A phenomenological description of critical slowing down at period-doubling bifurcations

TL;DR

This paper provides a universal phenomenological framework for understanding critical slowing down at period-doubling bifurcations in discrete dynamical systems, validated through numerical experiments on well-known maps.

Contribution

It derives a reduced description capturing critical slowing down near bifurcations and extends the phenomenology from one-dimensional to two-dimensional maps.

Findings

Universal critical exponents are identified for short-time and asymptotic behaviour.

The phenomenology extends from 1D to 2D maps via center manifold projection.

Numerical validation confirms the theoretical scaling laws and exponents.

Abstract

We present a phenomenological description of the critical slowing down associated with period-doubling bifurcations in discrete dynamical systems. Starting from a local Taylor expansion around the fixed point and the bifurcation parameter, we derive a reduced description that captures the convergence towards stationary state both at and near criticality. At the bifurcation point, three universal critical exponents are obtained, characterising the short-time behaviour, the asymptotic decay, and the crossover between these regimes. Away from criticality, a fourth exponent governing the relaxation time is identified. We show this phenomenology, well established for one-dimensional maps, extends naturally to two-dimensional mappings. By projecting the dynamics onto the centre manifold, we demonstrate that the local normal form of a two-dimensional period-doubling bifurcation reduces to the same universal structure found in one dimension. The theoretical predictions are validated numerically using the H\'enon and Ikeda maps, showing excellent agreement for all scaling laws and critical exponents.