We study the one-dimensional forced harmonic oscillator with relativistic effects. Under some conditions of the parameters, the existence of a unique stable periodic solution is proved which is of twist type. The results depend on a Twist Theorem for nonlinear Hill’s equations which is established and proved here.

In this paper, we study the existence of a stable periodic solution (periodic response) in the one-dimensional forced harmonic oscillator with relativistic effects:

The existence of chaotic behavior in the relativistic harmonic oscillator has been investigated numerically in [

From a more mathematical perspective, (

Chu et al. have proven in [

On the other hand, in recent years some results about similar oscillators with relativistic effects, like the relativistic forced pendulum [

Our aim is to study the existence and stability of periodic solutions for the relativistic harmonic oscillator (

It is a well known fact that the appearance of

The genericity is understood relative to certain topology constructed via jets of functions [

According to this, the proof of the existence of periodic solutions of twist type is a first step in the comprehension of the chaotic behavior numerically evidenced in [

Theorem

Assume that the parameters

Hypothesis (H3) may look rather weird; however, it gives some interesting corollaries in a direct way.

With fixed

With fixed

Both corollaries follow easily by passing to the limit in the conditions of Theorem

The rest of the paper is organized as follows. Section

As we mentioned in Introduction and Main Results, the existence of a periodic solution for (

For the Hamiltonian (

Let one assume that

It follows from the classical theory of upper and lower solutions [

Of course, the periodic solution

In this section, we study the stability of the periodic solution found in the previous section in the linear sense. Let us fix a

The following bounds over

Now we can formulate and prove the following result.

Assume that

From (

For the uniqueness, suppose that

The system under study is conservative, so the stability in the sense of Lyapunov can not be directly derived from the first approximation because of the possible synchronized influence of higher terms leading to resonance. After the works of Siegel and Moser [

From the point of view of KAM theory ([

More recently, these ideas have taken a renewal interest starting from some Ortega’s works [

Notice that the estimate (3.39) in [

We consider the nonlinear Hill equation:

The linearization of (

Let

Given

The Poincaré mapping associated with (

Note that

We say that the equilibrium

Notice that, according to this definition, all equilibrium of twist type is Lyapunov stable. Also, it is known, from the general theory, that an equilibrium of twist type exhibits

The twist coefficient

The main result of this section is as follows.

Assume that for (

Assume that the condition (

The function

The proof of Theorem

Condition (

On the other hand, from (

In Section

The third approximation for (

From Proposition

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

This work is supported by Capital Semilla 2014-2015 project 00004025 Pontificia Universidad Javeriana, Seccional Cali, Cali, Colombia.