Modiﬁed Emden Type Oscillator Equations with Exact Harmonic Solutions

. This paper is devoted to investigating the existence of exact harmonic solutions and limit cycles of certain modiﬁed Emden-type equations. The exact and general solutions obtained are in opposition to the predictions of classic existence theorems.


Introduction
The conservative Lienard equationẍ + u(x) = 0, (1.1) where the restoring term u(x) is a function of x and is presumed to have periodic solutions. When a frictional term is included in Equation (1.1), such as x + αxẋ + u(x) = 0, (1.2) where α is an arbitrary parameter, and the periodic behavior of the Lienard equation (1.2) is no longer assured. In this way, the modified Emden equation [1][2][3] is as follows: where λ is a constant and u(x) = λx 3 has been the subject of an intensive mathematical investigation in the literature using different approaches. While [ [4], p.393] asked to show that the form x + xẋ + x 3 = 0, (1.4) has a center at the origin, as an exercise, the exact and general solutions calculated in [2,5] are nonperiodic formulas. The existence of a center at the origin of Equation (1.4) is in fact established based on existence theorems [4,6,7], such as Theorem 11.3 in [ [4], p.390], using the phase plane.
According to these theorems, an equation of type (1.2) has a center at the origin when the function u(x) is continuous and odd, and u(x) 0 for x 0, which involves u(0) = 0, to ensure the existence of a single equilibrium point at the origin. Another interesting equation of type (1.2) can read as follows:ẍ where β is an arbitrary constant. This type of equation is known as a reversible system with a nonlinear center at the origin of the (x, y =ẋ) phase plane (see Theorem 6.6.1 of [ [8], p.164]).
where γ, a, c, and q are arbitrary constants. Equation (1.7) includes the previous equations as special cases. When q = 1, a = β = 0, Equation (1.7) reduces to the modified Emden Equation (1.3). When q = 1 and a = 0, Equation (1.7) becomes which has been widely investigated in the literature [3,9]. In [3], the authors claimed to present an and Equation (1.7) in an explicit way . The results are compared to numerical solutions obtained by using the fourth-order Runge-Kutta (RK4) method (Section 3). We additionally highlight the existence of modified Emden-type nonpolynomial differential equations with exact algebraic limit cycles (Section 4). Finally, we present a conclusion for the work.

Methods and solutions
2.1. Method and solution of Equation (1.9). To solve Equation (1.9), consider the general theory [11][12][13] of the following fundamental equation: where a, b and are arbitrary parameters, and f (x) and g(x) are functions of x and prime means derivative with respect to x, with the corresponding first integral Substituting = 0 and g(x) = exp(γx 2 ) leads tö where γ is an arbitrary constant. From f (x) = g(x) = exp(γx 2 ) and b=0 one can obtain In this situation, the general solution of Equation (2.4) is immediately obtained as follows: where k is an integration constant. Hence, the solution of Equation (1.9) takes following the form: where a = −1, and γ = − 1 2 . In this way, we have proven the following result. Therefore, the general solution of Equation (1.9) is not oscillatory, that is, nonperiodic, contrary to the assertion of Jordan and Smith in their books [4,10] on the basis of existence theorems.

Qualitative analysis.
In this part, the purpose is to study Equation (1.7) considering the classic theorems for the existence of a center at the origin [4,[6][7][8]. Indeed, Equation (1.7) is equivalent to the following system: where γ is an arbitrary constant. The equilibrium points are defined by Hence, there is an equilibrium point at the origin (0, 0) in the (x, y ) phase plane, and the other is given by where γc = 0. For simplicity, let c = 1. Then, Solution (2.10) becomes Whether the value of Solutions (2.11) is real or complex depends on the values of the parameters q, a, β and γ. When q = 1 2 , for example, Equation (2.11) takes the form This clearly shows that there are values of β, a, and γ for which Solution (2.12) is real. In other words, if a 2 β 2 γ 2 , then Solution (2.12) is real. In this case, there is not a single equilibrium point but several fixed points, so according to the above, Equation (1.7) of interest does not satisfy the theorem for the existence of an isochronous center at the origin. In contrast, if a 2 ≺ β 2 γ 2 , then Solution (2.12) is complex-valued, so the origin is a single equilibrium point, and Equation

