Nonexistence of Limit Cycles in Rayleigh System - m-hikari

4 downloads 0 Views 166KB Size Report
Oct 27, 2014 - Sandro-Jose Berrio-Guzman. Faculty of Exact and ... Ruben-Dario Ortiz-Ortiz. Faculty of ... S. J. Berrio, A. M. Marin and R. D. Ortiz. 1 Introduction.
International Journal of Mathematical Analysis Vol. 8, 2014, no. 49, 2427 - 2431 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.48283

Nonexistence of Limit Cycles in Rayleigh System Sandro-Jose Berrio-Guzman Faculty of Exact and Natural Sciences, University of Cartagena Campus San Pablo, Avenue of Consulado Cartagena de Indias, Bolivar, Colombia Ana-Magnolia Marin-Ramirez Faculty of Exact and Natural Sciences, University of Cartagena Campus San Pablo, Avenue of Consulado Cartagena de Indias, Bolivar, Colombia Ruben-Dario Ortiz-Ortiz Faculty of Exact and Natural Sciences, University of Cartagena Campus San Pablo, Avenue of Consulado Cartagena de Indias, Bolivar, Colombia c 2014 Sandro-Jose Berrio-Guzman, Ana-Magnolia Marin-Ramirez and RubenCopyright Dario Ortiz-Ortiz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract The equation which models the behavior of a violin’s string can be written as a dynamical system that naturally contain periodic orbits in the plane, after Poincar´e transformation is applied the resulting system does not have periodic orbits

Mathematics Subject Classification: 34A34, 34C25 Keywords: Bendixon–Dulac criterion, Dulac functions, Limit cycles, Periodic orbits, Poincar´e transformation

2428

1

S. J. Berrio, A. M. Marin and R. D. Ortiz

Introduction

A very important concern in the study of differential equation is the existence of periodic orbits, the Rayleigh equation related to the oscillation of a violin’s string(see [5]) can be written as a dynamical system, which naturally presents periodic orbits, a new system is shown by applying Poincar´e transformation, it will be considered the Bendixson-Dulac criterion.(See [4, 3]). The objective of this paper is to prove that the new system obtained does not contain periodic orbits in the phase portrait by using Dulac’s functions, in order to do this Bendixson’s criterion will be handled. Consider the system x˙ 1 = f1 (x1 , x2 ),

x˙ 2 = f2 (x1 , x2 )

(1)

where f1 and f2 are C 1 functions in a simply connected domain D ⊂ R2 . In order to discard the existence of periodic orbits of the system (1) in a simply connected region D, is necessary to use the Bendixson criterion.

2

Method to Obtain Dulac functions

A Dulac function ϕ for the system (1) satisfies the equation    ∂f1 ∂f2 ∂ϕ ∂ϕ + f2 = ϕ c(x1 , x2 ) − + f1 ∂x1 ∂x2 ∂x1 ∂x2

(2)

Theorem 2.1. (See [1]) Suppose that for some function c which does not change of sign and it vanishes only on a measure zero subset (2) has a solution ϕ on D such that ϕ conserves the sign and vanishes only on a measure zero subset, then ϕ is a Dulac function.

3

Main Result

Theorem 3.1. Let c1 (x2 ), c2 (x1 ) be functions C 1 in a simply connected domain D ⊂ R2 and ε > 0. Then the system  3  ( x x˙ 1 = −x21 x22 − ε 31 − x22 x1 + c1 (x2 ) x˙ 2 = c2 (x1 )x52 − x1 x32 does not have periodic orbits in R2 Proof. From (2.1) we have    ∂ϕ ∂f1 ∂f2 ∂ϕ + f2 = ϕ c(x1 , x2 ) − + f1 ∂x1 ∂x2 ∂x1 ∂x2

