非线性动力系统多项式首次积分的不存在性

On the Nonexistence of Polynomial First Integrals for Nonlinear Dynamical Systems

关键词:  多项式  首次积分  可积性

Key words:  polynomial  first integrals  integrability

摘要:

给出当f₁,…,f_n为非线性多项式时,系统dx_i(t)/dt=f_i(x₁(t),…,x_n(t))不存在多项式首次积分的1个充分条件:矩阵A的特征根λ₁,…,λ_n不满足任何非共振条件k₁λ₁+…+k_nλ_n=0,k₁,…,k_n∈Z⁺,∑_i=1ⁿk_i > 0.

Abstract:

The paper shows a sufficient condition that the system dx_i(t)/dt=f_i(x₁(t),…,x_n(t)) does not admit polynomial first intergals when f₁,…,f_n are nonlinear polynomial:the eigenvalues λ₁,…,λ_n of the matrix A are not satisfied with any nonresonant condition k₁λ₁+…+k_nλ_n=0,k₁,…,k_n∈Z⁺,∑_i=1ⁿk_i > 0.