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

TL;DR

This paper establishes necessary conditions for the existence of tensor invariants in general nonlinear dynamical systems, especially semi-quasihomogeneous ones, extending classical results by Poincaré and Kozlov.

Contribution

It provides a generalized framework for understanding tensor invariants in nonlinear systems, advancing the theoretical foundation of integrability conditions.

Findings

Derived necessary conditions for tensor invariants in nonlinear systems

Extended classical integrability results to semi-quasihomogeneous systems

Enhanced understanding of the topological structure of integrable 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 Poincar\'e and Kozlov's work.