# Quantifying Poincare’s Continuation Method for Nonlinear Oscillators

Authors: Daniel Núñez; Andrés Rivera
Journal: Abstract and Applied Analysis
 (2015)

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## 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.


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## Body

## 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 like(1)x¨+gx,x˙=ϵpt,x,x˙,ϵ,where g, p are continuous and p is T-periodic function in t and ϵ 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∗x¨+gx,x˙=0had an isolated equilibrium x≡0 and considered four cases according to the relative position of the Floquet multipliers of the variational equation(2)y¨+by˙+ay=0,with a=gx(0,0), b=gx˙(0,0). We denote by x(t,x0,v0) the general solution of ∗. Loud searched the solution of the implicit function systemLxT,x0,v0,ϵ-x0=0,x˙T,x0,v0,ϵ-x0=0,in order to obtain ϵ-family of T-periodic solutions as a continuation of the equilibrium x≡0 for ϵ=0. He used several versions of the implicit function theorem obtaining some orthogonality conditions involving the perturbation term p(t,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 ϵ small enough. For the frictionless and nonresonant case, that is,(3)b=0,a≠2nπT2,∀n∈N(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 equation∗∗x¨+gx=ϵpt,with g, p continuous, p being 2π-periodic function, and a=g′(0)>0 satisfying the above nonresonant condition. We assume that x≡0 is an elliptic equilibrium for the nonperturbed problem (ϵ=0) and we formulate the following two questions: (i)
How small is the perturbation parameterϵ to guarantee the linear stability for ∗∗?(ii)
How small is the perturbation parameterϵ to guarantee the nonlinear stability for ∗∗?A concrete example of oscillators like∗∗ is the forced pendulum(4)x¨+ω2sin⁡x=ϵpt,where 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 p(t) is 2π-periodic function, there exists 2π-periodic solution for the forced pendulum as a continuation of the trivial solution which is stable when ω∉N0 and ϵ 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 p(t) to guarantee the bifurcations of many periodic solutions from the hanging one.On the other hand, assuming appropriate symmetries on∗∗, g and p odd functions, the implicit system L could be reduced to a single scalar equation(5)fv0,ϵ=xT2,0,v0,ϵ=0,in order to find odd T-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 functionf), 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 arestricted three-body problem (see [10, 11]). In order to apply this tool it is necessary to computea priori bounds over the zeros of f (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 Section2 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 interestinga 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 ϵ-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 pendulum(6)x¨+ω2sin⁡x=ϵpt,where ϵ is a positive parameter and p(t) is an odd and 2π-periodic continuous function; that is, for all t∈R, we have(7)pt+2π=pt,p-t=-pt.The existence of odd and 2π-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(8)x¨+ω2sin⁡x=ϵpt,x0=xπ=0.See [8] for more references on this paradigmatic equation. Let x(t,η,ϵ) be the solution of (6) satisfying(9)x0=0,x˙0=η.This is a real analytic function in the arguments (t,η,ϵ)∈R3 (see [12]) and is globally defined in R. It is not difficult to prove that the research of odd and 2π-periodic solutions of (6) is equivalent to study (8). This follows by performing odd and 2π-periodic extensions improving the symmetries of (6) and its periodicity. So problem (8) can be reduced to the study of the implicit equation(10)xπ,η,ϵ=0.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 (η(s),ϵ(s)) solving (10) starting at (η,ϵ)=(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 functionf∈C1([a,b]) which does not vanish in a,b and has a finite number of nondegenerate zeros x1,…,xn in a,b it is possible to define theBrouwer degreedegf,a,b as(11)deg⁡f,a,b=∑i=1nsign⁡f′xi,where f′ denotes the derivatives of f. If x∗ is an isolated zero in the set of zeros of f, theBrouwer index of the zerox∗ is defined by(12)indf,x∗=deg⁡f,U,where U is a small neighbourhood of x∗. Now we are able to present the main theorem of this section and its application to (10).Theorem 1 (Leray-Schauder).
