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In this paper, we study the dynamics and control of a Caputo fractional difference form of the Duffing map. We use phase plots, bifurcation diagrams, and Lyapunov exponents to establish the existence of chaos over a wide range of fractional orders and examine the nature of the dynamics. Also, we present the 0–1 test to detect chaos and C 0 complexity, which is an alternative nonlinear statistical measure that can quantify the regularity of a time series. In addition, we measure the approximate entropy to see the performance of our numerical results. Through phase plots and bifurcation diagrams, it is shown that the proposed fractional map exhibits a range of different dynamical behaviors including chaos and coexisting attractors. A one-dimensional feedback stabilization controller is proposed. The asymptotic convergence of the proposed controller is established by means of the stability theory of linear fractional order discrete-time systems. Simulation results have been carried out to illustrate the findings of the study. [ABSTRACT FROM AUTHOR]

Chaotic systems with no equilibrium are a very important topic in nonlinear dynamics. In this paper, a new fractional order discrete-time system with no equilibrium is proposed, and the complex dynamical behaviors of such a system are discussed numerically by means of a bifurcation diagram, the largest Lyapunov exponents, a phase portrait, and a 0–1 test. In addition, a one-dimensional controller is proposed. The asymptotic convergence of the proposed controller is established by means of the stability theory of linear fractional order discrete-time systems. Next, a synchronization control scheme for two different fractional order discrete-time systems with hidden attractors is reported, where the master system is a two-dimensional system that has been reported in the literature. Numerical results are presented to confirm the results. [ABSTRACT FROM AUTHOR]

In this paper, we propose two control strategies for the Q–S synchronization of fractional-order discrete-time chaotic systems. Assuming that the dimension of the response system m is higher than that of the drive system n, the first control scheme achieves n-dimensional synchronization whereas the second deals with the m-dimensional case. The stability of the proposed schemes is established by means of the linearization method. Numerical results are presented to confirm the findings of the study. [ABSTRACT FROM AUTHOR]

This paper studies the synchronisation of integer- and fractional-order discrete-time chaotic systems with different dimensions. Control laws are proposed for the full-state hybrid projective synchronisation (FSHPS) of a master–slave pair, where the difference equations of the master have an integer order while those of the slave have a fractional order. Moreover, inverse FSHPS laws are proposed for a fractional-order master and an integer-order slave. The Lyapunov stability theory of integer-order maps and the stability theory of linear fractional-order maps are utilised to establish the asymptotic stability of the zero equilibrium corresponding to the synchronisation error system. Numerical results are presented to verify the findings of the study. [ABSTRACT FROM AUTHOR]