Universality classes of chaos in non Markovian dynamics (2512.22445v1)

Abstract: Classical chaos theory rests on the notion of universality, whereby disparate dynamical systems share identical scaling laws. Existing universality classes, however, implicitly assume Markovian dynamics. Here, a logistic map endowed with power law memory is used to show that Feigenbaum universality breaks down when temporal correlations decay sufficiently slowly. A critical memory exponent is identified that separates perturbative and memory dominated regimes, demonstrating that long range memory acts as a relevant renormalisation operator and generates a new universality class of chaotic dynamics. The onset of chaos is accompanied by fractional scaling of Lyapunov exponents, in quantitative agreement with analytical predictions. These results establish temporal correlations as a previously unexplored axis of universality in chaotic systems, with implications for physical, biological and geophysical settings where memory effects are intrinsic.

• The paper demonstrates that incorporating a power-law memory kernel into the logistic map leads to new critical scaling laws, deviating from classical Feigenbaum behavior.
• The analysis utilizes both analytic derivations and numerical investigations of Lyapunov exponents to classify stability regimes based on memory decay and strength.
• Practical implications include rethinking chaos indicators in real-world systems like climate, neuroscience, and viscoelastic media where memory effects are significant.

Universality Classes of Chaos in Non-Markovian Dynamics

Introduction

This work addresses a pivotal gap in the theory of dynamical systems: the classification of universality in routes to chaos for systems with long-range temporal correlations, i.e., non-Markovian dynamics. While classical universality—most notably Feigenbaum scaling—has been comprehensively established for Markovian low-dimensional dissipative maps, the consequences of introducing memory with slow temporal decay have been underexplored. The paper develops a minimal analytic model: a logistic map with a power-law memory kernel. It rigorously demonstrates that when temporal correlations decay sufficiently slowly (i.e., the memory exponent $\alpha \leq 1$), conventional period-doubling universality breaks down, and a qualitatively new critical scaling emerges. The analysis is grounded both in analytic derivations and direct numerical exploration of Lyapunov exponents and bifurcation structures.

Minimal Non-Markovian Model

The proposed model extends the classical logistic map by adding a nonlocal memory term:

$$x_{n+1} = r x_n (1 - x_n) + \epsilon \sum_{k=1}^n k^{-\alpha} x_{n-k}.$$

Here, $r$ is the canonical nonlinearity parameter, $\epsilon$ quantifies memory strength, and $\alpha$ controls the memory kernel's decay. For $\epsilon\to 0$, the model recovers the standard logistic map. The power-law kernel governs the range of history influence: summable for $\alpha > 1$, non-summable for $\alpha \leq 1$. In the non-summable case, all prior states affect the system's evolution with non-vanishing cumulative weight, and the system is truly infinite-dimensional.

Stability and Bifurcation Analysis

A detailed linear stability analysis reveals the existence of three distinct regimes, sharply demarcated by the critical value $\alpha=1$:

• Summable Memory ($\alpha > 1$): The system exhibits perturbative shrinkage of the stability window, with a renormalized but structurally identical period-doubling bifurcation cascade. The control parameters are rescaled by the finite memory effect, and classical Feigenbaum scaling is preserved.
• Marginal Memory ($\alpha = 1$): The memory term diverges logarithmically, leading to marginal destabilization. Even infinitesimal memory destroys fixed-point stability, introducing a qualitative departure from the classical scenario, characterized by logarithmic corrections.
• Non-Summable/Long-Range Memory ($\alpha < 1$): The fixed point is nonperturbatively destabilized. The onset of chaos and critical behavior are governed by the memory exponent, and the bifurcation structure is fundamentally altered. Chaos can emerge without the standard period-doubling route, and large driving can even suppress chaos, resulting in re-entrant stabilization.

The phase diagram thus reveals memory strength and memory exponent as independent control parameters for universality class switching. Notably, the analytic approach yields closed-form criterion for stability loss, grounded in the polylogarithmic structure of the characteristic equation.

Fractional Critical Scaling of Lyapunov Exponents

A prominent result is the emergence of non-classical, fractional scaling of the largest Lyapunov exponent near the chaotic transition, a direct consequence of the interplay between nonlocal memory and bifurcation dynamics:

• Markovian Case and Summable Memory: The Lyapunov exponent near the chaos threshold scales as $\lambda \sim |r - r_c|^{1/2}$, in agreement with classical intermittency theory.
• Long-Range Memory: The scaling exponent becomes $\beta(\alpha) = 1/(2-\alpha)$, i.e., $\lambda \sim |r - r_c|^{1/(2-\alpha)}$, which continuously interpolates with $\alpha$—recovering the classical square-root law only as $\alpha\to \infty$. For $\alpha \leq 1$, the system displays fractional critical behavior with memory-dominated exponents. The onset of chaos occurs with a tunable critical exponent, representing a new universality class unavailable in Markovian maps.

Numerical data confirm the analytic predictions, substantiating the existence, robustness, and universality of this fractional scaling in the non-Markovian regime.

Theoretical and Practical Implications

The findings establish temporal correlations as a new axis in the taxonomy of universality classes for chaos, analogous to spatial dimensionality in equilibrium statistical mechanics. The memory exponent effectively acts as a "temporal dimension": above the critical value, memory is irrelevant and universality is classical; below it, memory is relevant and universality shifts.

Practically, these results question the applicability of classical chaos indicators and transition scenarios in real-world systems where long memory is inherent, e.g., viscoelastic media, population dynamics with feedback, neuroscience, climate, and active matter systems. The fractional Lyapunov scaling suggests that fine structure in measured chaos indices can reveal latent memory effects, which standard Markovian modeling would miss. Additionally, the manifestation of chaos without Feigenbaum cascades and the presence of re-entrant stability are phenomena relevant for the control and prediction of real, history-dependent nonlinear phenomena.

Outlook and Open Problems

The research opens several avenues for future work:

• Extension to high-dimensional and continuous-time systems: The universality classification in higher dimensions, PDEs, or coupled map lattices with memory remains unsolved.
• Development of renormalization group frameworks: Most existing RG treatments are inherently finite-dimensional and Markovian; a general theory for non-Markovian functional phase space is open.
• Analysis of attractor geometry and entropy: How do memory-induced changes affect fractal dimensions and entropy production?
• Connections with stochastic processes: Including temporally correlated noise could further enrich the universality structure.

Conclusion

Long-range temporal correlations fundamentally redefine the universality of chaos, introducing new critical exponents, bifurcation structures, and dynamical regimes that fall outside the traditional Feigenbaum paradigm. The results demonstrate that universality in chaotic systems is sensitive, multi-dimensional, and contingent on the structure of memory, thus requiring a broader theoretical framework for memory-laden, real-world dynamics.

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Essay based on "Universality classes of chaos in non-Markovian dynamics" (2512.22445).