PPubMed27 Jun 2023Cited 5×

Exponentially Long Transient Time to Synchronization of Coupled Chaotic Circle Maps in Dense Random Networks

Hans Muller Mendonca, Ralf Tönjes, Tiago Pereira, Andrea Rapisarda

Abstract

We study the transition to synchronization in large, dense networks of chaotic circle maps, where an exact solution of the mean-field dynamics in the infinite network and all-to-all coupling limit is known. In dense networks of finite size and link probability of smaller than one, the incoherent state is meta-stable for coupling strengths that are larger than the mean-field critical coupling. We observe chaotic transients with exponentially distributed escape times and study the scaling behavior of the mean time to synchronization.

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Dense random networks of chaotic circle maps show exponentially long chaotic transients to synchronization, with escape times scaling as a power of size below a critical coupling and roughly inversely with link probability above it.

What’s new

i.Identified exponential distribution of transient times to synchronization in dense chaotic map networks.
ii.Discovered exponentially long chaotic transients above the mean‑field critical coupling in incomplete dense networks.
iii.Derived exact mean‑field equations and critical couplings (ε₁≈0.69, ε₀=2.0) for the Bernoulli circle map model.
iv.Showed that escape time scales as N^{1/3} below ε₀ and roughly as 1/p above ε₀ in random networks.

How they did it

The authors performed analytical mean‑field analysis of coupled Bernoulli circle maps and derived closed‑form dynamics for the order parameter. They then ran large‑scale numerical simulations on dense random networks with varying link probability p and coupling strength ε, measuring the first passage time to R≥0.5. Finite‑size scaling and maximum‑likelihood estimation were used to characterize the transient‑time statistics.

What they found

Critical coupling values: ε₁=ln(2)≈0.69 (bistability onset) and ε₀=2.0 (desynchronization‑synchronization transition).
Below ε₀, average escape time scales as T_esc ∝ N^{β} with β≈1/3.
Above ε₀, escape time is independent of N and scales roughly as 1/p.
Transient times follow an exponential (geometric) distribution, confirmed by semi‑log rank plots.

Caveats

✕Analysis is limited to dense, homogeneous random networks; effects of strong degree heterogeneity are not addressed.
✕Results are based on the specific Bernoulli map model; generality to other chaotic phase maps remains open.
✕Simulations are constrained by finite observation time T, potentially missing extremely long transients.

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