Abstract
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.
Highlights
Interest has grown in the subject of discrete fractional calculus and its application in science and engineering.1,2 This gave rise to many fractional order chaotic discrete-time systems.3–9 From what has been reported, the dynamical behaviors of fractional order discrete-time systems are heavily dependent on the fractional order, which introduces new degrees of freedom and makes them more suitable for secure communications and encryption.10 Research on control and synchronization of such systems has been widely investigated.11–15Until now, great consideration has been given to the study of no equilibrium fractional order chaotic systems due to their practical application in engineering systems
The attractors associated with these systems are called “hidden attractors.”. Such attractors are observed in nonlinear systems with no equilibrium point or with only stable points or in nonlinear systems with line equilibrium points
This paper has introduced a new fractional order discretetime system with hidden chaotic attractors
Summary
Interest has grown in the subject of discrete fractional calculus and its application in science and engineering. This gave rise to many fractional order chaotic discrete-time systems. From what has been reported, the dynamical behaviors of fractional order discrete-time systems are heavily dependent on the fractional order, which introduces new degrees of freedom and makes them more suitable for secure communications and encryption. Research on control and synchronization of such systems has been widely investigated.. Interest has grown in the subject of discrete fractional calculus and its application in science and engineering.. Interest has grown in the subject of discrete fractional calculus and its application in science and engineering.1,2 This gave rise to many fractional order chaotic discrete-time systems.. Great consideration has been given to the study of no equilibrium fractional order chaotic systems due to their practical application in engineering systems. There are only a few investigations of hidden attractors in discrete-time chaotic systems.. There are only a few investigations of hidden attractors in discrete-time chaotic systems.18–21 Based on this consideration, a new fractional order discretetime system with no equilibrium is developed.
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