On truly nonlinear oscillator equations of Ermakov-Pinney type

In this paper we present a general class of differential equations of ErmakovPinney type which may serve as truly nonlinear oscillators. We show the existence of periodic solutions by exact integration after the phase plane analysis. The related quadratic Lienard type equations are examined to show for the first time that the Jacobi elliptic functions may be solution of second-order autonomous non-polynomial differential equations.


Introduction
In the research field of periodic solution to Lienard nonlinear differential equations of the form (1.1)ẍ + f (x) = 0 where the overdot stands for the derivative with respect to time, and f (x) is a nonlinear function of x, it is less usual to notice differential equations with exact periodic solutions. It is again very less usual to find differential equations with exact periodic solutions in terms of trigonometric functions. This makes the Ermakov-Pinney equation unusual and underlines its high usefulness in science and engineering. In this way a lot of applications in classical mechanics as well as in quantum mechanics for example, has been carried out during these decades ( [1], [2]). The Ermakov-Pinney equation (1.2) has been studied in [2] to show the existence of new periodic solutions and non-periodic solutions. In [3] an exceptional Lienard equation with strong and high order nonlinearity is presented. A Lienard equation with a periodic solution in terms of a single trigonometric function, which may lead to a quadratic Lienard type equation with a periodic solution exhibiting harmonic oscillations, and contains several well-known equations like the Ermakov-Pinney equation [4], the Mickens truly nonlinear oscillators and the cubic Duffing equation as special cases, has never been highlighted in the literature despite the well established theory of differential equations, as it is carried out in [3]. According to ( [5], [6]) all Ermakov-Pinney equations may be reduced to using a variable change. One may say that the cubic singularity defines the nonlinear property of the Ermakov-Pinney equation. So a differential equation with cubic nonlinearity may be said of Ermakov-Pinney type. In this perspective consider the Lienard differential equation ( [2], [3]) Making q = 2, yields as equation In view of the above, the equation ( the general solution of (1.5) is written as the quadrature defined by As can be seen, the value of the integral in J could not be known exactly. A new change of variable in terms of trigonometric or hyperbolic functions may be also performed but this does not solve the problem.
However, it shows that the general solutions of some specific equations of the equation (1.5) are not periodic.
The equation (1.5) may be reduced to the form where α = 2n, n is an integer. The equation (1.11) may be of physical importance since it has the structure of truly nonlinear oscillators formulated by Mickens in his book [7], and contains the famous Ermakov-Pinney equation as special case. It is known also that differential equations with power nonlinearities are often encountered in mathematical modeling of physical problems. A vast literature exists on the topic of truly nonlinear oscillators. During the last decades many authors investigated these nonlinear differential equations. As nonlinear differential equations, they have no exact explicit solutions in general. Moreover they could not be solved by the well known standard approximate analytical techniques [7]. So the existence of periodic solutions of these equations is yet under some debate. This particularly, becomes an attractive research problem when the second order autonomous truly nonlinear equation has a singularity at the origin and can have no critical point, necessary condition, according to [8], for a planar autonomous systems to have a periodic solution. It was the case of the so-called pseudo-oscillator investigated in [8]. The authors [8] concluded that such a differential equation has no periodic solution. In contrast to this the author in [9] showed that periodic solution exists, at least a non-smooth solution. The author [9] carried out a theory to build such periodic solutions. In [10] the two general solutions predicted in [8] have been exactly calculated and the authors [10] concluded also to the non existence of smoth periodic solution. The equation (1.11) has a cubic singularity at the origin for a positive integer n, but may have fixed points. For n negative, singularities appear also. Choosing n = 1, that is α = 2, reduces the equation ( For b = 0, the equation (1.12) reduces to the well-known restricted cubic Duffing equation ( [7], [11], [12]) for which it is said that all the solutions are periodic. Contrary to these authors and to several others, it has been shown that such an equation may exhibit non-periodic solution, precisely complex-valued solutions [13].
Putting n = 4, into the equation (

