HIKARI Ltd, www.m-hikari.com. A Generalization of a Gradient System. A. M. Marin. Faculty of Exact and Natural Sciences, University of Cartagena. Sede Piedra ...
International Mathematical Forum, Vol. 8, 2013, no. 17, 803 - 806 HIKARI Ltd, www.m-hikari.com
A Generalization of a Gradient System A. M. Marin Faculty of Exact and Natural Sciences, University of Cartagena Sede Piedra de Bolivar, Avenue of Consulado Cartagena de Indias, Bolivar, Colombia R. D. Ortiz Faculty of Exact and Natural Sciences, University of Cartagena Sede Piedra de Bolivar, Avenue of Consulado Cartagena de Indias, Bolivar, Colombia J. A. Rodriguez Morelia Institute of Technology, Morelia, Mexico c 2013 A. M. Marin et al. This is an open access article distributed under Copyright the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
Mathematics Subject Classification: 34A34, 34C25 Keywords: Dulac functions; Quasilinear partial differential equations; Bendixon–Dulac criterion
1
Introduction
It is important to study in differential equations the analysis of the periodic orbits that there are in a system in the plane. It is well known certain sys-
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tems have no limit cycles as gradient systems. For this should be considered: Bendixson’s criteria, indices, invariant lines and critical points. See [1, 3, 6, 7]. In this paper we are interested in constructing a general system that not have periodic orbits using Dulac functions. We use the criterion of Bendixson– Dulac see [2] Theorem 1.1. (Bendixson–Dulac criterion)(See [4]) Let f1 (x1 , x2 ), f2 (x1 , x2 ) and h(x1 , x2 ) be functions C 1 in a simply connected domain D ⊂ R2 such that ∂(f1 h) 2 h) + ∂(f does not change sign in D and vanishes at most on a set of ∂x1 ∂x2 measure zero. Then the system x˙ 1 = f1 (x1 , x2 ), (1) x˙ 2 = f2 (x1 , x2 ), (x1 , x2 ) ∈ D, does not have periodic orbits in D. According to this criterion, to rule out the existence of periodic orbits of the system (1) in a simply connected region D, we need to find a function h(x1 , x2 ) that satisfies the conditions of the theorem of Bendixson–Dulac, such funtion h is called a Dulac function. In Saez and Szanto [5] was constructed Lyapunov functions using Dulac functions to assure the nonexistence of periodic orbits. Our goal is to find a general dynamical system on the plane that not have periodic orbits using Dulac functions.
2
Method to Obtain Dulac Functions
A Dulac function for the system (1) satisfies the equation ∂f1 ∂h ∂h ∂f2 + f2 = h c(x1 , x2 ) − + f1 ∂x1 ∂x2 ∂x1 ∂x2
(2)
(see [4]). Theorem 2.1. (see [4]) For the system of differential equations (1) a solution h of the associated system (2) (for some function c which does not change sign and vanishes only on a subset of measure zero) is a Dulac function for (1) in any simply connected region A contained in D\{h−1 (0)}. Theorem 2.2. (see [4]) For the system of differential equations (1), if (2) (for some function c which does not change of sign and it vanishes only on a subset of measure zero) has a solution h on D such that h does not change sign and vanishes only on a subset of measure zero, then h is a Dulac function for (1) on D.
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A generalization of a gradient system
3
Main Result Theorem 3.1. The system x˙ 1 = (αx2 + β)x31 − x1 , x˙ 2 = −x2 + (γx41 /4 + δ),
with α, β, γ, δ constants, does not have periodic orbits in a simply connected domain D ⊂ R2 when x1 > 0. Proof. Suppose that z depends on x1 , z = 1/x31 . Substituting in (2) we have ∂z ∂z ∂f2 ∂f1 c(x1 , x2 ) − = (1/z) f1 + + f2 ∂x1 ∂x2 ∂x1 ∂x2 we can take c := 1, then 1 ∂z ∂z + f2 = 3 f1 ∂x1 ∂x2 x1
1−
∂f2 ∂f1 + ∂x1 ∂x2
(3)
that it is a quasilinear partial differential equation, applying the method of the characteristic we obtain the associate system dx1 =f1 dt dx2 =f2 dt 1 ∂f2 ∂f1 dz = 3 1 − + dt x1 ∂x1 ∂x2 ∂f1 ∂f2 ∂f2 1 then −1 + ∂x + ∂x = 3f . Suppose ∂x = −1, then f2 = −x2 + c2 (x1 ). x1 1 2 2 ∂f1 3f1 3 But ∂x1 − x1 − 2 = 0, then f1 = c1 (x2 )x1 − x1 . Then the system x˙ 1 = c1 (x2 )x31 − x1 , x˙ 2 = −x2 + c2 (x1 )
As we want a gradient system x˙ 1 = −Vx1 = f1 , x˙ 2 = −Vx2 = f2 , then it holds ∂f1 ∂f2 = ∂x , then we have ∂x2 1
x˙ 1 = (c3 x2 + c4 )x31 − x1 , c x4 x˙ 2 = −x2 + 34 1 + c6
Hence c x4
∂(((c3 x2 + c4 )x31 − x1 )/x31 ) ∂((−x2 + 34 1 + c6 )/x31 ) ∂(f1 h) ∂(f2 h) + = + = 1/x31 > 0, ∂x1 ∂x2 ∂x1 ∂x2
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and the system does not contain periodic orbits in R2 . As it is a gradient system, there exists a function V such that x˙ 1 = x4 x2 x2 −Vx1 , x˙ 2 = −Vx2 , then −V = (c3 x2 + c4 ) 41 − 21 − 22 + c6 x2 . Example 3.2. Consider the equations with a sink at (0, 0) x˙ 1 = −x1 , x˙ 2 = −x2 . ∂V ∂V = −x1 , x˙ 2 = − ∂x = −x2 , This is a gradient system, because x˙ 1 = − ∂x 1 2
∂f1 2) 1) = ∂(−x = 0 = ∂(−x = ∂x , then V = x21 /2 + x22 /2 it is well known that ∂x1 ∂x2 2 periodic orbits cannot exist among others see . Taking x1 > 0 and c = 1, then the solution of the partial differential equation (2) is h = x13 . This example 1 provides a Dulac function h, and the system does not contain periodic orbits its domain. ∂f2 ∂x1
ACKNOWLEDGEMENTS. The authors express their deep gratitude to CONACYT-M´exico, Programa de Mejoramiento del Profesorado (PROMEP)M´exico and Universidad de Cartagena for financial support.
References [1] I. Bendixson, Sur les curbes definies par des equations differentielles, Acta Math., 24 (1901), 1–88. [2] C. C. McCluskey and J. S. Muldowney, Bendixson–Dulac criteria for difference equations, J. Dyn. Diff. Equ., 10 (1998), 567–575. [3] L. Perko, Differential Equations And Dynamical Systems, Springer, Berlin, 2006. [4] O. Osuna and G. Villase˜ nor, On the Dulac Functions. Qual. Theory Dyn. Syst. 10 1 (2011), 43–49. [5] E. Sa´ez and I. Sz´anto, On the construction of certain Dulac Function. IEEE Trans. Automat. Control 33 9 (1988), 856. [6] L. Stephen, Dynamical systems with applications using MAPLE, Birkh¨auser, Boston, 2001. [7] S. Strogatz, Nonlinear Dynamics and Chaos, Addison–Wesley, Reading, 1994. Received: February 24, 2013