LetF:R×[a,b]→R be analytic and let Z=(η,ϵ):F(η,ϵ)=0 be the set of zeros of F. Assume the following:(H1)
Z is bounded.(H2)
The setZa=(η,a):Fa(η)=0 with Fa(η)=F(η,a) is finite and there is (η0,a)∈Za with ind(Fa,η0)≠0.Then there is a continuum arcα:[0,1]→Z, α(s)=(η(s),ϵ(s)), with η(0)=η0, ϵ(0)=a, such that either ϵ(1)=b or ϵ(1)=a and η(1)≠η0.From Theorem1 we have the following consequence for the forced pendulum.Proposition 2.
GivenΔ>0, ω∈0,1 there exists a continuum arcη=η(s), ϵ=ϵ(s), s∈[0,1] such that(13)xst=xt,ηs,ϵsis an odd 2π-periodic solution of forced pendulum (6) with x(0)=0, ϵ(0)=0, and ϵ(1)=Δ.Proof.
Letx(t,η,ϵ) be the solution of (6) that satisfies initial conditions (9) and defines the real analytic function(14)Fη,ϵ=xπ,η,ϵwith ϵ∈[0,Δ]. The set of zeros of F is clearly bounded since the derivatives of the solutions of (6) are uniformly bounded in [0,2π] which reveals a simple integration over (6):(15)x˙t≤ϵp1+2πω2,∀t∈0,2π,where ·1 denotes L1-norm in the space L1([0,2π]); therefore(16)η≤Δp1+2πω2;then (H1) holds. On the other hand, since ω<1, the only 2π-periodic solution for the nonforced pendulum (ϵ=0) is the trivial one. This nonlinear center is surrounded by periodic solutions x(t,η,0) with a monotone increasing time period function T(η) with limη→0T(η)=2π/ω (see [13]); therefore if ω<1 we obtain T(η)>2π for all η≠0. As a consequence the zeros of the function F0(η)=x(π,η,0) are reduced to (0,0).
Now we compute the index atη0=0 by linearization; that is,(17)indF0,η0=sign⁡∂F0∂η0,0.Notice that (∂F0/∂η)(0,0)=y(π) where y(t) is the solution of the variational problem(18)y¨+ω2y=0,y0=0,y˙0=1.Since 0<ω<1, then(19)indF0,η0=sign⁡sin⁡ωπω=1,and this verifies (H2). As a consequence, we infer the existence of a continuous family ηs,ϵss∈[0,1] in R×[0,Δ] such that(20)Fηs,ϵs=0,η0=0,ϵ0=0,and either(21)e1=Δor(22)e1=0,η1≠η0,but this last alternative is not possible, again, because ω<1. Then we get the required global continuation xs(t) of odd 2π-periodic solutions for (6).Remark 3.
Note that the continuationxs(t) can be identified with a parametrized curve in ηϵ-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 η=ϕ(ϵ) globally defined on R+ when ω∈0,1/2.Figure 1
Graphic of the two possible options for the continuum arc which contains the point(η0,a) in the Leray-Schauder Continuation Theorem for real analytical functions.Remark 4.
Note that Leray-Schauder’s Theorem demandsa priori bounds over the continuation xs(t) 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 equation(23)x¨+gx=ϵpt,where p is an odd 2π-periodic function and g∈C∞(R) satisfies the following conditions for some positive numbers α0 and ω: (1)
g(-x)=-g(x), for all x∈R,(2)
g′x≤ω2, for all x∈R,(3)
g′(x)>0, for all x∈0,α0,(4)
g′α0=0, g′′≠0 in 0,α0.Without loss of generality, in the following we assume that(24)p1=∫02πptdt=1.The main result of this section is the following.Theorem 5.
