A class of vector fields on manifolds containing second order ODEs (Q1917222)

The author studies systems of differential equations on manifolds which are locally represented in the form \(x'= P(x) y\), \(y'= g(x, y)\), where \(P\) is a matrix-function. In the case \(P= \text{Id}\), the system corresponds to a second-order ODE. A lot of mechanical systems can be represented in this form. It is shown that if the manifold is compact, then a generic system has a finite number of equilibrium points, and these points are hyperbolic.