Periodic second-order systems and coupled forced Van der Pol oscillators

We present an existence and localization result for periodic solutions of second-order non-linear coupled planar systems, without requiring periodicity for the non-linearities. The arguments for the existence tool are based on a variation of the Nagumo condition and the Topological Degree Theory. The localization tool is based on a technique of orderless upper and lower solutions, that involves functions with translations. We apply our result to a system of two coupled Van der Pol oscillators with a forcing component.

Finding periodic solutions in second-order differential equations or systems can be delicate, specially when the non-linearities are non-periodic.The search for periodic solutions in such problems has had an increasing interest, as it can be seen by several papers from the last decades, for example, [7,8,9].The most common argument for obtaining periodic solutions is based on the assumptions of periodicity in the non-linearities.As an example, we refer to the work [8], where the authors study the problem with constants c i > 0, parameters s i , continuous functions Ξ i (t, x, u, v) : R 4 → R, 2π-periodic in t and x, obtaining an Ambrosetti-Prodi alternative for periodic solutions.
In [7], the following second-order differential equation is studied, where Ψ ∈ C((0, +∞), R), ϕ, a and b are T -periodic, and in L([0, T ], R), m, µ and y are constants with m ≥ 0 and µ ≥ y > 0. Here, asymptotic arguments are a key argument to obtain periodic solutions.The above problems, (3) and ( 4), require periodicity for the non-linearities.In [9], the authors study the second-order differential problem, without requiring periodicity for the non-linearities.However, the function Θ does not depend on the first derivative, thus making problem (5) a particular case of problem (1).Moreover, the obtained results require an order relation (for the well and inverse order cases) between lower and upper solutions.
Other attempts to find periodic solutions in ordinary differential problems may not present periodicity requirements but, typically, only approximate solutions are found, see [10,11,12].
To overcome such constraints, in this work we present an original methodology for finding periodic solutions in generalized second-order coupled systems.
Motivated by previous works existent in the literature, (see, for instance, [15,16,17,18]), it is the first time, to the best of our knowledge, where sufficient conditions are given to prove the existence of periodic solutions for coupled systems, without assuming periodicity in the non-linearities nor any order requirement between lower and upper solutions.Therefore, the set of admissible functions for possible lower and upper solutions has a wider range.
Within this line of research, the present work is somehow a continuation of [13,14], which aim to provide a complete study on the existence and localization of periodic solutions in first-order generalized coupled systems, with and without impulses, using this technique of orderless lower and upper solutions, where the lack of order is overcome with adequate translations.
The strategy employed consists of proving the existence of at least a periodic solution (z(t), w(t)) applying the Topological Degree Theory and to localize the existing solution in a strip bounded by lower and upper functions.
This paper is organized as follows.In Section 2 all the required definitions and lemmas are presented.In Section 3 the existence and localization result is formulated as the main theorem, together with the respective proof in four claims.A numerical example shows the applicability of the main theorem.In Section 4 we apply our methodology to the problem of two coupled Van der Pol oscillators with forcing terms.As all the requisites of the main theorem in Section 3 are verified, we suggest possible localizing functions for the existing periodic solutions of this problem.We draw the conclusions of our study in Section 5.

Definitions
Consider the Banach space X := C 1 [0, T ], equipped with the norm and the vectorial space X 2 := (C 1 [0, T ]) 2 , with the norm In second-order differential equations it is important to have a control on the first derivatives, which is achieved using a Nagumo-type condition: such that The a priori estimates for the first derivatives are given by the following lemma.
We present below the definition of upper and lower solutions.

Definition 3. The pair of real functions
and

Main Result
The following theorem is an existence and localization result using orderless lower and upper solutions.Suppose f, g satisfy a Nagumo-type condition, according to Definition 1 relative to the intervals min min and Then, there exists at least one pair (z(t), w(t)) ∈ (C 2 [0, T ]) 2 , solution of problem (1), (2) and, moreover, Proof.Define the truncated functions .
together with the periodic boundary conditions (2).The functions α i , β i : [0, 1] → R, i = 1, 2, given by are, respectively, lower and upper solutions of problem ( 29), ( 2), according to Definition 3, with Remark that the lower and upper functions, shown in Figure 1, are not ordered, as it is usual in the literature.
The above problem is a particular case of (1), ( 2), with T = 1, and where the non-linearities, according to Definition 1, satisfy a Nagumo-type condition in the set It is clear that functions φ and ψ satisfy (6).
As the assumptions of Theorem 4 are verified, then the system (29), ( 2) has, at least, a solution (z as shown in Figure 2.

Coupled forced Van der Pol oscillators
The equation for the damped harmonic motion is Balthazar Van der Pol (1889-1959) modified the damped harmonic oscillator by considering a negative quadratic term for the friction term to obtain self-sustained oscillations.This modification resulted in the Van der Pol oscillator [19,20], which can be represented by the following equation, x ′′ (t) − ϵ(1 − x 2 (t))x ′ (t) + x(t) = 0, where x(t) is the time-dependent variable and −ϵ(1 − x 2 (t)) is the non-linear damping term, with ϵ > 0.
A variant of this problem can be thought of, by coupling two Van der Pol oscillators.We consider the following system, for t ∈ [0, T ], together with the periodic boundary conditions (2).
The terms D 1 tanh(E 1 z(t) − F 1 w(t)) and D 2 arctan(E 2 w(t) − F 2 z(t)) correspond, respectively, to each coupling and G i cos(t) is a time-dependent periodic forcing.We chose T = 1 and the parameter set are, respectively, lower and upper solutions of problem (31), (2), with the numerical values (32) according to Definition 3, with The functions f and g verify the monotonicity requirements and the Nagumotype condition of Definition 1 in the set As all the assumptions of Theorem 4 are verified, then there is at least a periodic solution for the system of two coupled forced Van der Pol oscilators (31) with the periodic boundary conditions (2), and the parameter set of values (32).Moreover, remark that, from (31) and (32), this solution is a non-trivial one.
Figure 3 shows the shifted functions that localize existing periodic solutions in the system (31), (2), with parameter values (32).

Conclusions
When reviewing the literature, one realizes that there are several attempts to find periodic solutions in second-order differential problems.However, either periodicity requirements are made [7,8], or approximated solutions are presented [10,11,12], or the problem does not present complete non-linearities [9].
In this work we present a general second-order differential coupled system, with dependencies on both variables, z, w, their derivatives, z ′ , w ′ , and on time, t.We provide sufficient conditions to prove the existence of periodic solutions for problem (1), (2) without assuming any type of periodicity in the non-linearities.
To localize an existing periodic solution, no order requirement is made between lower and upper solutions.Therefore, we increase the range of possibilities when considering well-ordered and not well-ordered localizing functions α i , β i , see Example 5.
Our methodology is successfully applied to a system of coupled Van der Pol oscillators with forcing terms.Under the conditions of the main result, Theorem 4, we prove the existence of a periodic solution for problem (1), ( 2), and localize it within a strip bounded by shifted lower and upper solutions α 0 i , β 0 i , whose requirements follow Definition 3.
When applying Theorem 4 to a specific second-order system, one must verify that the non-linearities satisfy the Nagumo-type condition of Definition 1, as well as the required monotonicities.Note that Theorem 4 does not guarantee the existence of a non-trivial solution.That guarantee arises with the nature of the system in study, i.e., whether if it allows for constant solutions or not.
As we deal with a generalized system, our study has applicability in several realcase scenarios in Nature, and it can be an important tool for other mathematical problems.

Aknowledgements
This work research was supported by national funds through the Fundação para a Ciência e Tecnologia, FCT, under the project UIDB/04674/2020.