Introduction

Studying chaotic dynamics in fractional- and integer-order dynamical systems has let researchers understand and predict the mechanisms of related non-linear phenomena.

Objectives

Phase transitions between the fractional- and integer-order cases is one of the main problems that have been extensively examined by scientists, economists, and engineers. This paper reports the existence of chaotic attractors that exist only in the fractional-order case when using the specific selection of parameter values in a new hyperchaotic (Matouk’s) system.

Methods

This paper discusses stability analysis of the steady-state solutions, existence of hidden chaotic attractors and self-excited chaotic attractors. The results are supported by computing basin sets of attractions, bifurcation diagrams and the Lyapunov exponent spectrum. These tools verify the existence of chaotic dynamics in the fractional-order case; however, the corresponding integer-order counterpart exhibits quasi-periodic dynamics when using the same choice of initial conditions and parameter set. Projective synchronization via non-linear controllers is also achieved between drive and response states of the hidden chaotic attractors of the fractional Matouk’s system.

Results

Dynamical analysis and computer simulation results verify that the chaotic attractors exist only in the fractional-order case when using the specific selection of parameter values in the Matouk’s hyperchaotic system.

Conclusions

An example of the existence of hidden and self-excited chaotic attractors that appears only in the fractional-order case is discussed. So, the obtained results give the first example that shows chaotic states are not necessarily transmitted between fractional- and integer-order dynamical systems when using a specific selection of parameter values. Chaos synchronization using the hidden attractors’ manifolds provides new challenges in chaos-based applications to technology and industrial fields.

中文翻译：

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只存在于分数阶情况下的混沌吸引子

介绍

研究分数阶和整数阶动力系统中的混沌动力学让研究人员能够理解和预测相关非线性现象的机制。

目标

分数阶和整数阶情况之间的相变是科学家、经济学家和工程师广泛研究的主要问题之一。本文报告了在新的超混沌 (Matouk's) 系统中使用特定参数值选择时仅在分数阶情况下存在的混沌吸引子的存在。

方法

本文讨论了稳态解的稳定性分析、隐混沌吸引子和自激混沌吸引子的存在性。结果得到计算盆地景点集、分叉图和李雅普诺夫指数谱的支持。这些工具验证了分数阶情况下混沌动力学的存在；然而，当使用相同的初始条件和参数集选择时，相应的整数阶对应物表现出准周期动力学。分数 Matouk 系统的隐藏混沌吸引子的驱动和响应状态之间也实现了通过非线性控制器的投影同步。

结果

动力学分析和计算机仿真结果证实，在 Matouk 超混沌系统中使用特定参数值选择时，混沌吸引子仅存在于分数阶情况。

结论

讨论了仅在分数阶情况下存在的隐藏和自激混沌吸引子的示例。因此，所获得的结果给出了第一个示例，表明当使用特定的参数值选择时，混沌状态不一定在分数阶和整数阶动力系统之间传递。使用隐藏吸引子流形的混沌同步为基于混沌的技术和工业领域应用提供了新的挑战。