Abstract
A new finding is proposed for multi-fractional order of neural networks by multi-time delay (MFNNMD) to obtain stable chaotic synchronization. Moreover, our new result proved that chaos synchronization of two MFNNMDs could occur with fixed parameters and initial conditions with the proposed control scheme called sliding mode control (SMC) based on the time-delay chaotic systems. In comparison, the fractional-order Lyapunov direct method (FLDM) is proposed and is implemented to SMC to maintain the systems’ sturdiness and assure the global convergence of the error dynamics. An extensive literature survey has been conducted, and we found that many researchers focus only on fractional order of neural networks (FNNs) without delay in different systems. Furthermore, the proposed method has been tested with different multi-fractional orders and time-delay values to find the most stable MFNNMD. Finally, numerical simulations are presented by taking two MFNNMDs as an example to confirm the effectiveness of our control scheme.
Highlights
The concept and the knowledge of fractional-order calculus have existed since L’Hopital’s and Leibniz’s contribution in 1695s [1, 2], its applications to the real world of mathematics, physics engineering, and biology are only of interest recently [3, 4]
We introduced the global Mittag–Leffler MFNNMD model of projective synchronization in our work and designed a new multidelay sliding mode control (SMC) controller
Two definitions of the most conventional fractional-order calculus derivatives used are mentioned in the literature: the Caputo and the Riemann–Liouville. en, the Caputo derivative is implemented in this work because its initial conditions may be described in terms of integerorder derivatives, which is more practical in practice
Summary
The concept and the knowledge of fractional-order calculus have existed since L’Hopital’s and Leibniz’s contribution in 1695s [1, 2], its applications to the real world of mathematics, physics engineering, and biology are only of interest recently [3, 4]. E fractional-order chaotic system synchronization has recently garnered attention because of its capability of application in a diversity of physics and engineering science fields, including encryption and secure communication. The first study on the significance of time delay on chaotic behavior was in [34] based on the previous literature Most of these papers concern about ordinary differential equations or integer orders, and there are preliminary results concerning chaos synchronization of FNNs with multi-time-delay systems. Numerous researchers have attempted to employ the FNNs with multi-time-delay systems to research secure communication, and they have come up with some exciting results in [35,36,37,38].
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