In this paper, the non-linear dynamical behavior of a 3D autonomous dissipative system of Hopf–Langford type is investigated. Through the help of a mode transformation (as the system’s energy is included) it is shown that the 3D nonlinear system can be separated of two coupled subsystems in the master (drive)-slave (response) synchronization type. After that, based on the computing first and second Lyapunov values for master system, we have attempted to give a general framework (from bifurcation theory point of view) for understanding the structural stability and bifurcation behavior of original system. Moreover, a family of exact solutions of the master system is obtained and discussed. The effect of synchronization on the dynamic behavior of original system is also studied by numerical simulations.

中文翻译：

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评估 Hopf-Langford 类型系统的非线性动力学

在本文中，研究了 Hopf-Langford 型 3D 自主耗散系统的非线性动力学行为。通过模式转换（因为包含系统能量）的帮助，表明 3D 非线性系统可以在主（驱动）-从（响应）同步类型中分离两个耦合子系统。之后，基于计算主系统的第一和第二李雅普诺夫值，我们试图给出一个通用的框架（从分叉理论的角度）来理解原始系统的结构稳定性和分叉行为。此外，获得并讨论了主系统的一系列精确解。还通过数值模拟研究了同步对原始系统动态行为的影响。