Abstract：
BACKGROUND: Studying chaotic dynamics in fractional- and integer-order dynamical systems has let researchers understand and predict the mechanisms of related non-linear phenomena.
OBJECTIVE: 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.
OBJECTIVE: 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.
摘要：
背景：研究分数阶和整数阶动力系统中的混沌动力学使研究人员能够理解和预测相关非线性现象的机理。
目的：分数阶和整数阶情况之间的相变是科学家广泛研究的主要问题之一，经济学家,和工程师。本文报告了在新的超混沌（Matouk\'s）系统中使用特定的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。结果得到了流域景点集合计算的支持，分岔图和Lyapunov指数谱。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。在分数阶Matouk系统的隐混沌吸引子的驱动状态和响应状态之间，还可以通过非线性控制器实现投影同步。
结果：动力学分析和计算机仿真结果验证了在Matouk\的超混沌系统中使用参数值的特定选择时，混沌吸引子只存在于分数阶情况下。
结论：讨论了仅在分数阶情况下出现的隐藏和自激混沌吸引子的存在示例。所以,获得的结果给出了第一个示例，该示例表明，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输。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的应用提供了新的挑战。
目的：分数阶和整数阶情况之间的相变是科学家广泛研究的主要问题之一，
方法：本文讨论了稳态解的稳定性分析，
结果：动力学分析和计算机仿真结果验证了在Matouk\的超混沌系统中使用参数值的特定选择时，混沌吸引子只存在于分数阶情况下。
结论：讨论了仅在分数阶情况下出现的隐藏和自激混沌吸引子的存在示例。
