With both analytical and numerical methods, local dynamic behaviors including stability and Hopf bifurcation of a new four-dimensional hyper-chaotic system are studied in this paper. All the equilibrium points and their stability conditions are obtained with the Routh–Hurwitz criterion. It is shown that there may exist one, two, or three equilibrium points for different system parameters. Via Hopf bifurcation theory, parameter conditions leading to Hopf bifurcation is presented. With the aid of center manifold and the first Lyapunov coefficient, it is also presented that the Hopf bifurcation is supercritical for some certain parameters. Finally, numerical simulations are given to confirm the analytical results and demonstrate the chaotic attractors of this system. It is also shown that the system may evolve chaotic motions through periodic bifurcations or intermittence chaos while the system parameters vary.

中文翻译：

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一种新的四维超混沌系统的稳定性和Hopf分岔分析

本文采用解析和数值方法，研究了一种新的四维超混沌系统的局部动力学行为，包括稳定性和Hopf分岔。所有的平衡点及其稳定条件都是通过 Routh-Hurwitz 准则获得的。结果表明，对于不同的系统参数，可能存在一个、两个或三个平衡点。通过Hopf分岔理论，给出了导致Hopf分岔的参数条件。借助中心流形和第一李雅普诺夫系数，还提出了Hopf分岔对于某些参数是超临界的。最后，通过数值模拟证实了分析结果并证明了该系统的混沌吸引子。