Abstract
In this research paper, we solve the problem of synchronization and anti-synchronization of chaotic systems described by discrete and time-delayed variable fractional-order differential equations. To guarantee the synchronization and anti-synchronization, we use the well-known PID (Proportional-Integral-Derivative) control theory and the Lyapunov–Krasovskii stability theory for discrete systems of a variable fractional order. We illustrate the results obtained through simulation with examples, in which it can be seen that our results are satisfactory, thus achieving synchronization and anti-synchronization of chaotic systems of a variable fractional order with discrete time delay.
Highlights
We present in this research paper the solution to the problem of synchronization [1]and anti-synchronization [2] of discrete chaotic systems described by systems of differential equations of a variable fractional order [3] with time delay [4]
Anti-synchronization [2] of discrete chaotic systems described by systems of differential equations of a variable fractional order [3] with time delay [4]
The synchronization and anti-sinchronization problem of discrete fractional-order chaotic systems in time is solved by means of control laws (14) and (21), which are obtained using the stability analysis through the Lyapunov–Krasovskii and PID control laws for fractional-order systems, so we ensure that ∆(V (e)) < 0 ∀e 6= 0, and lim e(k ) = 0, ∆(V) < 0 k→∞
Summary
We present in this research paper the solution to the problem of synchronization [1]. We refer to the master–slave system, and even though the results obtained are for these two systems, the methodology can be used for other nonlinear discrete time systems of a variable fractional order with time delay in the Caputo sense. In this investigation, the Rossler system is forced to follow, or synchronize with, and anti-synchronize with the chaotic Chen system. The Rossler system is forced to follow, or synchronize with, and anti-synchronize with the chaotic Chen system Both systems are described, as mentioned above, by means of differential equations with a discrete and variable fractional order with time delay.
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