Exact and Sinusoidal Periodic Solutions of Lienard Equation Without Restoring Force

In this paper we study a Lienard equation without restoring force. Although this equation does not satisfy the classical existence theorems, we show, for the first time, that such an equation can exhibit harmonic periodic solutions. As such the usual existence theorems are not entirely adequate and satisfactory to predict the existence of periodic solutions.


Introduction
The Lienard equationẍ where overdot denotes derivative with respect to time, ϑ(x) and h(x) are arbitrary functions of x, is a generalization of the conservative equation where h(x) is a restoring force function, to take into account energy dissipation in real world dynamical systems. As such, the use of equation (1.2) constitutes a first approximation to model oscillations in dynamical systems so that the determination of periodic solutions of equation (1.1) has become a fundamental problem in mathematics and physics. In this way, a rich and various study has been carried out on the existence of periodic solutions of equation (1.1). One can find a vast literature on the theorems of existence and uniqueness ( [1], [2], [3], [4], [5], [6]) of periodic solutions of equation  , [2], [3], [4], [5], [6]) often the requirement xh(x) 0, for x = 0, that is to say has not been formulated and solved in any study published to date explicitly. Nevertheless, it is reasonable to investigate such a question due to the results obtained previously in ( [14], [15]). Indeed, the authors in ( [14], [15]) succeeded in proving explicitly the existence of harmonic and isochronous periodic solutions of equations of type where σ(x,ẋ) denotes the damping, as a function of x andẋ, and the restoring force is null, whereas equation (1.6), as such, can be viewed as a more general form of equation ( is carried out (section 4) and a conclusion is given finally for the work.

Theory
Let us consider the Lienard type equation ( [14], [15], [16]) where prime means differentiation with respect to x, a and are arbitrary parameters, and f (x) and g(x) = 0 are arbitrary functions of x. Equation (2.1) can be written Under the requirement g(x) = 1, equation (2.3) reduces tö which can lead to dy dx = −af (x) so that one can get that is, the integral curves where C is a constant of integration. The problem to solve becomes now the choice of function f (x) that reduces equation (2.7) to that of an ellipse centered at the origin if we expect to have harmonic periodic solution.  14], [15], [16]), defined in the form

Equation of interest
where K is an arbitrary constant and C = 0. From the expression of f (x), equation (3.2) takes the form −a(t + K) = dx so that, after integration, one can obtain In this context, by inverting, the exact harmonic and isochronous periodic solution of equation (3.1) is secured as where µ 0 and a ≺ 0. Solution (3.5) shows that equation ( As H is a constant, the damped equation (3.1) is then, a conservative nonlinear system.

Phase plane and existence theorem analysis
According to equation (2.5), equation (3.1) is equivalent to the dynamical systeṁ x = y ,ẏ = axy The equilibrium points in the (x, y ) phase plane are given by y = 0, and axy √ µ 2 −x 2 = 0. This means that if y = 0, then x = 0, and inversely, if x = 0, then y = 0. This is in opposition to results (3.5) and (3.7) showing that the origin is a center, that is a single equilibrium point. According, for example, to Theorem 11.3 of the book [ [2], p.390] equation (3.1) has a center at the origin when h(x) 0, for x 0, and is odd. Since h(x) = 0, then equation (3.1) cannot have a center at the origin. As previously, this prediction of the preceding existence theorem is in contradiction with the analytical results (3.5) and (3.7). As seen, the classical existence theorems exclude some cases of Lienard nonlinear differential equations while the exact and explicit results show the existence of harmonic and isochronous periodic solutions. Now, we can address a conclusion for the work.

Conclusion
In this paper a Lienard equation without restoring force is studied. Although the equation does not satisfy the usual existence theorems, we have successfully shown that it can exhibit sinusoidal and isochronous periodic solution as the linear harmonic oscillator.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.