We study the existence and stability of splay states in the coupled sine circle map lattice system using analytic and numerical techniques. The splay states are observed for very low values of the nonlinearity parameter, i.e., for maps which deviate very slightly from the shift map case. We also observe that depending on the parameters of the system the splay state bifurcates to a mixed or chimera splay state consisting of a mixture of splay and synchronized states, together with kinks in the phases of some of the maps and then to a stable globally synchronized state. We show that these pure states and the mixed states are all temporally chaotic for our systems, and we explore the stability of these states to perturbations. Our studies may provide pointers to the behavior of systems in diverse application contexts such as Josephson junction arrays and chemical oscillations.

We construct an equivalent cellular automaton (CA) for a system of globally coupled sine circle maps with two populations and distinct values for intergroup and intragroup coupling. The phase diagram of the system shows that the coupled map lattice can exhibit chimera states with synchronized and spatiotemporally intermittent subgroups after evolution from random initial conditions in some parameter regimes, as well as to other kinds of solutions in other parameter regimes. The CA constructed by us reflects the global nature and the two population structure of the coupled map lattice and is able to reproduce the phase diagram accurately. The CA depends only on the total number of laminar and burst sites and shows a transition from co-existing deterministic and probabilistic behavior in the chimera region to fully probabilistic behavior at the phase boundaries. This identifies the characteristic signature of the transition of a cellular automaton to a chimera state. We also construct an evolution equation for the average number of laminar/burst sites from the CA, analyze its behavior and solutions, and correlate these with the behavior seen for the coupled map lattice. Our CA and methods of analysis can have relevance in wider contexts.

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We study the existence of chimera states, i.e. mixed states, in a globally coupled sine circle map lattice, with different strengths of inter-group and intra-group coupling. We find that at specific values of the parameters of the CML, a completely random initial condition evolves to chimera states, having a phase synchronised and a phase desynchronised group, where the space time variation of the phases of the maps in the desynchronised group shows structures similar to spatiotemporally intermittent regions. Using the complex order parameter we obtain a phase diagram that identifies the region in the parameter space which supports chimera states of this type, as well as other types of phase configurations such as globally phase synchronised states, two phase clustered states and fully phase desynchronised states. We estimate the volume of the basin of attraction of each kind of solution. The STI chimera region is studied in further detail via numerical and analytic stability analysis, and the Lyapunov spectrum is calculated. This state is identified to be hyperchaotic as the two largest Lyapunov exponents are found to be positive. The distributions of laminar and burst lengths in the incoherent region of the chimera show exponential behaviour. The average fraction of laminar/burst sites is identified to be the important quantity which governs the dynamics of the chimera. After an initial transient, these settle to steady values which can be used to reproduce the phase diagram in the chimera regime.The study of chimera states, i.e. mixed states where synchronised and desynchronised dynamics coexist, has been at the forefront of studies in nonlinear dynamics involving both theoretical and experimental systems. A variety of classes of chimera states, i.e. states which contain coexisting domains of distinct kinds of spatiotemporal behaviour can be seen. These include multi-headed chimera states, travelling chimera states, amplitude chimera states, twisted chimera states etc, and have been seen in coupled oscillator models such as the Kuramoto model,coupled Ginzburg-Landau oscillators and other systems. Here, we investigate chimera and other states in a coupled sine circle map lattice which is a discrete version of coupled oscillator systems. The CML consists of two populations of globally coupled identical sine circle maps with distinct values for the intergroup and intragroup coupling. We observe spatiotemporally intermittent chimeras, i.e. states which consist of a synchronised subgroup, and a state where coherent (phase synchronised) and incoherent (phase incoherent) domains co-exist, at low values of the nonlinearity map parameter. Such STI chimeras have been observed earlier in coupled oscillators models such as Stuart-Landau oscillators, Ginzburg-Landau oscillators, coupled optical resonators, chemical reactions etc. We analyse the STI chimera seen in the CML system by plotting the phase diagram of the system using the global order parameter, and identifying the region where STI chimeras can be seen. The basin...

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