Nonintegrability of dynamical systems with homo- and heteroclinic orbits
The present paper concerns the integrability of a general differential equation \(\dot{x}=f(x)\) where \(f:\mathbb{R}^n\rightarrow \mathbb{R}^n\) is analytic. This equation is assumed to have equilibria connected by heteroclinic orbits on an \((n-2)\)-dimensional locally invariant manifold \(\mathcal{N}\), but it may be non-Hamiltonian and the equilibria do not have to be saddle centers. The local invariance means that the trajectory \(x(t)\) may pass through \(\partial \mathcal{N}\) and escape \(\mathcal{N}\) even if \(x(0)\in \mathcal{N}\). The equilibria are also allowed to be the same and connected by a homoclinic orbit. The main result is that, under some additional assumptions, the monodromy group of the \textit{normal variational equation} associated to the given differential equation is diagonalizable or infinitely cyclic. The used tools belong to differential Galois theory for linear differential equations.
scientific article; zbMATH DE number 6712369
scientific article