Ifω∈0,1/2, then there exist xϵ=xϵ(t) being 2π-periodic and odd continuation of the equilibrium x≡0 for (23), for all ϵ∈R. Moreover(25)xϵ∞≤σω2ωϵ1+sinh⁡ωπsin⁡ωπ,with(26)σω=max⁡sinh⁡ωπ,sinh⁡2ωπ-sin⁡2ωπ2.Remark 6.
As a particular case, this theorem gives the odd global continuation of the equilibrium solutionx≡0 for forced pendulum (6) for all frequencies ω∈0,1/2.The proof of Theorem5 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.
Leta∈C([0,T/2]) with |a(t)|≤ω2 for all t∈[0,T/2] and |a|≠ω2 with ω∈[0,π/T]. Let ϕ1(t) and ϕ2(t) be the solutions of(27)x¨+atx=0,satisfying the initial conditions(28)ϕ10=1,ϕ˙0=0,ϕ20=0,ϕ˙20=1,respectively. Then for all t∈0,T/2 we have the following inequalities:(29)cos⁡ωt<ϕ1t<cosh⁡ωt,1ωsin⁡ωt<ϕ2t<1ωsinh⁡ωt.Proof.
Letμi(t), i=1,2 be the canonical solutions of(30)x¨+ω2x=0that satisfy the initial conditions μ1(0)=μ˙2(0)=1 and μ2(0)=μ˙1(0)=0; that is,(31)μ1t=cos⁡ωt;μ2t=1ωsin⁡ωt.For i=1,2 observe that χi=ϕi-μi satisfies the boundary problem(32)x¨+atx+qit=0,x0=0,x˙0=0,with qi(t)=(a(t)-ω2)μi. So it is not difficult to prove that(33)χit=∫0tGt,sqisds,where(34)Gt,s=ϕ1tϕ2s-ϕ1sϕ2t.Now we want to know the sign of χi(t) on [0,T/2]. First, we remark that qi(t)≤0 since a(t)-ω2≤0 and μi≥0 for all 0,T/2 because ω≤π/T.
On the other hand, sincea(t)≤(π/T)2 on [0,T/2], from the Sturm Comparison Theorem, we deduce ϕ2(t)>0 in 0,T/2. This implies that G(t,s)<0 for all (t,s)∈(t,s):0<s<t,0<t<T/2 (see [14]). From the hypothesis on a(t) we know that qi(t) is not identically zero; then as a consequence we have χi(t)>0 on 0,T/2; that is,(35)cos⁡ωt<ϕ1t,1ωsin⁡ωt<ϕ2t,∀t∈0,T2.The proof for the other inequalities follows the same ideas if we consider the equation(36)x¨-ω2x=0and the canonical solutions(37)μ~1t=cosh⁡ωt,μ~2t=1ωsinh⁡ωt.As in the previous section, letx(t,η,ϵ) be the solution of (23) satisfying(38)x0=0,x˙0=η.From the symmetries over (23) the search of odd 2π-periodic solutions can be focused to solve the Dirichlet problem (in a half period)(39)x¨+gx=ϵpt,x0=0,xπ=0.In other words, we must find the zeros of the function(40)xπ,η,ϵ=0.This is a classical implicit function problem with initial solution of the trivial one η=ϵ=0. Let ω02≔g′(0). Since ω0<ω<1/2, then(41)∂ηxπ,0,0=1ω0sin⁡ω0π≠0;as a consequence there exists a local function η=η(ϵ) such that x(t,η(ϵ),ϵ)=0. Let V be an open neighborhood of (0,0) where ∂ηx(π,η,ϵ)≠0. By the Implicit Function Theorem problem (40) is equivalent to the Cauchy Problem(42)η′ϵ=Fη,ϵ,η0=0,which is well defined on V with(43)Fη,ϵ=-∂ϵxπ,η,ϵ∂ηxπ,η,ϵ.The next proposition gives sufficient conditions to guarantee that V=R2 and F is bounded.Proposition 8.
