Abstract and Applied Analysis

Volume 2015, Article ID 836312, 10 pages

http://dx.doi.org/10.1155/2015/836312

## Quantifying Poincare’s Continuation Method for Nonlinear Oscillators

Pontificia Universidad Javeriana, Cali 760031, Colombia

Received 16 June 2015; Accepted 5 August 2015

Academic Editor: Weinian Zhang

Copyright © 2015 Daniel Núñez and Andrés Rivera. 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

In the sixties, Loud obtained interesting results of continuation on periodic solutions in driven nonlinear oscillators with small parameter (Loud, 1964). In this paper Loud’s results are extended out for periodically driven Duffing equations with odd symmetry quantifying the continuation parameter for a periodic odd solution which is elliptic and emanates from the equilibrium of the nonperturbed problem.

#### 1. Introduction

After pioneering work of H. Poincare in celestial mechanics, the continuation analytical method will have a great relevance in applied problems in science and technology. Several versions of this approach for the searching of dynamic objects like periodic solutions and invariant manifolds have been very fruitful in dynamical systems and its applications; see [1, 2]. Perhaps there are perturbations of oscillators likewhere , are continuous and is -periodic function in and is a small parameter; this is one of the easiest environments on which we can apply the continuation methods. In the sixties, Loud [3] obtained interesting results of local continuation in driven nonlinear differential equations like (1). He assumed that the nonperturbed equationhad an isolated equilibrium and considered four cases according to the relative position of the Floquet multipliers of the variational equationwith , . We denote by the general solution of . Loud searched the solution of the implicit function systemin order to obtain -family of -periodic solutions as a continuation of the equilibrium for . He used several versions of the implicit function theorem obtaining some orthogonality conditions involving the perturbation term and the solutions of the variational equation. According to a sign over this orthogonality condition it is possible to know the direction of movement of the Floquet multipliers while the parameter increases. In this way the author is able to classify the stability properties for small enough. For the frictionless and nonresonant case, that is,(as will be considered in this paper), Loud’s result does not provide any stability information (see Theorem 2.9 in [3]).

From now on, we are interested in studying the periodically driven Duffing equationwith , continuous, being -periodic function, and satisfying the above nonresonant condition. We assume that is an elliptic equilibrium for the nonperturbed problem () and we formulate the following two questions: (i)How small is the perturbation parameter to guarantee the linear stability for ?(ii)How small is the perturbation parameter to guarantee the nonlinear stability for ?

A concrete example of oscillators like is the forced pendulumwhere many results with respect to the existence and stability of periodic solutions can be found in the literature [4–7]; see also [8] and the references therein. For instance, it is well known that if is -periodic function, there exists -periodic solution for the forced pendulum as a continuation of the trivial solution which is stable when and is small enough. This result is an easy consequence of the KAM theory.

There are at least three different types of analytical periodic continuation on the forced pendulum as follows: (i) the small oscillations previously mentioned, (ii) those emanating from certain periodic solutions of the nonlinear center for the nonforced case, and (iii) those emanating from the hanging solutions for the nonforced case [6]. In this last paper the author applied Loud’s techniques in order to find suitable to guarantee the bifurcations of many periodic solutions from the hanging one.

On the other hand, assuming appropriate symmetries on , and odd functions, the implicit system could be reduced to a single scalar equationin order to find odd -periodic solutions. This is an original idea by Hamel [4] in his research of periodic solutions on the forced pendulum.

Thereby in this work we focus on the global continuation problem of periodic solutions under this kind of symmetries for and their stability properties, starting from an elliptic equilibrium of the nonperturbed equation.

