Role of long-range memory in universality for higher-dimensional systems and continuous-time flows

Determine the role of long-range temporal memory in reshaping universality classes of chaotic dynamics for higher-dimensional deterministic systems and continuous-time flows, including how memory modifies routes to chaos and critical scaling relative to Markovian (Feigenbaum) behavior.

Background

The paper introduces a non-Markovian logistic map with a power-law memory kernel and shows that classical Feigenbaum universality is preserved only for summable memory (alpha > 1) and breaks down for long-range memory (alpha ≤ 1), where a new universality class with fractional Lyapunov scaling emerges.

All analyses are performed for a one-dimensional, discrete-time map. Extending these results to higher-dimensional dynamical systems and to continuous-time flows is identified as an outstanding direction, as memory could qualitatively alter bifurcation structures, stability, and critical exponents in those broader settings.

References

Several open theoretical questions arise naturally. The role of long‑range memory in reshaping universality in higher‑dimensional systems and continuous‑time flows remains largely unexplored, as does the formulation of a systematic renormalisation‑group theory for non‑Markovian dynamics in functional phase space.

— Universality classes of chaos in non Markovian dynamics (2512.22445 - Vijayan, 27 Dec 2025) in Section: Implications and Outlook