Abstract
We study the modified Liouville equation using various transformations to build dynamical systems and we use Dulac's criterion for give sufficient conditions of the non-existence of periodic orbits in the dynamical systems generated of the modified Liouville equation.
Highlights
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 (Marin et al, 2014; 2013a; Osuna and Villaseñor, 2011)
Example 4.2 If we consider that c2 has a first derivative and it is invertible such that c '2 c2−1(z) exists for all z in which c2−1(z) is defined, we have the generalized modified Liouville equation:
Summary
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 (Marin et al, 2014; 2013a; Osuna and Villaseñor, 2011). A Dulac function for a quadratic system was found in (Marin et al., 2013b). A Dulac function and a geometric method for a quadratic system was studied in (Marin-Ramirez et al., 2014). In this article our objective is construct dynamical systems that does not have periodic orbits using Dulac functions and we use the following criterion to show the non-existence of periodic orbits. The Dulac criterion was used in (Rana, 2015)
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