Existence of Running Solutions in a Relativistic Tricomi’s Equation Using Perturbation Theory

We use perturbation methods to establish the existence of a second kind periodic solution (running solution) of a nonlinear Tricomi’s equation type under relativistic effects. First, we estimate conditions for the existence of either an equilibrium point or a second-kind periodic solution through the average method, where we assumed the nonlinear part as a positive perturbation. Then, we use the Melnikov function to estimate conditions for the existence of running solutions, considering the persistence of the homoclinic orbits associated with the conservative equation.


Introduction
It is well known that Tricomi's equation: is part of the family of forced pendulum equations, where the parameters are related to constant torque, amplitude, and damping. The dynamics of these types of equations have been widely studied, see for example [12,26,27]. Some of those studies are focused on determining the existence of equilibrium solutions, and the existence of second-kind periodic solutions, i.e. solutions x(t) such that x(t + τ ) = x(t) + 2π, for some τ > 0, also known as running solutions.
We are interested in showing that the Tricomi's equation type has second kind periodic solutions with period τ = 2π, where b, c > 0 are parameters, and φ is the relativistic momentum defined by 187 Page 2 of 15 Z. Daniel Cortés and G. Alexander Gutierrez MJOM The operator is associated with a relativistic pendulum, see for example [13,15,24,28]. Particularly, Tung in [28], presented the deduction of the model of the relativistic generalization in the classical harmonic oscillator, and Llibre et. al. in [13], studied a non-autonomous system using the average method. In addition, in [2,3,9,11,19,21], it could be found other operators such as the p-Laplacian. Other Tricomi's generalized equations and pendulum equations can be found in [6,14,[16][17][18]25], for example in [14], the author studied the relativistic Liénard equations, where the problems of existence, non-existence, and uniqueness of homoclinic solutions are studied.
To establish conditions for the existence of second kind periodic solutions of (2), we use the perturbation theory. First, we use the Average method to find conditions for the existence of either equilibria or running solutions, see [7,23,29]. Second, we will use the Melnikov function, see [5,22], to establish conditions under the parameters to find the persistence of the homoclinic curve, and, therefore, conditions for the existence of the second kind periodic solutions.
This paper has three sections after the Introduction. In Sect. 2 we present the main theorems, where the Theorem 1 establishes conditions for the existence of asymptotically stable periodic solutions, which can be equilibria or second kind periodic solutions. Theorem 2 establishes conditions for the persistence of homoclinic orbits, and in consequence, Theorem 3 gives conditions for the existence of the second kind of periodic solution. Section 3 goes in two ways: in the first one, some results will be established about the technique of Average, which will be used to prove the first theorem; in the second one, it is presented some of the generalities of the Melnikov function, to prove the second and third theorem. Finally, we have included an Appendix where we present some computations, and we present some conclusions.

Main Results
Perturbation theory is applied to non-linear oscillations, based on a system from which the solutions are known and a perturbation of this system is made by introducing a small parameter ε, see [7,10,20,29], this method allows us to get an approximation that leads to asymptotic series, [1]. In this section, we will present the main results applying two perturbation methods: average method and the Melnikov function.
For that purpose, we must modify the eq. (2) in two ways: in the first way, we want to use the Average method, then we obtain the modified equation where h is a continuous 2π-periodic function, continuously differentiable, with an average zero, that is

