Patterns, Bifurcations, Multistability and Hysteresis in an Inhomogeneous Coupled Map Lattice

Abstract

We investigate local interaction between logistic maps in an inhomogeneous lattice numerically with respect to an inhomogeneity parameter [Formula: see text] and the coupling constant [Formula: see text]. In our model, the inhomogeneity appears in the form of different values of the map parameter at different sites. The phase diagram of the model in the [Formula: see text]–[Formula: see text] plane gives seven qualitatively different patterns. These are: synchronized patterns, steady (fixed in time) patterns with spatial period-two, spatially chaotic together with temporally periodic patterns, spatially coherent accompanied with temporally quasi-periodic patterns, spatial intermittency with temporal period-two patterns, spatiotemporal intermittency patterns or spatiotemporal chaotic patterns. Our system exhibits, tangent bifurcations in the transition from synchronized patterns to steady patterns, period-doubling bifurcation from steady patterns to temporal periodic patterns and Neimark–Sacker (Hopf) bifurcation from steady patterns to temporal quasi-periodic patterns. The system also shows the possibility of multistable attractors and the phenomena of hysteresis for some parameter values. We identify our results using techniques such as time series, space-time plot, Fourier transform, bifurcation diagram, stability analysis.