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 [47]; 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 thatis 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 .

Remark 4. Note that Leray-Schauder’s Theorem demands a priori bounds over the continuation which are generally coarse. A key subject for researching of the stability of the continuation is the possibility of obtaining more acute bounds over it by means of some constructive approach. This is the purpose in the next section, to present some basic procedure for quantifying a linear stable branch of odd periodic solutions emerging from an equilibrium on oscillators of pendulum type.

3. Odd Global Continuation of Equilibrium Solutions for Oscillators of Pendulum Type

This section is devoted to oscillators of pendulum type given by the equationwhere is an odd -periodic function and satisfies the following conditions for some positive numbers and : (1), for all ,(2), for all ,(3), for all ,(4), in .Without loss of generality, in the following we assume thatThe main result of this section is the following.

Theorem 5. If , then there exist being -periodic and odd continuation of the equilibrium for (23), for all . Moreoverwith

Remark 6. As a particular case, this theorem gives the odd global continuation of the equilibrium solution for forced pendulum (6) for all frequencies .

The proof of Theorem 5 requires several preliminary results. We start with an estimative over the canonical solutions of Hill’s equation as a consequence of Sturm’s Comparison Theorem.

Lemma 7. Let with for all and with . Let and be the solutions ofsatisfying the initial conditionsrespectively. Then for all we have the following inequalities:

Proof. Let , be the canonical solutions ofthat satisfy the initial conditions and ; that is,For observe that satisfies the boundary problemwith . So it is not difficult to prove thatwhereNow we want to know the sign of on . First, we remark that since and for all because .
On the other hand, since on , from the Sturm Comparison Theorem, we deduce in . This implies that for all (see [14]). From the hypothesis on we know that is not identically zero; then as a consequence we have on ; that is,The proof for the other inequalities follows the same ideas if we consider the equationand the canonical solutions

As in the previous section, let be the solution of (23) satisfyingFrom the symmetries over (23) the search of odd -periodic solutions can be focused to solve the Dirichlet problem (in a half period)In other words, we must find the zeros of the functionThis is a classical implicit function problem with initial solution of the trivial one . Let . Since , thenas a consequence there exists a local function such that . Let be an open neighborhood of where . By the Implicit Function Theorem problem (40) is equivalent to the Cauchy Problemwhich is well defined on withThe next proposition gives sufficient conditions to guarantee that and is bounded.

Proposition 8. If , then the function given in (43) is well defined in all andwith .

Proof. For each consider the variational equationFrom assumption   over the function , it follows that in uniformly in and for assumption   it follows that . Let and be the canonical solutions of (45). From Lemma 7 with we obtainfor all . On the other hand, the theorem of differentiability with respect to initial conditions and parameters implies that the functions and satisfy the following Cauchy problems:respectively. Therefore, for , we haveSince from the Comparison Sturm Theorem we obtain for all . Following the lemma in [14] we concluded thatfor all . Moreover since then and on ; thereforewhere . Since there are no critical points of in then has the maximum value on the boundary of . A simple calculation shows thatwithOn the other handFinally we obtain

Remark 9. For the autonomous Cauchy problemwhich has a bounded vector field ; we conclude that is well defined in (see [15]); therefore for all we obtain an odd, -periodic continuation of the equilibrium for all .

Now we are able to present the proof of the main theorem of this section.

Proof of Theorem 5. Let and consider the autonomous Cauchy problemIt follows from Proposition 8 that is bounded. From here we deduce that all solutions are globally defined; then is well defined for all ; (see [15]). As a consequence for all we obtain an odd, -periodic continuation of the equilibrium for all . Now by the Mean Value Theorem we haveOn the other hand,Since and are, respectively, the solutions of the initial value problems (47) and (48), with , they satisfy inequalities (46) and (54) in ; thereforeas a consequenceconcluding the proof.

4. Linear Stability of the Continuation

In this section we will obtain a proof of the linear stability of the continuation . We use techniques that are traditionally employed in the study of Hill’s equation. More precisely, the proof will be based on the Lyapunov-Zukovskii stability criteria for Hill’s equations. We start with a well-known result of the Sturm comparison theory for Hill’s equation.

Lemma 10. If satisfies on with strict inequality on a set of positive measures on , then (i)any solution of (27) has infinitely many zeros;(ii)if and are two consecutive zeros of any solution of (27), they satisfy

The proof of Lemma 10 follows easily from Sturm’s Comparison Theorems (see [16]).

Proposition 11 (Lyapunov-Zukovskii). Let such that , for all , with strict inequality on a set of positive measures in . Then (27) is elliptic.

Proof. By a contradiction argument suppose that there is a real Floquet multiplier . Let be the Floquet solution associated with ; thenSince it follows from Lemma 10 that such that ; then . Once again, from Lemma 10 and are consecutive zeros; as a consequence the distance is at most and this is a contradiction.

Finally we completed all the necessary arguments that we need in order to prove our second main result.

Theorem 12. Under the same hypothesis of Theorem 5, -periodic continuation of the equilibrium of (45) is elliptic if where

Proof. With and from (25) it is clear that . On the other hand, from the conditions on the function , we haveOn the other hand, if for all , then since is strict monotone in and this implies which is a contradiction. By continuity we deduce the existence of a set of positive measures in such that . The conclusion follows if we apply Proposition 11 to the variational equation along :where .