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Note that the function h is a generalization of function sin x. Then, we have the next theorem. Theorem 1. Given the differential eq. (3). Assume that c > 0, and b/c < 1 then, eq. (3) has an asymptotically stable periodic solution for ε small enough.
The previous theorem raises the existence of a periodic solution, that can be either an equilibrium or a second kind periodic solution. Therefore, for characterizing in which cases there are second-kind periodic solutions, we will use the Melnikov functions to find conditions on the parameters c, b. For this purpose, we perturbed the system (2) in the next way Note that for the conservative case, i.e. when ε = 0 in (4), the problem has three consecutive equilibria, with two saddles in x = ±π, and one center in x = 0. Additionally, note that we can define the energy function Moreover, there is a family of periodic orbits given by such that Γ e approaches to the center in (0, 0), when e → 0, and to a bounded curve, when e → 2 − , denoted as Γ 2 . In the last case, the trajectory is an heteroclinc orbit that connect (−π, 0) with (π, 0), see Fig. 1. Let be Γ + 2 and Γ − 2 the upper and lower heteroclinics respectively, all the loops situated between the separatrix Γ + 2 and Γ − 2 are closed loops surrounding the center (x, v) = (0, 0), which have energies E(x, v) = e, for 0 < e < 2. Now, before the next theorem, we define the following values then, the orbit Γ + 2 of (4) persist, for ε small enough.
It will be established that only the superior heteroclinic can persist, while the inferior one always breaks down. The persistence and rupture of homoclinic and heteroclinic, that the Melnikov function studies, allows to detect the appearance of chaos in the systems; see [7] or [22] for more details and examples.
Finally, we study the case Γ > Γ + 2 , and prove that it is possible to preserve a closed orbit in the cylinder C, when c is fixed and b > c ξ 2 ξ 1 , moreover, it is an attractor, and we will prove that this closed orbit is unique. When b c > ξ2 ξ1 , we have that the Melnikov function is positive, which implies that the upper branch of the stable manifold W s (π, 0) is below the  Figure 1. Energy level curves associated with the conservative equation of (4), note that in the red level curve, there are two heteroclinics upper branch of the unstable manifold W u (−π, 0), this gives rise to thinking about the possibility that one of the rotations may persist.

Proofs of Main Theorems and Some Comments
In this section we will present some necessary results about the methods used in our study, then we present the proofs of the main results. Additionally, some numerical computations will be presented.

Average
In this section, we are going to study the eq. (3), with the average technique, which is a computational method of approximation, directed in general to asymptotic series, so that, it is more natural in many problems, see [23]. Its first formulations are born with Lagrange to perturb the problem of the two bodies to generalize it to the problem of the three bodies, see [29].
In the theory of the Averaged method, the equation is studied, where x, ε) are T -periodic functions respect to the variable t. By making the appropriate variable changes to the eq. (7), the associated average equation (also known as Standard or Lagrange Form) in D is given by where Furthermore, it can be proved that if x(t) is a solution of eq. (7), and y(t) is a solution of eq. (3), then the difference of two solutions, associated with the error of the approximation is such that x(t) − y(t) = O(ε), on a time scale 1/ε; see [29, theorem. 11.1] and [23] for some examples.
The following theorem establishes the necessary conditions for the existence of a T -periodic solution of eq. (7).
Then there exists a T -periodic solution ϕ(t, ε) of eq. (7) which is close to p such that On the other hand, the same author allows us to characterize the stability of those periodic solutions in the following result.
and by dividing dy/dt over dx/dt, we obtain the equation that depends on the variable Then, we get that The critical point associated with the equation of the standard form associated to (3) is the points y = b c . Note that the eigenvalues associated with the equation of the standard form at the critical point is given by Then, from Theorem 4, there is a T -periodic solution, even more, as the eigenvalue associated with (12) is not positive, then by Theorem 5, we have a periodic asymptotically stable solution of (12). Now, since we have found a 2π-periodic solution in the xy plane, this can be a second kind asymptotically stable periodic solution or a fixed point of (3). Remark 1.
1. Note that the function is defined for all y in a neighborhood of the point b/c, and since it is continuous in a compact, we can define a constant M that bounded this function. Moreover, since φ (y) is always positive in the neighborhood, the function is well defined and its derivatives with respect to y exist and are continuous. 2. Note that when b > ||h|| ∞ , there is no a fixed point, then only a running solution can exist, while if b < ||h|| ∞ , either a running solution or a fixed point can exist, see Fig. 2. 3. Particularly, let be h(x) = sin x, then we have that the respective modified equation of (2), can have either a second kind asymptotically stable periodic solution or a fixed point. In the Fig. 3, the red curve is associated with the running solution and the blue is a curve which go to the equilibrium point.