Ifω∈0,1/2, then the function F(η,ϵ) given in (43) is well defined in all R2 and(44)Fη,ϵ≤σω2sin⁡πω,with σω=max⁡sinh⁡πω,(sinh⁡2πω-sin⁡2πω)/2.Proof.
For each(η,ϵ)∈R2 consider the variational equation(45)x¨+atx=0,at=aη,ϵt=g′xt,η,ϵ.From assumption  (2) over the function g, it follows that |a|≤ω2 in [0,π] uniformly in (η,ϵ) and for assumption  (3) it follows that a≠ω2. Let ϕ1 and ϕ2 be the canonical solutions of (45). From Lemma 7 with T=2π we obtain(46)cos⁡ωt<ϕ1t<cosh⁡ωt,1ωsin⁡ωt<ϕ2t<1ωsinh⁡ωt,for all t∈0,π. On the other hand, the theorem of differentiability with respect to initial conditions and parameters implies that the functions ϕ2(t)=∂ηx(t,η,ϵ) and ψ(t)=∂ϵx(t,η,ϵ) satisfy the following Cauchy problems:(47)x¨+atx=0,x0=x˙0=1,(48)x¨+atx=pt,x0=x˙0=0,respectively. Therefore, for t∈[0,π], we have(49)ψt=-∫0tGt,spsds.Since ω∈0,1/2 from the Comparison Sturm Theorem we obtain ϕ2(t)>0 for all t∈0,π. Following the lemma in [14] we concluded that(50)Gt,s=ϕ1tϕ2s-ϕ1sϕ2t<0,for all (t,s)∈Δ={(t,s)∣0<s<t,0<t<π}. Moreover since 0<ωπ<π/2 then cosωt>0 and sinωt>0 on 0,π; therefore(51)Gt,s=-Gt,s≤maxt,s∈Δ¯⁡Ht,s,where H(t,s)=(1/ω)(coshωtsinhωs-cosωssinωt). Since there are no critical points of H in Δ then H has the maximum value on the boundary of Δ. A simple calculation shows that(52)Gt,s=-Gt,s≤σωω,with(53)σω=max⁡sinh⁡πω,sinh⁡2πω-sin⁡2πω2.On the other hand(54)ψt≤σω2ω∀t∈0,π.Finally we obtain(55)Fη,ϵ=ψπϕ2π≤σω2sin⁡πω.Remark 9.
For the autonomous Cauchy problem(56)dηdϵ=Fη,t,dtdϵ=1,η0=0,t0=0,which has a bounded vector field X(η,ϵ)=(F(η,ϵ),1); we conclude that η(ϵ) is well defined in -∞,∞ (see [15]); therefore for all ω∈0,1/2 we obtain an odd, 2π-periodic continuation φϵ(t)=x(t,0,η(ϵ),ϵ) of the equilibrium for all ϵ∈R.Now we are able to present the proof of the main theorem of this section.Proof of Theorem5.
LetX(ϵ)=(η(ϵ),t(ϵ))tr and consider the autonomous Cauchy problem(57)X′=FX=Fη,ϵ,1,X0=0,0.It follows from Proposition 8 that F is bounded. From here we deduce that all solutions are globally defined; then η(ϵ) is well defined for all t∈-∞,∞; (see [15]). As a consequence for all ω∈0,1/2 we obtain an odd, 2π-periodic continuation φϵ(t)=x(t,0,η(ϵ),ϵ) of the equilibrium for all ϵ∈R. Now by the Mean Value Theorem we have(58)φϵ≤ϵmaxξ≤ϵ,t∈0,π⁡∂ϵφϵt,ηξ,ξ.On the other hand,(59)∂ϵφϵ≤∂ηxη′+∂ϵx.Since ∂ϵx and ∂ηx are, respectively, the solutions of the initial value problems (47) and (48), with a(t)=g′(φϵ(t)), they satisfy inequalities (46) and (54) in [0,π]; therefore(60)∂ϵφ≤σω2sin⁡πωsinh⁡ωπω+σω2ω=σω2ω1+sinh⁡ωπsin⁡ωπ;as a consequence(61)φϵ≤ϵσω2ω1+sinh⁡ωπsin⁡ωπ,concluding the proof.


