Articles written in Pramana – Journal of Physics

    • A scheme for designing extreme multistable discrete dynamical systems


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      In this paper, we propose a scheme for designing discrete extreme multistable systems coupling two identical dynamical systems. Existence of infinitely many attractors in the system is obtained via partial synchronization between two systems for a given set of parameters. We give a conjecture that extreme multistable systems can be designed by coupling two m-dimensional dynamical systems in such a way that $i (1 \leq i \leq m − 1)$ number of state variables of the two systems synchronize completely and $(m − i )$ number of state variables keep constant difference. We demonstrate the applicability of our scheme in two-dimensional (2D) as well as threedimensional (3D) discrete dynamical systems. In particular, we discuss our scheme taking coupled 2D Hénon maps, coupled 2D Duffing maps and coupled 3D Hénon maps. We have analytically shown the existence of fixed points and period-2 orbits in the coupled system with the variation of initial conditions. These analytically derived conditions matched very well with the numerical simulation results. Variation of the largest Lyapunov exponent with the initial conditions is shown to confirm the existence of extreme multistability in the model. Our scheme may be useful for designing physically, chemically and biologically useful multistable discrete dynamical systems.

    • Extreme multistable synchronisation in coupled dynamical systems


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      A rule for designing extreme multistable synchronised systems by coupling two identical dynamical systems has been proposed in this paper. The basic idea behind the proposed scheme is the existence of chaos in the coupled system in the presence of initial condition-dependent constants of motion. A new conjecture has been introduced according to which an extreme multistable synchronised system can be designed if all states of one system will synchronise with the corresponding states of the other system (of the two coupled systems) and the basin of the synchronised state depends on the difference between the initial conditions of the corresponding states of the individual systems. The proposed scheme has been illustrated with the help of coupled Rössler systems, coupled Hénon maps and coupled logistic maps. Moreover, the existence of flip bifurcation with the variation of initial conditions has been shown analytically as well as numerically in the case of coupled Hénon maps. Numerical results are reported to show the proficiency of the proposed scheme to design extreme multistable synchronisation behaviour. This work establishes a theoretical foundation for constructing extreme multistable synchronised continuous as well as discrete dynamical systems.

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