Nonexistence of limit cycles in Rayleigh system

we can take c = −ε(x21 − x22 ), ϕ =

1 x52

2429

must be

∂ϕ ∂ϕ 1 =0; = −5 6 ∂x1 ∂x2 x2 Now if

∂f1 ∂x1

= −2x1 x22 − ε(x21 − x22 ). Then   ∂f1 ∂f2 ∂h c− + = ϕ−1 f2 ∂x1 ∂x2 ∂x2 2x21 x22 −

∂f2 = x52 f2 (−5x−6 2 ) ∂x2

which is, ∂f2 − 5f2 x−1 2 − 2x1 x2 = 0 ∂x2 Now solving this differential equation is obtained f2 = c2 (x1 )x52 − x1 x32 Since

(3)

∂f1 = −2x1 x22 − ε(x21 − x22 ) ∂x1

integrating on both sides with respect to x1 , it follows that  3  x1 2 2 2 f1 = −x1 x2 − ε − x2 x1 + c1 (x2 ) 3

(4)

Now from (3) and (4) we have the following system  3  ( x x˙ 1 = −x21 x22 − ε 31 − x22 x1 + c1 (x2 ) x˙ 2 = c2 (x1 )x52 − x1 x32 Replacing in the equation (2) and taking c = −ε(x21 − x22 ) and assuming that ϕ = 1/x52 then the equality holds. Hence ∂(f1 ϕ) ∂(f2 ϕ) x2 − x2 + = −ε 1 5 2 ∂x1 ∂x2 x2 p Setting D1 := {(x1 , x2 ) ∈ R2 : 0 < x2 < x21 } it holds p that ∇ · (ϕf1 , ϕf2 ) < 0 2 for all (x1 , x2 ) ∈ D1 or D2 := {(x1 , x2 ) ∈ R : − x21 < x2 < 0} it holds that ∇ · (ϕf1 , ϕf2 ) < 0 for all (x1 , x2 ) ∈ D2 so, D1 ∪ D2 = R2 − nullset hence, applying Bendixson theorem the system does not have periodic orbits in R2 .

2430

S. J. Berrio, A. M. Marin and R. D. Ortiz

Example 3.2. Consider the system   3 ( u 2 2 2 2 uτ = −u z − z − ε 3 − z u zτ = −uz 3 .

(5)

It was obtained from Poincar´e map (6) (See [2]) dt u 1 (6) = dτ, x1 = , x2 = , (z 6= 0). 2 z z z and Rayleigh equation which describes the oscillation of a violin’s string (7)   1 2 (x) ˙ − 1 x˙ + x = 0 (7) x¨ + ε 3 By making a change of variables x˙ 1 = x2 , x¨1 = x˙ 2 , the following system is obtained ( x˙ 1 = x2 x˙ 2 = −x1 − ε



x22 3

 − 1 x2 .

(8)

Taking c = −ε(x21 − x22 ), with x2 > 0 in the system (5) and setting D1 and D2 as in the previous theorem it follows that the solution of the partial differential equation (2) is ϕ = 1/x52 . This example provides a Dulac function ϕ, and the system does not contain periodic orbits in R2 . It is very interesting that the system (8) has periodic orbits but, when Poincar´e map is applied (6) the new system (5) does not contain periodic orbits. Acknowledgements. The authors express their deep gratitude to Universidad de Cartagena for partial financial support.

References [1] F. Dumortier, J. Llibre and J. C. Art´es, Qualitative Theory of Planar Differential Systems, Springer–Verlag, New York, (2006). [2] Z. Feng, G. Chen and S.-B. Hsu, A qualitative study of the damped Duffing equation and applications, Discrete and Continuous Dynamical System B, 6 (5), (2006) 1097 - 1112. [3] S. Lynch, Dynamical Systems with Applications Using MAPLE, Birkh¨auser, Boston, (2001).

Nonexistence of limit cycles in Rayleigh system

2431

[4] L. Perko, Differential Equations and Dynamical Systems, Springer– Verlag, Berlin, (2006). [5] L. Rayleigh, 1878, The Theory of Sound, Macmillan and co, London, (1877). Received: September 20, 2014, Published: October 27, 2014