Jacobi Analysis of the Rabinovich System

Liu Yongjian,1, Huang Qiujian2

 基金资助: 国家自然科学基金.  11961074广西自然科学基金重点项目.  2018GXNSFDA281028广西高校高水平创新团队项目.  [2018]35玉林师范学院高层次人才启动项目.  G2019ZK51

 Fund supported: the NSFC.  11961074the NSF of Guangxi Province.  2018GXNSFDA281028the High Level Innovation Team Program from Guangxi Higher Education Institutions of China.  [2018]35the Senior Talent Research Foundation of Yulin Normal University.  G2019ZK51

Abstract

In this paper, the differential geometry technique is used to study the complexity of the system. Jacobi stability of the three-dimensional Rabinovich system is analyzed from any point of trajectory of the system. Based on KCC-theory, the Jacobi stable conditions of all equilibrium points of the system are obtained. On the basis of obtaining the time evolution of the deviation vector and its components near the equilibrium points of the system, the instability exponent and curvature are introduced, and the chaos mechanism of the system is analyzed tentatively by combining numerical simulation. Numerical results validate the existing theoretical analysis results.

Keywords： KCC theory ; Jacobi stability ; Deviation curvature tensor ; Equilibrium point ; Chaos

Liu Yongjian, Huang Qiujian. Jacobi Analysis of the Rabinovich System. Acta Mathematica Scientia[J], 2021, 41(3): 783-796 doi:

1 绪论

Rabinovich系统源自于国际知名学者Pikovski、Rabinovich和Trakhtengerts在研究等离子体中三种共振耦合波相互作用问题时, 发现下述动力学方程(其常被简称为PRT系统)能准确地描述平行于磁场的等离子体波与离子声波的相互作用, 以及等离子体在低杂化共振附近的振荡的等离子体动力学行为[1]:

$\begin{eqnarray} \left\{ \begin{array}{lll} \dot{x} = hy-v_{1}x-yz, \\ \dot{y} = hx-v_{2}y+xz, \\ \dot{z} = xy-v_{3}z, \end{array} \right. \end{eqnarray}$

图 1

KCC理论是源于Kosambi[10], Cartan[11]和Chern[12]的开创性工作所提出的一种研究动力系统稳定性的几何动力学方法(geometrodynamical approach), 其基本思想是考虑到二阶动力系统与相关Finsler空间中的测地线方程之间存在一一对应关系. 准确而言, 其是构建整体轨迹偏离附近轨迹的变分方程的微分几何理论. 从动力系统的几何描述而言, 其将非线性联络和Berwald联络关联到微分系统中, 从而获得五个几何不变量. 其中, 第二个不变量, 也被称为偏离曲率张量, 能准确刻画系统的Jacobi稳定性(KCC理论也常被称为Jacobi分析方法). Jacobi稳定性事实上是将测地线流在具有度量(黎曼(Riemann)或芬斯勒(Finsler))的可微流形上的稳定性推广至非度量上的一种自然演化, 进而构成了KCC理论的重要组成部分[13]. Jacobi分析方法是从系统轨迹的任何一点出发, 通过计算系统轨迹偏离曲率张量所构成的不变量: 线性化特征值, 根据特征值的实部的符号来判断系统的稳定性. 可见它不仅能分析系统平衡点的稳定性, 也能分析没有平衡点的系统的轨线上任意一点处的稳定性情况. 这给含隐藏吸引子[14]系统的复杂性分析提供了一种很好的几何方法. 事实上, Maria等人自1997年开始建立了一个理论框架[15], 致力于动力系统行为的几何描述以及它们的混沌特性.

2 KCC理论和Jacobi稳定性

$(x) = (x_{1}, x_{2}, \ldots, x_{n})\in{\Bbb R}^n$, $(y) = (y_{1}, y_{2}, \ldots, y_{n})\in{\Bbb R}^n$, $y_{i} = {\rm d}x_{i}/{\rm d}t$. 考虑二阶微分方程组

$$$\frac{{\rm d}^{2}x_{i}}{{\rm d}t^{2}}+2G_{i}(x, y, t) = 0, i = 1, 2, \ldots, n,$$$