## 4. Linear Stability of the Continuation

In this section we will obtain a proof of the linear stability of the continuationφϵ. 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.
Ifa∈CT(R) satisfies 0<m2≤a(t)≤M2 on [0,T] with strict inequality on a set of positive measures on [0,T], then (i)
any solution of (27) has infinitely many zeros;(ii)
ift1 and t2 are two consecutive zeros of any solution of (27), they satisfy(62)πM<t2-t1<πm.The proof of Lemma10 follows easily from Sturm’s Comparison Theorems (see [16]).Proposition 11 (Lyapunov-Zukovskii).
Leta∈CT(R) such that 0<a(t)≤(π/T)2, for all t∈R, with strict inequality on a set of positive measures in [0,T]. Then (27) is elliptic.Proof.
By a contradiction argument suppose that there is a real Floquet multiplierλ. Let φ(t) be the Floquet solution associated with λ; then(63)φt+T=λφt,∀t∈R.Since 0<a(t) it follows from Lemma 10 that ∃t0∈R such that φ(t0)=0; then φ(t0+T)=0. Once again, from Lemma 10t0 and t0+T are consecutive zeros; as a consequence the distance is at most T 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 Theorem5, 2π-periodic continuation φϵ of the equilibrium x=0 of (45) is elliptic if |ϵ|<I=I(ω) where(64)Iω=2α0ωsin⁡πωσωsinh⁡πω+sin⁡πω.Proof.
Withϵ∈]-I,I[ and from (25) it is clear that |φϵ(t)|≤α0. On the other hand, from the conditions on the function g, we have(65)0<at=g′φϵt≤ω2≤14,∀t∈R.On the other hand, if at=1/4 for all t∈R, then φϵ≡0 since g′ is strict monotone in 0,α0 and this implies p≡0 which is a contradiction. By continuity we deduce the existence of a set of positive measures in [0,T] such that 0<a<1/4. The conclusion follows if we apply Proposition 11 to the variational equation along φϵ:(66)y¨+aty=0,where a(t)=g′(φϵ(t)).Corollary 13.
Forp∈C2π(R), p1=1, and odd ω∈0,1/2, the equilibrium solution x≡0 of the oscillator (23) can be continued like an odd 2π-periodic solution φϵ(t) of the forced pendulum(67)x′′+ω2sin⁡x=ϵpt,which is elliptic (therefore linear stable) for |ϵ|<γ(ω) where(68)γω=πωsin⁡πωσωsinh⁡πω+sin⁡πω;moreover,(69)φϵ≤ϵσω2ω1+sinh⁡πωsin⁡πω.


## 5. Odd Global Continuation for any Positiveω

In order to remove the restrictionω∈0,1/2 in Theorem 5, we follow a different approach to estimate an upper bound for |∂ϵx(π;η,ϵ)| and a lower bound for |∂ηx(π;η,ϵ)| in (42)-(43). As before we assume the condition p1=1 on oscillator (23).We start this new approach by setting new conditions over the functiong. Suppose that there exist positive values ω and ω0∉N such that(a)
|g′(x)|, |g′′(x)|≤ω2, for all x∈R.(b)
g′(0)=ω02.The main result of this section is the following.Theorem 14.
