Evidence for directed percolation universality at the onset of spatiotemporal intermittency in coupled circle maps

Abstract

We consider a lattice of coupled circle maps, a popular model for the study of mode-locked phenomena. We find that the onset of spatiotemporal intermittency (STI) in this system is analogous to directed percolation (DP), with the transition being to a unique absorbing state for low nonlinearities, and to weakly chaotic absorbing states for high nonlinearities. We find that the complete set of static exponents and spreading exponents at all critical points match those of DP very convincingly. Further, hyperscaling relations are fulfilled, leading to independent controls and consistency checks of the values of all the critical exponents. These results provide an example in support of the conjecture that the onset of STI in deterministic models belongs to the DP universality class. Nonuniversal spreading exponents are seen only for the cases where the initial state is homogeneous with symmetrically placed seeds leading to strictly symmetric spreading. However, very small departures from homogeneity are sufficient to restore the DP exponents.

Spatiotemporal intermittency and scaling laws in the coupled sine circle map lattice

“…It has been shown convincingly that STI with synchronized laminar state interspersed with turbulent bursts (seen at points marked with ✸ in Figure 1) belongs to the DP universality class [17,18]. The infective dynamics of the turbulent bursts wherein the turbulent site either infects the adjacent laminar sites, or dies down to the laminar state, is similar to the behaviour seen in directed percolation [22].…”

“…Earlier studies of the diffusively coupled sine circle map lattice showed regimes of STI which were completely free of solitons [17,18]. Two types of STI were seen along the bifurcation boundaries of the bifurcation from the synchronized solution.…”

“…These solutions are seen over a large section of the phase diagram and are stable against perturbations. Cluster solutions, in which x i (t) = x j (t) for all i, j belonging to a particular class [17,18]. However, the other types of intermittency seen in the parameter space do not belong to the DP class.…”

“…However, a CML specially designed to exhibit STI by Chaté and Manneville showed critical exponents significantly different from the DP universality class [11]. This led to a long debate in the literature [14,15,16,17]. It was concluded that the presence of coherent structures, called solitons, were responsible for spoiling the analogy with DP.…”

Chimera states in coupled sine-circle map lattices

“…Coupled map systems show many of the varied phenomena observed in the case of continuous extended systems, but are far more numerically and analytically tractable. Coupled map systems have been seen to show synchronised solutions, spatiotemporal intermittency, spatiotemporal chaos [13,14]. However, to the best of our knowledge, chimera states have not been reported earlier in coupled map lattice systems.…”