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Necessary conditions for existence of tensor invariants for general nonlinear dynamical systems

Zitong Zhao, Shaoyun Shi, Wenlei Li, Zhiguo Xu, Kaiyin Huang

Abstract

The integrability has been playing an essential role in the field of differential equations. This property may better help us obtain the topological structure and even the global dynamics for the considered system. A system is called integrable if it has a number of tensor invariants, which can comprehensively define the integrability problem. In this paper, we give necessary conditions for existence of tensor invariants for general nonlinear systems, especially semi-quasihomogeneous systems. Our results may be viewed as a generalization of Poincaré and Kozlov's work.

Necessary conditions for existence of tensor invariants for general nonlinear dynamical systems

Abstract

The integrability has been playing an essential role in the field of differential equations. This property may better help us obtain the topological structure and even the global dynamics for the considered system. A system is called integrable if it has a number of tensor invariants, which can comprehensively define the integrability problem. In this paper, we give necessary conditions for existence of tensor invariants for general nonlinear systems, especially semi-quasihomogeneous systems. Our results may be viewed as a generalization of Poincaré and Kozlov's work.
Paper Structure (8 sections, 14 theorems, 159 equations)

This paper contains 8 sections, 14 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Tensor invariants for general nonlinear systems
  3. Trivial tensor invariant
  4. Necessary condition for the existence of analytic tensor invariants
  5. Tensor invariants for semi-quasihomogeneous systems
  6. Quasihomogeneous systems and semi-quasihomogeneous systems
  7. Necessary condition for tensor invariants of semi-quasihomogeneous systems
  8. Examples

Key Result

Proposition 2.2

If the tensor field $T$ given by TF is trivial, then all of components $T_{j_{1}\cdots j_{q}}^{i_{1}\cdots i_{p}}(x)$ are constant and $p=q$. Moreover, $T_{j_{1}\cdots j_{p}}^{i_{1}\cdots i_{p}}=0$ for any $(j_1,\cdots,j_p)\notin \Theta_p$, where $\Theta_p= \{(j_1,\cdots,j_p)|(j_1,\cdots,j_p)=\sigma

Theorems & Definitions (35)

  • Definition 1.1
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 25 more