$\begin{eqnarray} \frac{{\rm D}\xi_{i}}{{\rm d}t} = \frac{{\rm d}\xi_{i}}{{\rm d}t}+N^{i}_{j}\xi_{j}, \end{eqnarray}$

$$$\frac{{\rm D}y_{i}}{{\rm d}t} = N^{i}_{j}y_{j}-2G_{i} = -\epsilon_{i}.$$$

$\epsilon_{i}$称为方程(2.1)的第一个KCC不变量. 方程(2.1)轨线$x_{i}(t)$一个微小扰动, 记为

$\begin{eqnarray} \tilde{x}_{i}(t) = x_{i}(t)+\eta\xi_{i}(t), \end{eqnarray}$

$|\eta|$是小参数, $\xi_{i}(t)$是沿着轨线$x_{i}(t)$定义的逆变矢量场的分量. 将方程(2.4)代入方程(2.1) 中, 取极限$\eta\rightarrow0$, 可获得下述偏离方程

$\begin{eqnarray} \frac{{\rm d}^{2}\xi_{i}}{{\rm d}t^{2}}+2N^{i}_{j}\frac{{\rm d}\xi_{j}}{{\rm d}t}+2\frac{\partial G_{i}}{\partial x_{j}}\xi_{j} = 0. \end{eqnarray}$

$\begin{eqnarray} \frac{{\rm D}^{2}\xi_{i}}{{\rm d}t^{2}} = P^{i}_{j}\xi_{j}, \end{eqnarray}$

$$$P^{i}_{j} = -2\frac{\partial G_{i}}{\partial x_{j}}-2G_{l}G^{i}_{jl}+y_{l}\frac{\partial N^{i}_{j}}{\partial x_{l}}+N^{i}_{l}N^{l}_{j}+\frac{\partial N^{i}_{j}}{\partial t},$$$

$$$P^{i}_{jk} = \frac{1}{3}\left(\frac{\partial P^{i}_{j}}{\partial y_{k}}-\frac{\partial P^{i}_{k}}{\partial y_{j}}\right), P^{i}_{jkl} = \frac{\partial P^{i}_{jk}}{\partial y_{l}}, D^{i}_{jkl} = \frac{\partial G^{i}_{jk}}{\partial y_{l}}.$$$

$\begin{eqnarray} \left\{ \begin{array}{lll} \ddot{x} = h\dot{y}-v_{1}\dot{x}-\dot{y}z-y\dot{z}, \\ \ddot{y} = h\dot{x}-v_{2}\dot{y}+\dot{x}z+x\dot{z}, \\ \ddot{z} = \dot{x}y+x\dot{y}-v_{3}\dot{z}. \end{array} \right. \end{eqnarray}$

$x = x_{1}$, $\dot{x} = y_{1}$, $y = x_{2}$, $\dot{y} = y_{2}$, $z = x_{3}$, $\dot{z} = y_{3}$, 则Rabinovich系统可化为

$\begin{eqnarray} \left\{ \begin{array}{lll} \ddot{x_{1}}-hy_{2}+v_{1}y_{1}+x_{3}y_{2}+x_{2}y_{3} = 0, \\ \ddot{x_{2}}-hy_{1}+v_{2}y_{2}-x_{3}y_{1}-x_{1}y_{3} = 0, \\ \ddot{x_{3}}-x_{2}y_{1}-x_{1}y_{2}+v_{3}y_{3} = 0. \end{array} \right. \end{eqnarray}$

$\begin{eqnarray} \lambda^{3} -(P_{1}^{1}+P_{2}^{2}+P_{3}^{3})\lambda^{2}+(P_{1}^{1}P_{2}^{2}+P_{1}^{1}P_{3}^{3}+P_{2}^{2}P_{3}^{3} +P^{1}_{3}P^{3}_{1}+P^{2}_{3}P^{3}_{2}+P^{1}_{2}P^{2}_{1})\lambda {}\\ -(P_{1}^{1}P_{2}^{2}P_{3}^{3}+P^{1}_{2}P^{2}_{3}P^{3}_{1}+P^{1}_{3}P^{2}_{1}P^{3}_{2}+ P^{1}_{3}P_{2}^{2}P^{3}_{1}+P_{1}^{1}P^{2}_{3}P^{3}_{2}+P^{1}_{2}P^{2}_{1}P_{3}^{3}) = 0. \end{eqnarray}$

 参数取值 平衡点 线性稳定性态 Jacobi稳定性态 $h = 5.5$ $S_0$ 不稳定 不稳定 $S_1, S_2$ 不稳定 不稳定 $h = 6.75$ $S_0$ 不稳定 不稳定 $S_1, S_2$ 不稳定 不稳定 $h = 10$ $S_0$ 不稳定 不稳定 $S_1, S_2$ 不稳定 不稳定 $h = 15$ $S_0$ 不稳定 不稳定 $S_1, S_2$ 不稳定 不稳定

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