Under assumptions (a) and (b), there existφϵ=φϵ(t) being 2π-periodic and odd continuation of the equilibrium x≡0 for (23) for all |ϵ|<Δ0 with(70)Δ0=Δ0ω,ω0=1+ω-2ω0Aω3Aω2-1sin⁡ω0π.Moreover(71)φϵt≤ϵAω1+AωΥω,ω0,ϵ,where Aω2=e(1+ω2)π and Υ(ω,ω0,ϵ) is a positive constant for |ϵ|<Δ0 given by(72)Υω,ω0,ϵ=1ω0sin⁡ω0π-ϵω2Aω31+ω2Aω2-1.The proof of Theorem14 follows similar arguments and steps are given in Section 2 for proving Theorem 5. We start by considering again the solutions ϕ2(t)=∂ηx(t,η,ϵ) and ψ(t)=∂ϵx(t,η,ϵ) of Cauchy problems (47) and (48), respectively.Before the proof of Theorem14 we point out some preliminary results. The first one is about the growing of the solutions for Hill’s equations.Lemma 15.
Leta∈C([0,T]) with |a(t)|≤σ for all t∈[0,T] and let ϕ1(t), ϕ2(t) be the unique solutions of⋄x¨+atx=0satisfying(73)ϕ10=ϕ˙20=1,ϕ˙10=ϕ20=0.Then, for i=1,2, we have(74)ϕit≤e1+σ/2T∀t∈0,T.Proof.
Letx(t) be a solution of ⋄ and define the function(75)Vt=12x2t+x˙2t.By a direct computation we obtain(76)V˙t=xx˙1-at≤x2+x˙221+σ=Vt1+σ.Solving this differential inequality with initial condition V(0) we get(77)Vt≤V0e1+σt.In particular for the canonical solutions ϕi(t)   i=1,2 we have V(0)=1/2 and therefore(78)ϕit≤ϕit,ϕ˙it≤e1+σ/2t,∀t∈0,T.From (49), hypothesis (a), and Lemma 15 taking a(t)=g′(x(t,η,ϵ)) on [0,π] we have the following estimative:(79)∂ϵxπ,η,ϵ≤e1+ω2π,∀η,ϵ∈R2.The second preliminary result is the classical fundamental inequality in ordinary differential equations (see [16]).Proposition 16.
Suppose thatx(t), y(t) satisfy (80)x˙=ft,x,xt0=x0,y˙=gt,y,yt0=y0,where f,g:U⊂R×Rn→Rn, are continuous and Lipschitz with respect to the second variable with Lipschitz constant K. If(81)ft,u-gt,u≤δ,t,u∈U,then(82)xt-yt≤x0-y0eKt-t0+δKeKt-t0-1,for all t in a common interval of existence.Proof of Theorem14.
The existence of nontrivial odd,2π-periodic solutions of (23) is equivalent to the solving of problem (40). Furthermore, from assumption (b)(83)∂ηxπ,0,0=1ω0sin⁡ω0π≠0;therefore we can consider the Cauchy problem(84)η′ϵ=-∂ϵxπ,η,ϵ∂ηxπ,η,ϵ,η0=0,in a small rectangle of the form ΩΔ,Γ=(ϵ,η):|ϵ|≤Δ,|η|≤Γ.
Our objective is to solve initial value problem (84) in a concrete ϵ-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 |∂ηx(t,η,ϵ)| on ΩΔ,Γ.
Consider variational equation (45) given by (85)x¨+aη,ϵtx=0,aη,ϵt=g′xt,η,ϵ.The vector field of the first order system associated with this variational equation is given by (86)Fη,ϵt,x,y=y,-aη,ϵtx,with y=x′. Notice that this vector field is continuous and (x,y)-Lipschitz, with Lipschitz constant K=1+ω2 uniformly in (ϵ,η)∈R2.
