Closed-Form Exact Solutions for the Unforced Quintic Nonlinear Oscillator

Closed-form exact solutions for the periodic motion of the one-dimensional, undamped, quintic oscillator are derived from the first integral of the nonlinear differential equation which governs the behaviour of this oscillator. Two parameters characterize this oscillator: one is the coefficient of the linear term and the other is the coefficient of the quintic term. Not only the common case in which both coefficients are positive but also all possible combinations of positive and negative values of these coefficients which provide periodic motions are considered.The set of possible combinations of signs of these coefficients provides four different cases but only three different pairs of period-solution.The periods are given in terms of the complete elliptic integral of the first kind and the solutions involve Jacobi elliptic function. Some particular cases obtained varying the parameters that characterize this oscillator are presented and discussed. The behaviour of the periods as a function of the initial amplitude is analysed and the exact solutions for several values of the parameters involved are plotted. An interesting feature is that oscillatory motions around the equilibrium point that is not at x = 0 are also considered.


Introduction
Mathematical models based on nonlinear oscillators have been widely used in physics, engineering, applied mathematics, and related fields [1,2].These nonlinear systems have been the focus of attention for many years and several methods have been used to find approximate solutions to them [1,3].In conservative nonlinear oscillators the restoring force is not dependent on time, the total energy is constant [1], and any oscillation is stationary.The aim of this paper is to obtain closed-form exact periodic solutions to the quintic equation corresponding to a nonlinear oscillator described by a differential equation with fifthpower nonlinearity.Due to the presence of the fifth-power nonlinearity, this oscillator is difficult to handle and has not been studied as extensively as the Duffing oscillator with cubic nonlinearity.For this reason, several techniques have been used to obtain analytical approximate expressions for the period and the solution of the quintic oscillator.Lin [4] proposed a new parameter iteration technique to solve the Duffing equation with strong and higher-order nonlinearity.Ramos [5] approximately solved the quintic equation using some Lindstedt-Poincaré techniques.Pirbodaghi et al. [6] obtained an accurate analytical approximate solution to Duffing equations with cubic and quintic nonlinearities using the homotopy analysis method and homotopy Padé technique.Wu et al. [7] approximately solved this nonlinear oscillator using a method that incorporates salient features of both Newton's method and the harmonic balance method.Later, Lai et al. [8] used a Newton-harmonic balancing approach to obtain accurate approximate analytical higherorder solutions for strong nonlinear Duffing oscillators with cubic-quintic restoring force.They also discussed the effect of strong quantic nonlinearity on accuracy as compared to cubic 2 Advances in Mathematical Physics nonlinearity.Beléndez et al. [9] approximately solved the quintic oscillator using a cubication method which allowed them to obtain approximate analytical expressions for the period and the solution in terms of elementary functions.Scarpello and Ritelli [10] exactly solved the quintic oscillator, but only when the coefficient for the linear term is equal to one and the coefficient for the nonlinear term is positive.Elías-Zúñiga [11] derived the exact solution of the cubicquintic Duffing equation based on the use of Jacobi elliptic functions by the same method that he used to obtain the exact solution of the mixed-parity Helmholtz-Duffing oscillator [12].However, in both cases he did not solve the nonlinear differential equation but assumed that its exact solution is given by a rational equation which includes the cn Jacobian elliptic function and five unknown parameters that need to be determined.Based on his pervious results, Elías-Zúñiga obtained the analytical approximate solution of the damped cubic-quintic Duffing oscillator [13] and also developed a "quintication" method [14] to obtain approximate analytical solutions of conservative nonlinear oscillators.Recently, Beléndez et al. [15] have exactly solved the unforced cubicquintic Duffing oscillator, providing exact expressions for the period and the solution, but only for oscillations around  = 0 and taken into account that the coefficients for the linear and the nonlinear terns are positive.
In this paper we obtain the closed-form exact expressions for the period and the solution of the quintic Duffing nonlinear oscillator modelled by the second-order differential equation  2 / 2 +  1  +  5  5 = 0, where  1 and  5 are the coefficients of the linear and the nonlinear terms, respectively, considering all possible combinations of signs of  1 and  5 that provide oscillatory motions.Unlike the procedure considered by Elías-Zúñiga [11,12], we do not assume an expression for the solution but solve the nonlinear differential equation exactly as was done in [15].This is done using elliptic functions so that, after inversion, the solution  is provided as an explicit function of time t.When  1 = 0 and  5 > 0, the system becomes a truly nonlinear oscillator [16] for which the exact expressions for the period and the solution have been already obtained [17].The particular situation in which coefficients  1 and  5 are both positive is the most common case analysed.However, we obtain closedform exact solutions not only for the case in which both coefficients are positive, but also for all possible combinations of positive and negative values of these coefficients which provide periodic motions.The set of possible combinations of signs of these coefficients gives rise to four different cases.In three of these combinations ((a)  1 ≥ 0,  5 > 0, and  0 > 0, (b)  1 < 0,  5 > 0 and  0 > (−3 1 / 5 ) 1/4 , and (d)  1 > 0,  5 < 0 and 0 <  0 < (− 1 / 5 ) 1/4 ) the system oscillates around the equilibrium position  = 0 with  ∈ [− 0 ,  0 ], where  0 > 0 is the oscillation amplitude.However, there is still one more case ((c)  1 > 0,  5 < 0, and 0 <  0 < (−3 1 / 5 ) 1/4 or −(−3 1 / 5 ) 1/4 <  0 < 0) in which the system does not oscillate around the position x = 0 with  ∈ [− 0 ,  0 ], but around the equilibrium position  = (− 1 / 5 ) 1/4 with  ∈ [ 1 ,  0 ].Three different sets of closed-form expressions for the exact period and solution were obtained.Following the procedure considered by Lai and Chow [18], some examples are analysed and plots including periods, solutions, or phasediagrams are presented and discussed.

