Nonexistence of Limit Cycles in Rayleigh System

Transcription

1 International Journal of Mathematical Analysis Vol. 8, 014, no. 49, HIKARI Ltd, Nonexistence of Limit Cycles in Rayleigh System Sandro-Jose Berrio-Guzman Ana-Magnolia Marin-Ramirez Ruben-Dario Ortiz-Ortiz Copyright c 014 Sandro-Jose Berrio-Guzman, Ana-Magnolia Marin-Ramirez and Ruben- 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 Poincaré transformation is applied the resulting system does not have periodic orbits Mathematics Subject Classification: 34A34, 34C5 Keywords: Bendixon Dulac criterion, Dulac functions, Limit cycles, Periodic orbits, Poincaré transformation

2 48 S. J. Berrio, A. M. Marin and R. D. Ortiz 1 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 Poincaré 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 ẋ 1 = f 1 (x 1, x ), ẋ = f (x 1, x ) (1) where f 1 and f are C 1 functions in a simply connected domain D R. 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. Method to Obtain Dulac functions A Dulac function ϕ for the system (1) satisfies the equation ( ( ϕ ϕ f1 f 1 + f = ϕ c(x 1, x ) + f )) x x () Theorem.1. (See [1]) Suppose that for some function c which does not change of sign and it vanishes only on a measure zero subset () 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 c 1 (x ), c (x 1 ) be functions C 1 in a simply connected domain D R and ε > 0. Then the system ẋ 1 = x 1x x 3 ε 1 3 x x 1 + c 1 (x ) ẋ = c (x 1 )x 5 x 1 x 3 does not have periodic orbits in R Proof. From (.1) we have ( ( ϕ ϕ f1 f 1 + f = ϕ c(x 1, x ) + f )) x x

3 Nonexistence of limit cycles in Rayleigh system 49 we can take c = ε(x 1 x ), ϕ = 1 x 5 must be Now if f 1 which is, ϕ = 0 ; = x 1 x ε(x 1 x ). Then ( f1 c + f ) x ϕ x = 5 1 x 6 = ϕ 1 f h x x 1x f x = x 5 f ( 5x 6 ) f x 5f x 1 x 1 x = 0 Now solving this differential equation is obtained Since f = c (x 1 )x 5 x 1 x 3 (3) f 1 = x 1 x ε(x 1 x ) integrating on both sides with respect to x 1, it follows that x f 1 = x 1x 3 ε 1 3 x x 1 + c 1 (x ) (4) Now from (3) and (4) we have the following system ẋ 1 = x 1x x 3 ε 1 3 x x 1 + c 1 (x ) ẋ = c (x 1 )x 5 x 1 x 3 Replacing in the equation () and taking c = ε(x 1 x ) and assuming that ϕ = 1/x 5 then the equality holds. Hence (f 1 ϕ) + (f ϕ) x = ε x 1 x x 5 Setting D 1 := (x 1, x ) R : 0 < x < x 1} it holds that (ϕf 1, ϕf ) < 0 for all (x 1, x ) D 1 or D := (x 1, x ) R : x 1 < x < 0} it holds that (ϕf 1, ϕf ) < 0 for all (x 1, x ) D so, D 1 D = R nullset hence, applying Bendixson theorem the system does not have periodic orbits in R.

4 430 S. J. Berrio, A. M. Marin and R. D. Ortiz Example 3.. Consider the system u τ = u z z u ε 3 z u z τ = uz 3. (5) It was obtained from Poincaré map (6) (See []) dt z = dτ, x 1 = 1 z, x = u, (z 0). (6) z and Rayleigh equation which describes the oscillation of a violin s string (7) 1 ẍ + ε 3 (ẋ) 1 ẋ + x = 0 (7) By making a change of variables ẋ 1 = x, ẍ 1 = ẋ, the following system is obtained ẋ1 = x x ẋ = x 1 ε 1 x 3. (8) Taking c = ε(x 1 x ), with x > 0 in the system (5) and setting D 1 and D as in the previous theorem it follows that the solution of the partial differential equation () is ϕ = 1/x 5. This example provides a Dulac function ϕ, and the system does not contain periodic orbits in R. It is very interesting that the system (8) has periodic orbits but, when Poincaré 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. Artés, Qualitative Theory of Planar Differential Systems, Springer Verlag, New York, (006). [] 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), (006) [3] S. Lynch, Dynamical Systems with Applications Using MAPLE, Birkhäuser, Boston, (001).

5 Nonexistence of limit cycles in Rayleigh system 431 [4] L. Perko, Differential Equations and Dynamical Systems, Springer Verlag, Berlin, (006). [5] L. Rayleigh, 1878, The Theory of Sound, Macmillan and co, London, (1877). Received: September 0, 014, Published: October 7, 014