Quantifying Poincare’s Continuation Method for Nonlinear Oscillators

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.


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 likë where , 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 equation had an isolated equilibrium ≡ 0 and considered four cases according to the relative position of the Floquet multipliers of the variational equation̈+̇+ with = (0, 0), =(0, 0). We denote by ( , 0 , V 0 ) the general solution of ( * ). Loud searched the solution of the implicit function system in order to obtain -family of -periodic solutions as a continuation of the equilibrium ≡ 0 for = 0. He used several versions of the implicit function theorem obtaining some orthogonality conditions involving the perturbation term ( , 0, 0, 0) 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]).
with , continuous, being 2 -periodic function, and = (0) > 0 satisfying the above nonresonant condition. We assume that ≡ 0 is an elliptic equilibrium for the nonperturbed problem ( = 0) 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 pendulum̈+ where many results with respect to the existence and stability of periodic solutions can be found in the literature [4][5][6][7]; see also [8] and the references therein. For instance, it is well known that if ( ) is 2 -periodic function, there exists 2periodic solution for the forced pendulum as a continuation of the trivial solution which is stable when ∉ N 0 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 equation in 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 threebody 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.

The Forced Pendulum and a Global Implicit Function Theorem
Consider the forced pendulum where is a positive parameter and ( ) is an odd and 2periodic continuous function; that is, for all ∈ R, we have The existence of odd and 2 -periodic solutions of (6) was proved for the first time by Hamel [4] in 1922 by means of a reduction to the boundary value problem See [8] for more references on this paradigmatic equation. Let ( , , ) be the solution of (6) satisfying This is a real analytic function in the arguments ( , , ) ∈ R 3 (see [12]) and is globally defined in R. It is not difficult to prove that the research of odd and 2 -periodic solutions of (6) is equivalent to study (8). This follows by performing odd and 2 -periodic extensions improving the symmetries of (6) and its periodicity. So problem (8) can be reduced to the study of the implicit equation Therefore 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 ( , ) = (0, 0). We present the following version of this result. The complete proof can be found in [10]. First, we recall that for a given function ∈ 1 ([ , ]) which does not vanish in { , } and has a finite number of nondegenerate zeros 1 , .
where denotes the derivatives of . If * is an isolated zero in the set of zeros of , the Brouwer index of the zero * is defined by where is a small neighbourhood of * . Now we are able to present the main theorem of this section and its application to (10). From Theorem 1 we have the following consequence for the forced pendulum.
is an odd 2 -periodic solution of forced pendulum (6) with Proof. Let ( , , ) be the solution of (6) that satisfies initial conditions (9) and defines the real analytic function with ∈ [0, Δ]. The set of zeros of is clearly bounded since the derivatives of the solutions of (6) are uniformly bounded in [0, 2 ] which reveals a simple integration over (6): where ‖ ⋅ ‖ 1 denotes 1 -norm in the space 1 ([0, 2 ]); therefore then (H1) holds. On the other hand, since < 1, the only 2periodic solution for the nonforced pendulum ( = 0) is the trivial one. This nonlinear center is surrounded by periodic solutions ( , , 0) with a monotone increasing time period function ( ) with lim → 0 ( ) = 2 / (see [13]); therefore if < 1 we obtain ( ) > 2 for all ̸ = 0. As a consequence the zeros of the function 0 ( ) = ( , , 0) are reduced to {(0, 0)}. Now we compute the index at 0 = 0 by linearization; that is, Notice Since 0 < < 1, then and this verifies (H2). As a consequence, we infer the existence of a continuous family and either or but this last alternative is not possible, again, because < 1.
Then we get the required global continuation ( ) of odd 2periodic 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 R + when  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.

Odd Global Continuation of Equilibrium Solutions for Oscillators of Pendulum Type
This section is devoted to oscillators of pendulum type given by the equation̈+ where is an odd 2 -periodic function and ∈ ∞ (R) satisfies the following conditions for some positive numbers 0 and : (1) (− ) = − ( ), for all ∈ R, Without loss of generality, in the following we assume that The main result of this section is the following.

Theorem 5.
If ∈ ]0, 1/2], then there exist = ( ) being 2 -periodic and odd continuation of the equilibrium ≡ 0 for (23), for all ∈ R. Moreover satisfying the initial conditions respectively. Then for all ∈ ]0, /2[ we have the following inequalities: Proof. Let ( ), = 1, 2 be the canonical solutions of that satisfy the initial conditions 1 (0) =̇2(0) = 1 and 2 (0) =̇1(0) = 0; that is, with ( ) = ( ( ) − 2 ) . So it is not difficult to prove that where Now we want to know the sign of ( ) on  [14]). From the hypothesis on ( ) we know that ( ) is not identically zero; then as a consequence we have ( ) > 0 on ]0, /2]; that is, The proof for the other inequalities follows the same ideas if we consider the equation̈− 2 = 0 (36) and the canonical solutions As in the previous section, let ( , , ) be the solution of (23) satisfying The next proposition gives sufficient conditions to guarantee that V = R 2 and is bounded.
Now we are able to present the proof of the main theorem of this section.

(57)
It follows from Proposition 8 that F 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 ∈ ]0, 1/2] we obtain an odd, 2 -periodic continuation ( ) = ( , 0, ( ), ) of the equilibrium for all ∈ R. Now by the Mean Value Theorem we have On 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 [0, ]; therefore as a consequence concluding the proof.

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. (ii) if 1 and 2 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]). Proof. By a contradiction argument suppose that there is a real Floquet multiplier . Let ( ) be the Floquet solution associated with ; then Since 0 < ( ) it follows from Lemma 10 that ∃ 0 ∈ R such that ( 0 ) = 0; then ( 0 + ) = 0. Once again, from Lemma 10 0 and 0 + 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.
On the other hand, if ( ) = 1/4 for all ∈ R, then ≡ 0 since is strict monotone in ]0, 0 ] and this implies ≡ 0 which is a contradiction. By continuity we deduce the existence of a set of positive measures in [0, ] such that 0 < < 1/4. The conclusion follows if we apply Proposition 11 to the variational equation along : where ( ) = ( ( )).

Odd Global Continuation for any Positive
In order to remove the restriction ∈ ]0, 1/2[ 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 ‖ ‖ 1 = 1 on oscillator (23). We start this new approach by setting new conditions over the function . Suppose that there exist positive values and 0 ∉ N such that (a) | ( )|, | ( )| ≤ 2 , for all ∈ R.
(b) (0) = 2 0 . The main result of this section is the following.
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.
Then, for = 1, 2, we have Proof. Let ( ) be a solution of (⬦) and define the function By a direct computation we obtaiṅ (76) Solving this differential inequality with initial condition V(0) we get In particular for the canonical solutions ( ) = 1, 2 we have V(0) = 1/2 and therefore The second preliminary result is the classical fundamental inequality in ordinary differential equations (see [16]). 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 Ω Δ,Γ .