As the core problem has been reduced to an implicit one, that is, the study of the set of zeros of a continuous real function (e.g., the function ), some topological tools, like the Leray-Schauder Continuation Theorem [9], help us to understand its structure. This approach has been successfully applied in the study of periodic solutions on a* restricted three-body problem* (see [10, 11]). In order to apply this tool it is necessary to compute* a priori* bounds over the zeros of (see Theorem 1 in Section 2 for more details) but the conclusion of the Leray-Schauder Theorem says nothing about the linear stability of the associated periodic solutions. For this study it is necessary to obtain more refined bounds over the periodic solution in order to apply some classical stability results on the variational equation (like Hill’s equation).

The rest of the paper is divided in four sections. In Section 2 we illustrate how the Leray-Schauder Continuation Theorem can be applied to the forced pendulum in order to get a global family of periodic solutions from the equilibrium and remark its limitations for the stability analysis of this family. In Section 3 we consider oscillators of pendulum type with odd symmetries and present our first main result (Theorem 5); a family of odd periodic solution is obtained for all parameter values, and furthermore we present some interesting* a posteriori bounds* for its amplitude (see (25)). In Section 4 we review some basic facts about the stability of Hill’s equation and we present the second main result, namely, the determination of a computable -interval, where we guarantee the linear stability for the periodic continuation obtained in Theorem 5. Finally, Section 5 is devoted to point out some open questions about the nonlinear stability of the obtained periodic family.

#### 2. The Forced Pendulum and a Global Implicit Function Theorem

Consider the forced pendulumwhere is a positive parameter and is an odd and -periodic continuous function; that is, for all , we haveThe existence of odd and -periodic solutions of (6) was proved for the first time by Hamel [4] in 1922 by means of a reduction to the boundary value problemSee [8] for more references on this paradigmatic equation. Let be the solution of (6) satisfyingThis is a real analytic function in the arguments (see [12]) and is globally defined in . It is not difficult to prove that the research of odd and -periodic solutions of (6) is equivalent to study (8). This follows by performing odd and -periodic extensions improving the symmetries of (6) and its periodicity. So problem (8) can be reduced to the study of the implicit equationTherefore we want to apply some global version of the Implicit Function Theorem in order to solve (10), namely, the analytical version of the Leray-Schauder Continuation Theorem (see [9]), which provides parametrized curves solving (10) starting at . We present the following version of this result. The complete proof can be found in [10].

First, we recall that for a given function which does not vanish in and has a finite number of nondegenerate zeros in it is possible to define the* Brouwer degree * aswhere denotes the derivatives of . If is an isolated zero in the set of zeros of , the* Brouwer index of the zero * is defined bywhere is a small neighbourhood of . Now we are able to present the main theorem of this section and its application to (10).

Theorem 1 (Leray-Schauder). *Let be analytic and let be the set of zeros of . Assume the following:*(H1)* is bounded.*(H2)*The set with is finite and there is with .**Then there is a continuum , , with , , such that either or and .*

From Theorem 1 we have the following consequence for the forced pendulum.

Proposition 2. *Given , there exists a continuum , , such that**is an odd -periodic solution of forced pendulum (6) with , , and .*

*Proof. *Let be the solution of (6) that satisfies initial conditions (9) and defines the real analytic functionwith . The set of zeros of is clearly bounded since the derivatives of the solutions of (6) are uniformly bounded in which reveals a simple integration over (6):where denotes -norm in the space ; thereforethen (H1) holds. On the other hand, since , the only -periodic solution for the nonforced pendulum () is the trivial one. This nonlinear center is surrounded by periodic solutions with a monotone increasing time period function with (see [13]); therefore if we obtain for all . As a consequence the zeros of the function are reduced to .

Now we compute the index at by linearization; that is,Notice that where is the solution of the variational problemSince , thenand this verifies (H2). As a consequence, we infer the existence of a continuous family in such thatand eitherorbut this last alternative is not possible, again, because . Then we get the required global continuation of odd -periodic solutions for (6).

*Remark 3. *Note that the continuation can be identified with a parametrized curve in -plane and it could have turning points. See Figure 1. In the next section we will show that this is not the case and actually this curve is a graph of a differentiable function globally defined on when .