Abstract
In this paper, we study the synchronization of a nonlinear fractional system, and analyze its time response and chaotic behaviors. We represent a solution for considered system by employing the Mittag-Leffler matrix function and Jacobian matrix. Thereafter, we prove synchronization of error system between drive-response systems using stability theory and linear feedback control methods. Finally, numerical simulations are presented to show the effectiveness of the theoretical results.
Highlights
Fractional calculus has over 300 years of history
Sufficient conditions for synchronization of fractional-order chaotic systems via linear control were investigated in [9], synchronization of fractional chaotic systems with different orders was studied in [10] using stability concepts and synchronization of the Lotka-Volterra chaotic system was studied using the active control method in [11]
Fractional chaotic system, global asymptotic synchronization and adaptive sliding mode synchronization have been studied in [13,14,15] and sufficient conditions were presented for exponential synchronization of fractional order chaotic systems in [16]
Summary
Fractional calculus has over 300 years of history. Recently, it has attracted increasing interest due to its potential applications physics and engineering such as viscoelastic systems [1], dielectric polarization [2], electrode-electrolyte polarization [3], electromagnetic waves [4], quantitative finance [5], and quantum evolution of complex systems [6]. Robust synchronization for chaotic and hyperchaotic fractional order systems with model uncertainties and disturbances was investigated in [12]. Finite time stability and synchronization of fractional order chaotic system with uncertainties and disturbance was studied in [17]. We present sufficient conditions for synchronization of nonlinear fractional order chaotic system by using asymptotic stability theory for Mittag-Leffler matrix function, Jacobian matrix and linear feedback controller. We introduce a Jacobian matrix for the nonlinear term and represent the solution for error system between drive-response systems. We numerically check the time response for drive-response systems for different fractional orders and employ the linear feedback controller to synchronize the drive-response systems based on the chaotic behavior and state trajectories
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