Invariant hyperplanes and Darboux integrability for d-dimensional polynomial differential systems

AbstractFor a class of polynomial differential systems of degree (m1,…,md) in Rd which is open and dense in the set of all polynomial differential systems of degree (m1,…,md) in Rd, we study the maximal number of invariant hyperplanes. This is a well known problem in dimension d=2 (see for instance [1,12,16]). Furthermore, using the Darboux theory of integrability we analyse when can be possible to find a first integral of a polynomial vector field of degree (m1,…,md) in Rd by knowing the existence of a sufficient number of invariant hyperplanes

A survey on stably dissipative Lotka-Volterra systems with an application to infinite dimensional Volterra equations

For stably dissipative Lotka{Volterra equations the dynamics on the attractor are Hamiltonian and we argue that complex dynamics can occur. We also present examples and properties of some infinite dimensional Volterra systems with applications related with stably dissipative Lotka-Volterra equations. We finish by mentioning recent contributions on the subject

Final evolutions of a class of May-Leonard Lotka-Volterra systems

We study a particular class of Lotka-Volterra 3-dimensional systems called May-Leonard systems, which depend on two real parameters a and b, when a + b = −1. For these values of the parameters we shall describe its global dynamics in the compactification of the non-negative octant of ℝ3 including its infinity. This can be done because this differential system possesses a Darboux invariant