Original research articleFractional chaos synchronization schemes for different dimensional systems with non-identical fractional-orders via two scaling matrices
Introduction
The study of fractional order chaos has become an interdisciplinary field of many scientific disciplines. Meanwhile, the fractional order chaotic system applications has been found to be very useful and has great potential in many engineering-oriented applied fields such as dielectric polarization, electromagnetic waves, quantitative finance, quantum evolution of complex systems, viscoelastic systems, and electrode–electrolyte polarization [1], [2], [3], [4], [5].
Synchronization of fractional order chaotic dynamical systems has starts to attract increasing attention of many researchers since the synchronization of chaotic system with integer order is understood well [6], [7] and widely explored [8], [9]. The synchronization problem means making two systems oscillate in a synchronized manner. The dynamical behavior of two copies of a fractional order chaotic system may be identical after a transient when the second system is driven by the first one. Several different approaches, have already been successfully applied to the problem such as active control [10], [11], [12], projective synchronization [13], [14], [15], [16], [17], adaptive synchronization [18], [19], [20], [21], complete and generalized synchronization [22], [23], [24], [25], [26], exponential synchronization [27], Q-S synchronization [28], linear and nonlinear feedback synchronization [29], [30] and single state fractional-order controller technique [31], [32], etc. To the best of our knowledge, most of the existing papers discuss the synchronization between two fractional order chaotic or hyperchaotic with the same dimension and which also unable to synchronize the fractional order chaotic systems in different dimension since, in many real physics systems, the synchronization is carried out through the oscillators with different dimension.
Recently, Ouannas and Al-sawalha [33], [34] introduced a new approach to synchronize different dimensional chaotic system in continuous-time and discret-time using two scaling matrices. This work is to further develop the new approach to synchronize different dimensional fractional order chaotic system and hyperchaotic even though they have different dimensions using two scaling matrices. However, the synchronization controller is designed based on fractional Lyapunov method. An analytic expression of the controller is shown. Finally, illustrative examples of fractional order chaotic and hyperchaotic systems are used to show the effectiveness of the proposed method.
Section snippets
Preliminaries
There are some definitions for fractional derivative. The commonly used definition is the Caputo fractional derivative [35], [36], [37], defined bywhere m = [p], i.e., m is the first integer which is not less than p, xm is the m-order derivative, and Jq(q > 0) is the q-order Reimann–Liouville integral operator with expression:where Γ denotes Gamma function.
Problem description and control design for synchronization
The purpose of this paper is to discuss two schemes of synchronization: the first scheme is propose when the synchronization dimension d = m in dimension m, and the second one is construct when the synchronization dimension d = n in dimension n.
Numerical simulations
In this section, to verify and demonstrate the effectiveness of the proposed method, we consider two numerical examples.
Conclusion
In this paper, new control approaches were presented to study the problem of Θ − Φ synchronization with scaling constant matrix and scaling function matrix between n-dimensional fractional master system and m-dimensional fractional slave system was investigated. The new control schemes were proved theoretically using nonlinear controllers fractional Lyapunov function method. Firstly, to achieve Θ − Φ synchronization with respect to dimension m, the synchronization criterion was obtained via
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