Globally coupled map lattice models are particularly useful for the study of spatially extended systems which can model coupled oscillator system such as Josephson junction oscillator arrays in solid state physics, multi-mode lasers in nonlinear optics etc. Here we study a globally coupled sine circle map lattice with different strengths of intragroup and intergroup coupling and explore the existence and stability of splay phase states and chimera states. This model is motivated by coupled oscillator systems with two populations and represents a discrete version of such systems.
The splay phase state and its multiple copies are observed in this CML evolving from a single splay phase initial condition spanning the entire system, for low values of sinusoidal nonlinearity i.e. where the system deviates slightly from the coupled shift map case. We show that the pure splay state is temporally chaotic, via linear stability analysis. We also show that depending on the parameters of the system the splay states bifurcate to splay chimera states consisting of a mixture of splay like structure and synchronised states, together with kinks in the phases of some of the maps and then to a stable globally synchronised states and are all found to be temporally chaotic in nature.
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 shows structures similar to spatiotemporally intermittent regions. We obtain a phase diagram, identifying the region of the parameter space which supports such chimera state as well as other solutions such as globally phase synchronised states, two phase clustered states and fully desynchronised states. We estimate the basin stability 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 average fraction of laminar/burst sites in the incoherent region of the chimera state is identified to be a crucial quantity as it settles to an unique value for each solution and is used to reproduce the phase diagram. An equivalent cellular automaton is obtained which shows space time behaviour similar to the CML. The chimera regions show coexisting deterministic and probabilistic behaviour in the subgroup probabilities which show a transition to purely probabilistic behaviour at the boundaries of the region. We also derive an evolution equation for the average fraction of laminar sites for this cellular automaton and compare its solutions with the corresponding phase configurations obtained in the parameter space, and show that its behaviour matches with the behaviour seen for the CML.