An exploration of global stability of a three-dimensional dynamical system with a strong nonlinear term

机译：具有强大非线性术语的三维动力系统全球稳定性探讨

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In this short paper, we present an exploration of global stability of a seemingly simple yet complex three-dimensional dynamical system with a strong nonlinear term. We start with some physical backgrounds of the dynamical system in question, then, based on its dynamical behaviors near the stationary point and some physical intuition, we discuss certain aspects of existence and uniqueness of its corresponding Liapunov functions. The key contributions of this work are twofold: firstly, we introduce a straightforward numerical strategy to identify the stability areas within a finite distance from the stationary point; and secondly, we design a specific linear transformation to convert the original dynamical system derived from control theory to a dynamical system which belongs to a particular set of systems linked to the dynamics of mechanical systems containing a gyroscopic pendulum. The combination of both numerical and analytical approaches finally enables us to make some observations with respect to global stability and the prospect of numerical search algorithms for Liapunov functions. In particular, with the guidance of the discovered theorem, we hope we are able to generate new controllers for global asymptotic stabilization.

机译：在这篇短文中，我们探讨了具有强大非线性术语的看似简单而复杂的三维动态系统的全球稳定性。我们从有问题的动态系统的一些物理背景开始，然后，基于其在静止点附近的动态行为和一些物理直觉，我们讨论了其相应的Liapunov函数的存在和唯一性的某些方面。这项工作的主要贡献是双重的：首先，我们引入了一种直接的数值策略，以识别距静止点的有限距离内的稳定性区域;其次，我们设计了一种特定的线性变换，以将从控制理论导出的原始动态系统转换为属于与包含陀螺柱形的机械系统的动态组织的特定系统组的动态系统。两种数字和分析方法的组合最终使我们能够对Liapunov函数的数值搜索算法的全局稳定性和前景进行一些观察。特别是，随着发现定理的指导，我们希望我们能够为全球渐近稳定产生新的控制器。

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《MIT Conference on Computational Fluid and Solid Mechanics》|2001年||共4页
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作者
Kai Liu; Xiaodong Wang; Zhongping Jiang;
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中图分类 O35-53;
关键词
Global stability; Nonlinear dynamics; Liapunov function; Numerical method;

机译：全球稳定性;非线性动力学;Liapunov函数;数值方法;

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专利

1. NUMERICAL MODELING OF THE DYNAMICS AND STABILITY OF THREE-DIMENSIONAL GEOMETRICALLY NONLINEAR ROD SYSTEMS [J] . A. A. Lukyanov Russian engineering research . 2005,第4期

机译：三维几何非线性杆系统动力学和稳定性的数值模拟
2. The Analysis of Global Stability Boundary and Multistability in the Nonlinear Dynamical System of an Advanced Heavy Water Reactor [J] . Vikas Pandey, Suneet Singh Nuclear science and engineering . 2017,第2期

机译：先进重水反应堆非线性动力系统的整体稳定性边界和多重稳定性分析
3. Exponential and global stability of nonlinear dynamical systems relative to initial time difference [J] . Li A., Lv W., Ye J. Applied mathematics and computation . 2011,第12期

机译：非线性动力系统相对于初始时间差的指数和全局稳定性
4. An exploration of global stability of a three-dimensional dynamical system with a strong nonlinear term [C] . Kai Liu, Xiaodong Wang, Zhongping Jiang MIT Conference on Computational Fluid and Solid Mechanics Vol.2 Jun 12-15, 2001 . 2001

机译：具有强非线性项的三维动力学系统的全局稳定性的探讨
5. Aperiodic response and bifurcation of dynamic systems with strong local nonlinearities. [D] . Choi, Sang-Kyu. 1992

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6. An LMI Based Criterion for Global Asymptotic Stability of Discrete-Time State-Delayed Systems with Saturation Nonlinearities [O] . Priyanka Kokil 2014

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An exploration of global stability of a three-dimensional dynamical system with a strong nonlinear term

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