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

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.

💡 Research Summary

The paper develops a phenomenological framework to describe critical slowing down (CSD) associated with period‑doubling (PD) bifurcations in discrete dynamical systems. Starting from a local Taylor expansion of a one‑dimensional map (x_{n+1}=f(x_n,r)) around a fixed point (x^) and the critical parameter value (r_c), the authors retain terms up to third order in the deviation (\varepsilon_n=x_n-x^) and first order in the control‑parameter shift (\mu=r-r_c). At a PD bifurcation the linear term satisfies (f_x(x^*,r_c)=-1), leading to the discrete update
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