Exact harmonic and isochronous solution.
This section is devoted to the integrability of Equation (1.7) in terms of exact harmonic and isochronous periodic solutions. In this context, consider the general solution of Equation (1.7) in the form where A, ω, and φ are parameters to be determined. From Equation (2.13) which easily holds when q = 1 2 , a = ±A and γ = αω. With these parameters, the desired Equation (1.7) takes the formẍ where ε = ±1, which is completely integrable with the exact harmonic and isochronous solution to Equation (2.13) when ε = +1 and with the following solution: when ε = −1. The constant φ and φ 1 can be obtained using the initial conditions. Solutions (2.13) and (2.19) show that Equation (2.18) has an isochronous center for any arbitrary value of α = 0, or γ = 0, in contrast to the predictions of the qualitative theory of differential equations. Then, the following theorem is proved.
result in the following: In this situation, the general solution (2.13) becomes Hence, Figure 3 shows the graphical comparison of the numerical solution in the solid line of Equation

3.2.
Case of solution (2.19). Using the following initial conditions:   Figure 3 and Figure 4, the numerical solutions match the analytical results. Now, using Equation (2.18), we formulate the nonpolynomial differential equations of the modified Emden-type with exact algebraic limit cycles.

Nonpolynomial differential equations of the modified Emden-type
Although the existence of limit cycles of differential equations has been widely investigated, there are a few works devoted to exhibiting their exact and explicit expressions in the literature. In addition, the existence of limit cycles of nonpolynomial differential systems or equations [14,15] is investigated much less often, contrary to the vast literature that can be found on the study of polynomial differential systems in connection with the second part of the Hilbert 16th problem [16]. In this context, consider the equationẍ (4.1) When α = 0, Equation (4.1) reduces to the well-known hybrid Rayleigh-Van der Pol equation, which is as follows:ẍ with following the exact harmonic solution: which corresponds to the algebraic limit cycle of degree 2 given by where y =ẋ. Using Equation (   If ε = −1, one can easily verify that Equation (4.1) has the following exact harmonic solution: x(t) = si nt. It is worth noting that Equation (4.1) does not satisfy the Lienard-Levinson-Smith theorem for the existence of at least one limit cycle, but can exhibit an algebraic limit cycle of degree 2 given by x 2 + y 2 − 1 = 0 for ε = 1 and α 0. Indeed, Equation (4.1) has the form of the generalized Lienard equation, as follows:ẍ and where α 0 and ε = 1. This theorem [4,[17][18][19] primarily requires that (i) g(0) = 0, xg(x) 0 for |x| 0, (iv) there exists x 0 0 such that h(x,ẋ) 0 for |x| x 0 . Thus, let The results is that    As shown by these figures, the limit cycle shape is controlled by the parameter α. When ε = −1, the phase paths are not closed trajectories, as shown in Figures 8 and 9 for α = 0.1, and α = 10, respectively, while Equation (4.1) admits the exact harmonic solution sin(t). However, for small enough α, such as 0 ≺ α ≺ 0.1, the phase portraits show the existence of limit cycles. In this way, Figure 10 exhibits the phase portrait and vector field of Equation (4.1) for ε = −1, and α = 0.001, showing the existence of an algebraic limit cycle of degree 2 given by Equation (4.4). Now, the objective is to investigate a more general nonpolynomial differential equation. Therefore, consider the following theorem.
The phase portraits and vector field of Equation (4.13) are exhibited in Figures 11, 12 and 13 for α = 0.1 and n = 0, 1 and 2, respectively, which show the existence of an algebraic limit cycle of degree 2. The above allows us to present a conclusion for the work.

Conclusion
This work has been devoted to investigating some modified Emden-type equations. We successfully calculated their exact and general solutions and showed the existence of algebraic limit cycles of degree 2. Numerical simulations were performed to illustrate the validity of the solution methods.
Authors' Contributions: All authors have equal contributions in this paper. All authors read and approved the final manuscript.
Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publication of this paper.