Jacobi stability for dynamical systems of two-dimensional second-order differential equations and application to overhead crane system
Jacobi stability for dynamical systems of two-dimensional second-order differential equations and application to overhead crane system
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DOI：
10.1142/s0219887816500456
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发表时间：
2016-03
影响因子：
1.8
通讯作者：
T. Yajima;K. Yamasaki
中科院分区：
文献类型：
--
作者：
T. Yajima;K. Yamasaki
基于微分几何方法(KCC理论的Jacobi稳定性)研究了动力系统的几何结构。本文主要研究了二维二阶微分方程解的Jacobi稳定性与一维二阶微分方程解的Jacobi稳定性的差异。与一维情形不同的性质之一是由不同符号的偏离曲率的本征值给出的雅可比不稳定条件。然后，将该几何理论作为二维动力学系统应用于桥式起重机系统。分析了线性化桥式起重机的Hopf分叉与雅可比稳定性之间的关系。特别地，对于二维线性化系统的稳定螺线和不稳定螺线，找到了雅可比稳定轨迹。对于线性化桥式起重机系统，雅可比稳定螺线比雅可比不稳定螺线更快地逼近平衡点。这意味着雅可比稳定性与偏离弹道在过渡状态下的弹性有关。此外，对于非线性桥式起重机系统，极限环的雅可比稳定性随时间变化为稳定和不稳定。
Geometric structures of dynamical systems are investigated based on a differential geometric method (Jacobi stability of KCC-theory). This study focuses on differences of Jacobi stability of two-dimensional second-order differential equation from that of one-dimensional second-order differential equation. One of different properties from a one-dimensional case is the Jacobi unstable condition given by eigenvalues of deviation curvature with different signs. Then, this geometric theory is applied to an overhead crane system as a two-dimensional dynamical system. It is shown a relationship between the Hopf bifurcation of linearized overhead crane and the Jacobi stability. Especially, the Jacobi stable trajectory is found for stable and unstable spirals of the two-dimensional linearized system. In case of the linearized overhead crane system, the Jacobi stable spiral approaches to the equilibrium point faster than the Jacobi unstable spiral. This means that the Jacobi stability is related to the resilience of deviated trajectory in the transient state. Moreover, for the nonlinear overhead crane system, the Jacobi stability for limit cycle changes stable and unstable over time.