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

TL;DR

The paper addresses critical slowing down at period-doubling bifurcations in discrete-time dynamical systems by deriving a universal, phenomenological description from a local Taylor expansion around the fixed point and the bifurcation parameter. A reduced one-dimensional dynamics at the bifurcation and near-critical regimes reveals four universal exponents, three ($alpha=1$, $beta=-1/2$, $z=-2$) characterizing short-time, long-time, and crossover behavior at criticality, plus a fourth ($delta=-1$) governing the relaxation time away from criticality. The authors extend the framework to two-dimensional maps via center-manifold projection, showing the second-iterate normal form collapses to the same cubic structure as in 1D, preserving universality. Numerical validations on the Hénon and Ikeda maps demonstrate excellent agreement with the predicted scaling laws and exponents, reinforcing the universality of critical slowing down across dimensions and map types.

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 Hénon and Ikeda maps, showing excellent agreement for all scaling laws and critical exponents.

Figures (7)

• Figure 1: (a) Orbit diagram for the Hénon map (\ref{['eq15']}). (b) Largest Lyapunov exponent corresponding to the orbits shown in panel (a). The initial condition used was $(x_0,y_0)=(0.952723,0.2857)$ for a fixed $b=0.3$ and $a\in[-0.1,1.4]$. Each initial condition was iterated for $10^9$ iterations and only the last $100$ points of the orbit were recorded.
• Figure 2: Orbit diagram for the Hénon map (\ref{['eq15']}) -- a zoom of Fig. \ref{['Fig1']}(a) -- together with the largest Lyapunov exponent near the first period-doubling bifurcation. The dashed line is shown as a guide to the eye. The first period-doubling bifurcation is observed at $a=0.367495$.
• Figure 3: (a) Distance $d(n)$ as a function of the iteration number $n$ for different initial conditions near the fixed point at the period-doubling bifurcation of the Hénon map. (b) Collapse of the curves shown in panel (a) onto a universal plot after the appropriate scaling transformations. The control parameters used were $b=0.3$ and $a=0.367495$.
• Figure 4: Crossover iteration number $n_x$ as a function of the initial distance $d_0$. A power-law fit yields $z=-2.036(6)$.
• Figure 5: Relaxation time $\tau$ as a function of $\mu$ for the Hénon map. A power-law fit yields $\delta=-1.00(3)$, in excellent agreement with the theoretical prediction.