A generalization of a gradient system

Abstract

It is well known that gradient systems cannot have periodic orbits. In this work we find a general dynamical system on the plane without periodic orbits. We use Dulac's criterion that gives sufficient conditions for the non-existence of periodic orbits of dynamical systems in simply connected regions of the plane. Using a Dulac function we can rule out periodic orbits.

A Generalization of the Modified Liouville Equation

“…The Bendixson-Dulac criterion consists of a sufficient number of conditions for the nonexistence of periodic orbits in planar dynamical systems (Farkas, 1994). The modified Liouville equation (Abdelrahman et al, 2015;Salam et al, 2012) plays an important role in various areas of mathematical physics, from plasma physics and field theoretical modeling to fluid dynamics, using various transformations the differential equation can be written as a dynamic system that under some conditions does not have periodic orbits 2013a;Osuna and Villaseñor, 2011). The system in (Marin-Ramirez et al, 2015) coincides to our system.…”

Stability in a Holling-Tanner predator-prey model like a Kolmogorov system without periodic orbits via Dulac functions

“…In [6] it were studied quadratic systems using certain Dulac functions and a geometric method. In [7] it was found a general dynamical system on the plane without periodic orbits. In [8] it were studied the quadratic systems by means of the use of particular form of the Dulac functions.…”