Universality classes of chaos in non Markovian dynamics

TL;DR

This work addresses how non‑Markovian temporal memory alters the classical universality of chaos, notably the Feigenbaum period‑doubling scenario. By introducing a minimal non‑Markovian logistic map with a power‑law memory kernel, it derives a nonlocal stability condition via $\lambda = a + \epsilon \mathrm{Li}_{\alpha}(\lambda^{-1})$ and analyzes finite‑time Lyapunov scaling across memory regimes. The main contributions are the identification of a critical memory exponent $\alpha$ separating perturbative and memory‑dominated chaos, the persistence of Feigenbaum universality under a memory‑dependent rescaling for $\alpha>1$, and a genuinely new universal class with fractional scaling for $\alpha\leq1$ (including the marginal $\alpha=1$ case). The results reveal temporal correlations as a new axis of universality with potential relevance for systems where memory effects are intrinsic, and lay groundwork for extending renormalization concepts to non‑Markovian chaotic dynamics.

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.

Figures (3)

• Figure 1: Bifurcation diagrams and Lyapunov exponents.(a) Classical logistic map ($\epsilon=0$). (b) Summable memory ($\epsilon=0.02$, $\alpha=1.5$). (c) Long-range memory ($\epsilon=0.02$, $\alpha=0.7$). Top row: bifurcation diagram; bottom row: Lyapunov exponent $\lambda(r)$. Vertical lines mark the first ($r_1=3.0$) and second ($r_2\approx 3.45$) period-doubling bifurcations of the classical map; the dashed line at $\lambda=0$ marks the onset of chaos.
• Figure 2: Phase diagram of chaos universality classes in the non-Markovian logistic map. The three regions correspond to: classical universality (period-doubling cascade), deformed universality (memory-renormalized cascade), and memory-dominated chaos (fractional scaling, no Feigenbaum cascade). The dashed line at $\alpha=1$ separates summable (short-range) and non-summable (long-range) memory. The solid white curve shows the analytic stability boundary $|a| + \epsilon\,\zeta(\alpha) = 1$ for $\alpha > 1$. Parameters: $r = 3.2$, $x_0 = 0.2$, $n_{\mathrm{iter}} = 8000$, $n_{\mathrm{transient}} = 5000$, memory cutoff $= 2000$.
• Figure 3: Memory‑induced change in Lyapunov scaling. (a) Lyapunov exponent $\lambda_{\max}$ vs. $r - r_c(\alpha)$ for the classical logistic map ($\alpha=\infty$), summable memory ($\alpha=1.5$), and long-range memory ($\alpha=0.7$). Classical and summable memory show the same square-root scaling $\lambda \sim (r - r_c)^{1/2}$, while long-range memory exhibits a different, fractional scaling. (b) Critical exponent $\beta(\alpha)$ vs. memory exponent $\alpha$. The numerical data (circles) are well described by $\beta(\alpha) = (2-\alpha)^{-1}$ for long-range memory, while classical and summable memory fall on the universal value $\beta = 1/2$.