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Modeling of nonlinear dynamical systems based on deterministic learning and structural stability

基于确定学习和结构稳定理论的非线性动态系统建模

Abstract

Recently, a deterministic learning (DL) theory was proposed for accurate identification of system dynamics for nonlinear dynamical systems. In this paper, we further investigate the problem of modeling or identification of the partial derivative of dynamics for dynamical systems. Firstly, based on the locally accurate identification of the unknown system dynamics via deterministic learning, the modeling of its partial derivative of dynamics along the periodic or periodic-like trajectory is obtained by using the mathematical concept of directional derivative. Then, with accurately identified system dynamics and the partial derivative of dynamics, a C 1-norm modeling approach is proposed from the perspective of structural stability, which can be used for quantitatively measuring the topological similarities between different dynamical systems. This provides more incentives for further applications in the classification of dynamical systems and patterns, as well as the prediction of bifurcation and chaos. Simulation studies are included to demonstrate the effectiveness of this modeling approach.

摘要

创新点

本文的创新点主要体现在以下两方面:

  1. 1)

    基于结构稳定和确定学习理论, 提出了 C1 -范数下的动力学建模方法。 在已有研究中, 结构稳定概念主要限于定性分析系统在参数微小扰动下, 系统结构的改变; 而本文给出的 C1 范数模型为定量计算非线性系统之间的动力学差异提供了工具。 该模型可进一步用于非线性系统及动态模式的分类, 分岔和混沌的预测等实际问题。

  2. 2)

    通过引入沿轨迹的方向导数这一数学概念, 实现了对系统偏导部分动力学的准确建模。 偏导部分的动力学信息, 是判断受扰系统结构稳定与否的必要部分。 确定学习算法获得了对系统动力学的准确辨识和建模, 在此基础上, 通过方向导数完成了偏导部分动力学的准确建模, 继而实现了 C1 范数意义下的系统动力学度量。

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Correspondence to Cong Wang.

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Cite this article

Chen, D., Wang, C. & Dong, X. Modeling of nonlinear dynamical systems based on deterministic learning and structural stability. Sci. China Inf. Sci. 59, 92202 (2016). https://doi.org/10.1007/s11432-015-5498-0

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Keywords

  • system modeling
  • system identification
  • deterministic learning
  • nonlinear dynamics
  • structural stability
  • topological equivalence

关键词

  • 系统建模
  • 系统辨识
  • 确定学习
  • 非线性动力学
  • 结构稳定
  • 拓扑等价