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A novel non-equilibrium fractional-order chaotic system and its complete synchronization by circuit implementation. (English) Zbl 1243.93033

Summary: In this paper, we construct a novel four dimensional fractional-order chaotic system. Compared with all the proposed chaotic systems until now, the biggest difference and most attractive place is that there exists no equilibrium point in this system. Those rigorous approaches, i.e., Melnikov’s and Shilnikov’s methods, fail to mathematically prove the existence of chaos in this kind of system under some parameters. To reconcile this awkward situation, we resort to circuit simulation experiment to accomplish this task. Before this, we use an improved version of the Adams-Bashforth-Moulton numerical algorithm to calculate this fractional-order chaotic system and show that the proposed fractional-order system with the order as low as 3.28 exhibits a chaotic attractor. Then an electronic circuit is designed for order \(q=0.9\), from which we can observe that chaotic attractor does exist in this fractional-order system. Furthermore, based on the final value theorem of the Laplace transformation, synchronization of two novel fractional-order chaotic systems with the help of one-way coupling method is realized for order \(q=0.9\). An electronic circuit is designed for hardware implementation to synchronize two novel fractional-order chaotic systems for the same order. The results for numerical simulations and circuit experiments are in very good agreement with each other, thus proving that chaos exists indeed in the proposed fractional-order system and the one-way coupling synchronization method is very effective to this system.

MSC:

93B51 Design techniques (robust design, computer-aided design, etc.)
34H10 Chaos control for problems involving ordinary differential equations
34A08 Fractional ordinary differential equations
93C15 Control/observation systems governed by ordinary differential equations
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