Lorenz系统的Jacobi稳定性分析
Jacobi stability analysis of the Lorenz system
摘要：
我们使用Jacobi稳定性分析或Kosambi-Cartan-Chern（KCC）理论来研究Lorenz系统的稳定性。Lorenz模型对于理解流体动力学不稳定性和湍流的本质起着重要的作用，也是研究非线性效应的重要测试对象。KCC理论是分析动力系统的一种强有力的数学方法。在这种方法中，我们描述的Lorenz系统在几何方面的演变，通过考虑它作为一个Finsler空间中的测地线。通过将一个非线性联络与一个Berwald型联络联系起来，得到了五个几何不变量，其中第二个不变量给出了系统的Jacobi稳定性. Jacobi（in）稳定性是赋以度量（黎曼或芬斯勒）的可微流形上的测地线流的（in）稳定性到非度量设置的自然推广。为了应用KCC理论，我们将Lorenz系统重新表示为两个二阶非线性微分方程组。几何不变量与此系统（非线性和Berwald连接），偏差曲率张量，以及它的特征值，明确获得。研究了Lorenz系统平衡点的Jacobi稳定性，得到了平衡点稳定的条件。最后，我们考虑了平衡点附近的偏差向量的分量的时间演化。
英文摘要：
We perform the study of the stability of the Lorenz system by using the Jacobi stability analysis, or the Kosambi-Cartan-Chern (KCC) theory. The Lorenz model plays an important role for understanding hydrodynamic instabilities and the nature of the turbulence, also representing a non-trivial testing object for studying non-linear effects. The KCC theory represents a powerful mathematical method for the analysis of dynamical systems. In this approach we describe the evolution of the Lorenz system in geometric terms, by considering it as a geodesic in a Finsler space. By associating a non-linear connection and a Berwald type connection, five geometrical invariants are obtained, with the second invariant giving the Jacobi stability of the system. The Jacobi (in)stability is a natural generalization of the (in)stability of the geodesic flow on a differentiable manifold endowed with a metric (Riemannian or Finslerian) to the non-metric setting. In order to apply the KCC theory we reformulate the Lorenz system as a set of two second order non-linear differential equations. The geometric invariants associated to this system (nonlinear and Berwald connections), and the deviation curvature tensor, as well as its eigenvalues, are explicitly obtained. The Jacobi stability of the equilibrium points of the Lorenz system is studied, and the condition of the stability of the equilibrium points is obtained. Finally, we consider the time evolution of the components of the deviation vector near the equilibrium points.
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