Corollary 13. For , , and odd , the equilibrium solution of the oscillator (23) can be continued like an odd -periodic solution of the forced pendulumwhich is elliptic (therefore linear stable) for wheremoreover,

5. Odd Global Continuation for any Positive

In order to remove the restriction in Theorem 5, we follow a different approach to estimate an upper bound for and a lower bound for in (42)-(43). As before we assume the condition on oscillator (23).

We start this new approach by setting new conditions over the function . Suppose that there exist positive values and such that(a), , for all .(b).The main result of this section is the following.

Theorem 14. Under assumptions (a) and (b), there exist being -periodic and odd continuation of the equilibrium for (23) for all withMoreoverwhere and is a positive constant for given by

The proof of Theorem 14 follows similar arguments and steps are given in Section 2 for proving Theorem 5. We start by considering again the solutions and of Cauchy problems (47) and (48), respectively.

Before the proof of Theorem 14 we point out some preliminary results. The first one is about the growing of the solutions for Hill’s equations.

Lemma 15. Let with for all and let , be the unique solutions ofsatisfyingThen, for , we have

Proof. Let be a solution of and define the functionBy a direct computation we obtainSolving this differential inequality with initial condition we getIn particular for the canonical solutions     we have and therefore

From (49), hypothesis , and Lemma 15 taking on we have the following estimative:

The second preliminary result is the classical fundamental inequality in ordinary differential equations (see [16]).

Proposition 16. Suppose that , satisfy where , are continuous and Lipschitz with respect to the second variable with Lipschitz constant . Ifthenfor all in a common interval of existence.

Proof of Theorem 14. The existence of nontrivial odd, -periodic solutions of (23) is equivalent to the solving of problem (40). Furthermore, from assumption therefore we can consider the Cauchy problemin a small rectangle of the form .
Our objective is to solve initial value problem (84) in a concrete -interval and on an appropriate rectangle. For this purpose we will estimate an upper bound over the absolute value of the right hand side of (84). In accordance with (79) we only have to find a positive lower bound for on .
Consider variational equation (45) given by The vector field of the first order system associated with this variational equation is given by with . Notice that this vector field is continuous and -Lipschitz, with Lipschitz constant uniformly in .
Next we apply the fundamental inequality to the vector fields , and the respective solutions , (both in the same initial conditions) on the domain , where is an upper bound for the canonical solutions of (45) given by Lemma 15. Henceforthwhere is an upper bound on forFrom the mean value theorem we getwhere is any positive number such thatBesides,and from (79) and condition (a) over the function we havefor all , , . In conclusion on the domain we haveNow we impose the restriction (see (87))The restriction is equivalent to with given by (70). Combining (87) and we obtain (for all if , then )with given by (72). Finally, an upper bound for the modulus of the right hand side in (84) is given byon . For classical Peano’s Theorem we obtain a solution of (10) for . Note that our estimatives do not depend on ; therefore we can take so the solution is defined on . So we obtain the odd -periodic family . Following the lines in the proof of Theorem 5 (inequalities (58) and (59)) is a straightforward task to obtain (71) using the above estimative.

6. Concluding Remarks

Applying the techniques in [7] to forced pendulum (6) we obtain a finite -interval for frequencies far away of the strong resonances. In contrast, our first main result (Theorem 5) provides an infinite -interval for low frequencies . Moreover for high frequencies we obtain -periodic continuation of the equilibrium in a quantified -interval (Theorem 14) for general oscillators of pendulum type.

The nonlinear stability in the Lyapunov sense of the family obtained before remains an open problem. This can be studied by means of the KAM Theory. More precisely, the Twist Theorem of Möser [17] gives sufficient conditions to ensure that Poincare’s mapping of (23) has invariant curves in all neighbourhood of the elliptic fixed point .

From Möser’s work this approach has been referenced in the literature like a third-order method, because the stability is generically decided by the nonvanishing of a certain third-order coefficient of called the first twist coefficient. The Möser Theorem essentially claims that if , then the fixed point is stable and therefore is also stable. We will explain this a little bit more.

By means of a translation to the origin and making Taylor’s expansion up to the third order, we arrive to the equivalent equation with equilibrium :where , , and and the remaining terms are periodic functions of period ; moreoverThe twist coefficient is a meromorphic function which depends on , , and in a not easy way (see [7, 14, 18, 19]). In [14] Ortega gave for the first time an explicit formula for and subsequently Zhang in [7, 19] provided some refinements of it making the dependence with respect to more synthetic and very practical for applications (see [5, 7, 1921]). The open problem consists of studying the first zero of in order to estimate -range for the stability of . In this direction the results in [7] could be useful. Finally, it is important to notice that when is really analytical the recently Hanßmann and Simó’s results in [22] could be useful to study the nonlinear stability for the continuation.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


This work was supported by Capital Semilla (2014-2015), Project 00003967, and Pontificia Universidad Javeriana, Seccional Cali, Cali, Colombia.