Next we apply the fundamental inequality to the vector fieldsFη,ϵ(t,x,y), F0,0(t,x,y) and the respective solutions ∂ηx(t,η,ϵ), ∂ηx(t,0,0)=(1/ω0)sin(ω0t) (both in the same initial conditions) on the domain U=R×-Aω,Aω×R, where Aω=e(1+ω2/2)π is an upper bound for the canonical solutions of (45) given by Lemma 15. Henceforth(87)∂ηxπ,η,ϵ-1ω0sin⁡ω0π≤δ1+ω2Aω2-1,where δ is an upper bound on U for(88)Fη,ϵt,x,y-F0,0t,x,y=aη,ϵt-ω02x.From the mean value theorem we get(89)aη,ϵt-ω02≤ϵm,where m is any positive number such that(90)m≥sup⁡∂aη,ϵt∂ϵ:t∈0,π,η∈R,ϵ>0.Besides,(91)∂aη,ϵt∂ϵ=g′xt,η,ϵ∂ϵxt,η,ϵ,and from (79) and condition (a) over the function g we have(92)∂aη,ϵt∂ϵ≤ω2Aω2,for all t∈[0,π], η∈R, ϵ>0. In conclusion on the domain U we have(93)Fη,ϵt,x,y-F0,0t,x,y=aη,ϵt-ω02x≤δ≔ϵω2Aω3.Now we impose the restriction (see (87))Rδ1+ω2Aω2-1<1ω0sin⁡ω0π.The restriction R is equivalent to |ϵ|<Δ0 with Δ0 given by (70). Combining (87) and R we obtain (for all u,v,γ∈R if |u-v|<γ<|v|, then |u|>|v|-γ)(94)∂ηxπ,η,ϵ≥Υω,ω0,ϵ>0,with Υ(ω,ω0,ϵ) given by (72). Finally, an upper bound for the modulus of the right hand side in (84) is given by(95)∂ϵxπ,η,ϵ∂ηxπ,η,ϵ≤M≔Aω2Υω,ω0,ϵ,on ΩΔ0,Γ. For classical Peano’s Theorem we obtain a solution η=ϕ(ϵ) of (10) for |ϵ|≤minΔ0,Γ/M. Note that our estimatives do not depend on η; therefore we can take Γ≥MΔ0 so the solution ϕ(ϵ) is defined on -Δ0,Δ0. So we obtain the odd 2π-periodic family φϵ(t)=x(t,ϕ(ϵ),ϵ). 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 ϵ-interval for frequencies ω far away of the strong resonances. In contrast, our first main result (Theorem 5) provides an infinite ϵ-interval for low frequencies ω∈[0,1/2]. Moreover for high frequencies we obtain 2π-periodic continuation of the equilibrium x≡0 in a quantified ϵ-interval (Theorem 14) for general oscillators of pendulum type.The nonlinear stability in the Lyapunov sense of the familyφϵ 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 Pϵ of (23) has invariant curves in all neighbourhood of the elliptic fixed point z0=(φϵ(0),φ˙ϵ(0)).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 coefficientβ of Pϵ called the firsttwist coefficient. The Möser Theorem essentially claims that if β≠0, then the fixed point z0 is stable and therefore φϵ is also stable. We will explain this a little bit more.By means of a translationφϵ to the origin y=x-φϵ and making Taylor’s expansion up to the third order, we arrive to the equivalent equation with equilibrium y≡0:(96)y¨+aϵty+bϵty+cϵty3+⋯=0,where aϵ, bϵ, and cϵ and the remaining terms are periodic functions of period 2π; moreover(97)aϵt=g′φϵt,bϵt=g′′φϵt2,cϵt=g′′′φϵt6.The twist coefficient β=β(ϵ) is a meromorphic function which depends on aϵ, bϵ, and cϵ in a not easy way (see [7, 14, 18, 19]). In [14] Ortega gave for the first time an explicit formula for β and subsequently Zhang in [7, 19] provided some refinements of it making the dependence with respect to aϵ more synthetic and very practical for applications (see [5, 7, 19–21]). The open problem consists of studying the first zero of β(ϵ) in order to estimate ϵ-range for the stability of φϵ. In this direction the results in [7] could be useful. Finally, it is important to notice that when g is really analytical the recently Hanßmann and Simó’s results in [22] could be useful to study the nonlinear stability for the continuation.



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