三维Chen系统的动力学行为分析

Analysis of Dynamical Behavior in a Three-Dimensional Chen System

摘要: 本文利用谱理论和动态跃迁理论研究了三维Chen系统的动力学行为。利用谱理论，得到了该系统发生两次动态跃迁的临界控制参数c₁和c₂，并给出了相应的PES条件。基于动态跃迁理论，得到了该系统发生两次跃迁的相关结论。首先，该系统在c₁处发生连续型跃迁，原有平衡态失稳，并分歧出两个渐近稳定的奇点。其次，随着控制参数c的持续增大，该系统在c₂处发生两种不同类型的跃迁。在一定条件下，该系统发生跳跃型跃迁，并分歧出不稳定的周期轨道；而在相反的条件下发生连续型跃迁，分歧出渐近稳定的周期轨道。

Abstract: This paper investigates the dynamical behavior of the three-dimensional Chen system using spectral theory and dynamic transition theory. By applying spectral theory, the critical control parameters c₁ and c₂ for the two dynamic transitions of the system are obtained, along with the corresponding PES conditions. Based on dynamic transition theory, conclusions regarding the two transitions are derived. First, the system undergoes a continuous transition at c₁, where the original equilibrium state loses stability and bifurcates into two asymptotically stable singular points. Second, as the control parameter c continues to increase, the system undergoes two different types of transitions at c₂. Under certain conditions, the system undergoes a jump transition and bifurcates into an unstable periodic orbit; under the opposite conditions, a continuous transition occurs, bifurcating into an asymptotically stable periodic orbit.