processes, where time-dependent variability is fundamental.

Further analysis could involve examining high-dimensional extensions and dependencies beyond simplified fluctuation characterizations, integrating concepts such as fractal dimensions and cross-component correlations to yield broader applicability across chaotic and complex dynamical systems.

Conclusion

Olmi and Politi's contribution lies in adapting the concept of crises to non-autonomous systems and offering an analytical basis for understanding escape dynamics under fluctuating conditions. The presented scaling law provides valuable insights and practical tools for future explorations in dynamical systems facing complex, real-world influences, opening pathways for diverse applications in synchronization studies and chaotic theory in applied science domains.