Abstract
In this paper we derived some simple criteria for non-existence and partial non-existence Laurent polynomial rst integrals for a general nonlinear systems of ordinary dieren tial equations _ x = f(x); x 2 R n with f(0) = 0. We show that if the eigenvalues of the Jacobi matrix of the vector eld f(x) are Z-independent, then the system has no nontrivial Laurent polynomial
Highlights
IntroductionIs called completely integrable, if it has a sufficiently rich set of first integrals such that its solutions can be expressed by these integrals
A system of differential equations x = f (x) is called completely integrable, if it has a sufficiently rich set of first integrals such that its solutions can be expressed by these integrals
Liouville-Arnold theorem, a Hamiltonian system with n degrees of freedom is integrable if it possesses n independent integrals of motion in involution
Summary
Is called completely integrable, if it has a sufficiently rich set of first integrals such that its solutions can be expressed by these integrals. In [3], Furta suggested a simple and verifiable criterion of non-existence of nontrivial analytic integrals for general analytic autonomous systems He proved that if the eigenvalues of the Jacobi matrix of the vector field f (x) at some fixed point are N-independent, the system x = f (x) has no nontrivial integral analytic in a neighborhood of this fixed point. (k1,···,kn)∈A where Pk1···kn ∈ C and A, the support of P (x), is a finite subset of integer group Zn. We will give a simple criterion for non-existence of Laurent polynomial first integrals for general nonlinear analytic systems.
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