Coexistence of attractors in integer- and fractional-order three-dimensional autonomous systems with hyperbolic sine nonlinearity: Analysis, circuit design and combination synchronisation

This paper reports the results of the analytical, numerical and analogical analyses of integer- and fractional-order chaotic systems with hyperbolic sine nonlinearity (HSN). By varying a parameter, the integer order of the system displays transcritical bifurcation and new complex shapes of bistable double-scroll chaotic attractors and four-scroll chaotic attractors. The coexistence among four-scroll chaotic attractors, a pair of double-scroll chaotic attractors and a pair of point attractors is also reported for specific parameter values. Numerical results indicate that commensurate and incommensurate fractional orders of the systems display bistable double-scroll chaotic attractors, four-scroll chaotic attractors and coexisting attractors between a pair of double-scroll chaotic attractors and a pair of point attractors. Moreover, the physical existence of chaotic attractors and coexisting attractors found in the integer-order and commensurate fractional-order chaotic systems with HSN is verified using PSIM software. Numerical simulations and PSIM results have a good qualitative agreement. The results obtained in this work have not been reported previously in three-dimensional autonomous system with HSN and thus represent an enriching contribution to the understanding of the dynamics of this class of systems. Finally, combination synchronisation of such three-coupled identical commensurate fractional-order chaotic systems is analysed using the active backstepping method.