Periodic Solutions of a Class of Second-Order Differential Equation

We study the periodic solutions of the second-order differential equations of the form ( )( ) ( ) ( ) x xx x F t x x G t x H t    + + + + + + = 3 2 3 0, where the functions ( ) F t , ( ) G t and ( ) H t are periodic of period π 2 in the variable t.


Introduction and Statement of the Main Results
In this paper we shall study the existence of periodic solutions of the second-order differential equation of the form ( )( ) ( ) ( ) where the dot denotes derivative with respect to the time t, and the functions ( ) F t , ( ) G t and ( ) H t are periodic of period 2π in the variable t.
We note that the second-order differential Equation (1), when 0 , appears in the Ince's catalog of equations possessing the Painlevé property (see [1]).Moreover, the differential equation well known in many areas of mathematics and physics, and it possesses the algebra ( ) sl 3,  of Lie point symmetries (see for more details in the paper [2] and the references quoted there).
In a recent paper [3] (see also [4] [5]), the second-order differential Equation (1) has been studied when 0 F H = = .A study of coupled quadratic unharmonic oscillators in terms of the Painlevé analysis and inte- grability can be seen in [6], and studies on the second-order differential equations can be seen in [7].Other approach to the periodic solutions of second-order differential equations can be found in [8].
Here we study the periodic solutions of the second-order differential Equation (1) when ( ) ( ) , and . Our main results are the following ones.Theorem 1.We define the functions where are 2π -periodic.Then for 0 ε ≠ sufficiently small and for every ( ) the differential Equation (1) has a 2π -periodic solution ( ) ( ) ( ) . Theorem 1 is proved in section 3 using the averaging theory described in section 2. Two applications of Theorem 1 are the following.

H t t ε =
. Then for 0 ε ≠ sufficiently small, this differential equation has a 2π -periodic solution We consider the differential Equation (1) with ( ) ( ) with are 2π -periodic functions.Then for 0 ε ≠ sufficiently small and for every ( ) Corollaries 3 and 4 also proved in section 4.

Basic Results on Averaging Theory
We state the results from the averaging method that we shall use for proving the results of this work.
We consider differential systems of the form where ε is a small parameter, and the functions 0 1 , : ( )  functions, T-periodic in the variable t, and Ω is an open subset of n  .Suppose that the unperturbed system has a submanifold of dimension n of T-periodic solutions, i.e. of periodic solutions of period T.
We denote by ( ) , ,0 t x z the solution of system (6) such that ( ) 0, , 0 = x z z.We consider the first variational equation of system (6) on the periodic solution ( ) x z y (7) where y is an n n × matrix.Let

( )
M t z the fundamental matrix of system (7) such that ( ) is the identity matrix of n  .By assumption there exists an open set V such that ( ) Cl V ⊂ Ω and for each ( ) ) is T-periodic.Therefore we have the following result.
Theorem 3. We suppose that there is an open and bounded set V with ( ) is T-periodic, and let ( ) If there is x z .Theorem 3 is due to Malkin [9] and Roseau [10], for a new and shorter proof (see [11]).

Proof of Theorem 1 and Its Two Corollaries
Proof of Theorem 1. Introducing the variable y x =  , we can write the second-order differential Equation (1) as the following first-order differential system x y Doing the rescaling ( ) ( ) , we obtain the system ( ) 3 .
System (10) with 0 ε = is the unperturbed system, otherwise system (10) is the perturbed system.The unperturbed system has a unique singular point, the origin of coordinates.The solution Note that all these periodic orbits have period 2π .Using the notation introduced in section 2. We have that ( ) given in ( 8), and we get the functions (2) of the statement of Theorem 1.
By Theorem 3 each zero ( )

Proof of Theorem 2 and Its Corollaries
Proof of Theorem 2. As in the proof of Theorem 1, the second-order differential Equation (1) can be written as the first order differential system (9).Doing the rescaling ( ) ( ) 3 .11)