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A novel non-equilibrium fractional-order chaotic system and its complete synchronization by circuit implementation

机译：一种新颖的非平衡分数阶混沌系统及其电路实现的完全同步

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摘要

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 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.
机译：在本文中，我们构建了一个新颖的四维分数阶混沌系统。与迄今为止提出的所有混沌系统相比，最大的区别和最吸引人的地方是该系统中不存在平衡点。这些严格的方法，即梅尔尼科夫（Melnikov）方法和席尔尼科夫（Shilnikov）方法，无法在数学上证明在某些参数下这种系统中存在混沌。为了调解这种尴尬局面，我们借助电路仿真实验来完成此任务。在此之前，我们使用改进的Adams-Bashforth-Moulton数值算法来计算该分数阶混沌系统，并表明所提出的阶数低至3.28的分数阶系统表现出混沌吸引子。然后设计了一个q = 0.9阶的电子电路，从中我们可以观察到该分数阶系统中确实存在混沌吸引子。此外，基于拉普拉斯变换的最终值定理，在阶数q = 0.9的情况下，借助单向耦合方法实现了两个新颖的分数阶混沌系统的同步。为硬件实现设计了一种电子电路，以将两个新颖的分数阶混沌系统同步为同一阶。数值仿真和电路实验的结果彼此非常吻合，从而证明了所提出的分数阶系统确实存在混乱，并且单向耦合同步方法对该系统非常有效。

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