The two-parameter Hamiltonian system with the autonomous perturbation is considered. Via the Mel’nikov method, existence and uniqueness of a limit cycle of the system in a certain region of a two-dimensional space of parameters is proved.
This paper deals with the following question: does the asymptotic stability of the positive equilibrium of the Holling-Tanner model imply it is also globally stable? We will show that the answer to this question is negative. The main tool we use is the computation of Poincaré-Lyapunov constants in case a weak focus occurs. In this way we are able to construct an example with two limit cycles.
We prove that in quadratic perturbations of generic Hamiltonian vector fields with two saddle points and one center there can appear at most two limit cycles. This bound is exact.
The class of linear differential systems with coefficient matrices which are commutative with their integrals is considered. The results on asymptotic equivalence of these systems and their distribution among linear systems are given.
Given a second order differential equation on a manifold we find necessary and sufficient conditions for the existence of a coordinate system in which the system is linear. The main tool to be used is a linear connection defined by the system of differential equations.