Qualitative properties of solutions
The qualitative properties of solutions to (1.11) are investigated in this section using the phase plane method. Therefore the equation (1.11) is equivalent to the planar autonomous dynamical system The fixed point is defined by y = 0 and . As one may see, for b a = −1, the critical point is real, but for b a = 1, the coordinate x may become complex. From (2.1) one may write The separation of variable leads to By integration, one may obtain the integral curves given by This means, according to the equation (1.6) that the integration constant can be choosen as c = 0, so that the Hamiltonian of the system can be written The Hamiltonian ( where B is the arbitrary constant. The authors [6] observe that the analysis of Lie point symmetries is not adequate for (3.3) which requires, rather than nonlocal symmetry calculation. As the evaluation of nonlocal symmetries may be complicated, the authors [6] apply the Jacobi last multiplier approach to find the solution of (3.1) in terms of time dependent integral. On the other hand, the equation (1.3) is also investigated in [14]. The authors [14] succeed to calculate a general solution of (1. from which, using the previous relation X 2 = b − ax 2 , one may secure the general solutions of (1.3) as where K is an integration constant.

Solution using the auxiliary equation (3.1). Let us consider the generalized Sundman transformation
theory introduced recently in the literature by Akande and coworkers [15]. In fact the generalized Sundman transformation is a powerfull change of variables which allows solving differential equations with a few mathematical manipulations. In the theory introduced by Akande et al. [15] the oscillator harmonic equation to the second order differential equation where A 0 , β, l, γ are arbitrary parameters, g(z) = 0, and ϕ(z) are arbitrary functions of z, and prime denotes differentiation with respect to z. The application of ϕ(z) = ln(f (z)), leads to Putting g(z) = z, and f (z) = z 2 , into (3.13), allows one to obtain Substituting (3.19) into (3.9) yields the general solution to (3.15) as Using (3.20) one may deduce the solution of (3.1) in the form Therefore the solution of (1.3) where b = −1, becomes for ν ≺ 0, then the general solutions (3.23) reduce to (3.24) x(t) = ± 2 1 +

Exact periodic and complex-valued solutions.
3.2.1. Periodic and complex-valued solutions of (1.12). The equation (1.12) is obtained when n = 3, that is when α = 6, from the equation (1.11). Two cases may be investigated.

Periodic solution
For reason of simplicity we choose a = b = 1. In this case the integral J becomes which may be rewritten as [17] (3.26) where 0 ≺ φ ≺ ∞, and c 1 is an arbitrary parameter.
By integration, (3.26) reduces to [17] (3.27) J = 1 3 , and k 2 = 2+ √ 3 4 . Using (3.27), one may write from which one may get In this situation φ may be written as which becomes x α = 1 − X 2 , as a = b = 1, the solution x takes the definitive form This case corresponds to where i is the purely imaginary number. The equation (3.32) gives [17] (3.33) where 0 ≺ φ ≺ ∞, and c 2 is an arbitrary parameter.
The evaluation of the integral in (3.33) leads to The equation (3.34) may be rewritten in the form From (3.36) one may get φ as In the present case, φ = X √ −1 , which is rewritten as φ = −iX, that is X = iφ, so that from which one may secure the complex-valued solution x(t) in the form
Periodic solution: a = b = 1 In this case, the integral J is written in this form Using (1.7) one may get the equation ( [16], [17]) where c 3 is an arbitrary parameter. Therefore, by integration, the equation (3.41) reduces to where 0 ≺ φ 1. From (3.42) one may ensure the following equation which is written in this form From the equation (3.44), one may secure Using the relation φ = X √ b , that is X = φ, the equation (3.45) is rewritten as This condition leads to the equation [17] (3.48) where 0 ≺ x 1. The integral in (3.48) is known as hyperelliptic integral and its evaluation gives [18] (3.49) which may be rearranged in the form Thus, the above shows that under the conditions that a b = 1, and 2 n 4, the explicit general solutions of (1.11) are periodic.
In the sequel of this work, the related quadratic Lienard type equations to the equation (1.13) is examined.

Quadratic Lienard type equations
To determine the quadratic Lienard type equations related to (1.13), consider the change of variable which shows for the first time that the Jacobi elliptic function cn [18] may be solution of second order autonomous non-polynomial differential equations. Now a conclusion of this work may be addressed.

Conclusion
In this paper a general class of truly nonlinear oscillator equations is presented. The conditions of existence of periodic solutions are shown and periodic and explicit general solutions are examined. The general solutions of a well known Ermakov-Pinney type equation are also calculated. Finally it has been shown that the Jacobi elliptic function cn may be solution of second-order autonomous non-polynomial differential equations.

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