Collective behavior of a coupled-map system with a conserved quantity

Abstract

We present the results of a numerical investigation of, and supporting analytic arguments for, the behavior of coupled sine maps with a locally conserved variable on a square lattice, with and without noise. By varying parameters we find two distinct conservation-induced routes to chaos: One is a first-order transition from a periodic 2-cycle phase with broken time-translational invariance and long-range, striped, spatial order, to a chaotic, spatially disordered phase in which time-translation invariance is restored; this transition is accompanied by a discontinuous jump in the maximum Lyapunov exponent, ${\lambda }_{\max }$. The other is a continuous onset of temporal chaos occurring without any change in the spatial symmetry of the phase, which has striped (‘‘antiferromagnetic’’) long-range order on both sides of the chaotic onset. In this latter case, ${\lambda }_{\max }$ increases continuously from zero as the control parameter R is increased above the value R=${\mathit{R}}^{\mathrm{*}}$ where chaos sets in; in the presence of noise, the numerics show ${\lambda }_{\max }$ growing linearly with R-${\mathit{R}}^{\mathrm{*}}$, in agreement with simple arguments. Numerical and analytic evidence (strong for the nonchaotic and weaker for the chaotic phases) is presented for the algebraic decay in time of autocorrelations. The asymptotic behavior of spatial correlations is also discussed. We also verify numerically the absence of ‘‘collective chaos,’’ i.e., of chaotic fluctuations of appropriately normalized Fourier amplitudes in the system, consistent with earlier arguments. We briefly discuss the relationship of these results to those of recent experiments on surface waves.