Melnikov Function
The objective of this subsection is to apply the Melnikov method to (4), and prove that there exists a finite positive number c(b) such that , all solutions converge to one of the equilibrium points.  = c(b), instead of a periodic solution there is a separatrix in the phase space, the unique trajectory connecting the unstable equilibrium points. All solutions starting above the separatrix converge to it, whereas motions initially below the separatrix converge to one of the stable equilibrium points.
Before approaching our problem, we are going to establish some aspects about the Melnikov function. Consider the equation where f is a vector field Hamiltonian in R 2 and g(x, t) ∈ C ∞ (R 2 × R/(T Z)), where its components are given by g = (g 1 , g 2 ) and ε ≥ 0. The first approximation of Γ e is given by the zeros of the Melnikov function M e , defined as: Therefore, it is necessary to know the number of zeros of (4). Thus, we will use the next result. 1. If M e0 (t 0 ) = 0, then, there are no limit cycles near Γ e0 for ε + |t 0 + t| sufficiently small. 2. If M e0 (t 0 ) = 0 is a simple zero there is exactly one limit cycle Γ e0 (t 0 , ε) for ε+|t 0 +t| sufficiently small that approaches Γ e0 when (t, ε) → (t 0 , 0).
Remark 2. The Melnikov function can be interpreted as the first approximation in ε of the distance between the stable and unstable manifold, measured along the direction perpendicular to the undisturbed connection, that is, In particular, when M e (t 0 ) > 0 (resp. < 0) the unstable manifold is above (resp. below) the stable manifold; see [7] or [22] for more details.
Let us denote by T (e) the period of librations, associated to the energy when e ∈]0, 2[. Solving for y in the equation E(x, y) = e, and using the symmetries of the unperturbed system, we get the following expression Using the change of variable u = 1− 1 e (1 − cos x), we can find the relationship between the period T (e), and energy e as follows

Existence of Running Solutions in a Relativistic
Page 9 of 15 187 The computations are quite laborious and have been placed in Appendix. Note that T (0) = 2π, which is equal to the period of the linearization of the unperturbed system. Moreover, for e small enough, satisfying e 2 (1 − u) < 1, then, we can expand the Taylor series of around e = 0, and we obtain the following expression On the other hand, using (14) we have T (e) → ∞, as e → 2 − . Besides, the period T (e) on the interior of Γ + 2 ∪ Γ − 2 , is an increasing function in e.
Proof of Theorem 2. Note that, when ε = 0, we have a fixed point in (0, 0) corresponding to a center and another pair in (±π, 0), where two heteroclinics orbits close. Let be Γ + 2 , Γ − 2 defined as before, that is, the upper and lower heteroclinics, respectively, that correspond to the level curves associated with the energy function of the system when ε = 0 given by: The Melnikov function associated with Γ + 2 is given by: Now, let be y = (2 + cos(x)) 2 − 1 2 + cos(x) .
Replacing in (4) and recalling that y = dx/dt, we obtain that the associated Melnikov function is given by let be define ξ 1 , ξ 2 as in (5), we get that which has simple zeros when Then [8,Theorem 6.4] implies that the Γ + 2 survives, in a homoclinic bifurcation on a unique curve tangent. We present a numerical example in Fig. 4 where the conditions on Theorem 2 are fulfilled. Note that the upper homoclinic persists (near the red curve), while the lower goes to the fixed point (near to the green one).
Finally, we conclude that there is a unique attracting closed orbit encircling the phase cylinder for each c ∈ 0, ξ 1 ξ 2 b , so that the vector field is directed into a band B, and the unique orbit which persists belongs to such band B.
In Figs. 5 and 6 , we present numerical examples in the phase plane, for the non-existence of the running solution and the existence respectively. Particularly, Fig. 5 shows that any curve goes toward the fixed point. On the other hand, Fig. 6 shows the existence of a band where the solutions approximated to the running solution.