Formulation and Solution Procedure
A quintic oscillator is an example of a conservative autonomous oscillatory system, which is modelled by the following second-order differential equation: with initial conditions In (1)  and  are generalized dimensionless displacement and time variables, and we assume that the coefficients for the linear and the nonlinear terms satisfy.In order to obtain the exact period and periodic solution for (1), we take into account that this is a conservative system and has the following first integral: which can be written as follows: The dynamical study [19] of the nonlinear differential equation given in (1) showed that its motion is periodic in the following situations: (a)  1 ≥ 0,  5 > 0, and  0 > 0: the system oscillates around the equilibrium position  = 0 and the periodic solution  satisfies  ∈ [− 0 ,  0 ].

Exact Solution
When  1 ≥ 0,  5 > 0, and  0 > 0 In this situation all the solutions are periodical and the phase space diagram is made up of an infinite number of closed orbits, each of them for each value of the initial amplitude  0 (as can be seen in Figure 1(a)).This system oscillates around the equilibrium position  = 0 with  ∈ [− 0 ,  0 ], where  0 > 0 is the initial amplitude and the period, T, and periodic solution, , are dependent on the oscillation amplitude  0 .This system corresponds to a nonlinear oscillator for which the nonlinear function () =  1  +  3  3 +  5  5 is odd; that is, (−) = −().From (4) we obtain Advances in Mathematical Physics and then where the sign (±) is chosen taking into account the sign of / in each quadrant.
Taking into account that  5 > 0, from ( 6) we can write and after some mathematical operations we obtain Integrating ( 8) we obtain The change of variable  2 =  gives This is an improper integral which contains a square root of a four-degree polynomial in the denominator and so its solution can be expressed as a function of an elliptic integral.

Calculation of the Exact Period.
The symmetry of the problem indicates that the period of the oscillation T is four times the time taken by the oscillator to go from  = 0 to  =  2 0 .Therefore, from (10) it follows that We consider the definite integral [20, section 3.145, formula 2, pages 270-271] where  <  < , (, ) is the incomplete elliptic integral of the first kind [20] and , , and  are defined as By comparing the integrals in (11) and ( 12) we obtain  =  2 0 ,  =  2 0 , and = 0, as well as the following values for the different parameters which appear in ( 12) As  =  =  2 0 , then (2 arccot 0, ) = (, ) = 2(), where () is the complete elliptic integral of the first kind defined as [20] From (11) to (16) we conclude that the exact period of the quintic nonlinear oscillator can be written in the compact form where   takes the form and coefficients   -which depend on  1 ,  5 , and  0 -are defined as follows:
In order to introduce an "arccos" function in (24) we take into account that where From (28) we obtain

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Taking into account the relation we can finally write which allows us to write (24) as follows: and from ( 25), (26), and (32) we can write, after some simplifications, where the relation cn(2() − , ) = −cn(, ) [22] has been taken into account.
Finally, for trajectory 4 → 1 we have and we obtain the same value for the solution as that given in (34).
The exact solution of the quintic oscillator can be written as follows: Taking into account the relation [22, formula 16.18.4,page 574] and ( 34) and (38), the exact periodic solution of the quintic oscillator can be written in compact form as follows: Advances in Mathematical Physics which is valid for all value of  and where cn, sn, and dn are the basic Jacobi elliptic functions [20].Figure 3 shows the plot of the displacement  as a function of time  when  1 = 1,  5 = 3, and  0 = 1.This displacement was obtained using (40).This figure corresponds to the phase space portrait shown in Figure 1(a).

Calculation of the Exact Period.
We shall now obtain the period and the solution for the right orbit in Figure 1(c) for which  > 0 taking into account that  0 is the highest value for .From ( 7) and considering that  > 0, we obtain  as a function of  as follows: which can be written as where  1 and  2 are defined by the following equations: where  2 and  4 are defined in (19).
For the right orbit in Figure 1(c) it is easy to verify that it satisfies Now the period, T, and the periodic solution, , are dependent on  0 and  ∈ [ 1 ,  0 ], where  1 = √ 1 and  1 is given in (43).The following integral is valid for  ≥  ≥  >  >  [20, 6 th edition, Formula 3.147, Integral 7, page 272]: where (, ) is the incomplete elliptic integral of the first kind defined in (13) and  and  are given by the following equations: We have  =  2 0 ,  =  1 ,  = 0, and  =  2 .Then the value of the integral in (42) is ) . ( As can be seen in Figure 1(c), the period of the oscillation is twice the time taken by the oscillator to go from  =  2 0 to  =  1 .Therefore, where   is given by 0 ,  2 , and  4 are defined in (19) and () is the complete elliptic integral of the first kind defined in (16).Figure 6 shows the period T as a function of the initial amplitude  0 (49) when Figure 7 shows the variation in the period as a function of the initial position when  1 = −1 and  5 = 3, (a)  0 > (−3 1 / 5 ) 1/4 = 1, Section 4 (17), and (b) 0 <  0 < (−3 1 / 5 ) 1/4 = 1, Section 3 (49).As can be seen the motion is not periodic when the initial position is  = (−3 1 / 5 ) 1/4 .
The expressions for the period and the exact solution for the left orbit in Figure 1(c), that is, for  < 0, can be obtained following the same procedure as that used for  > 0 and are not included here.
As can be seen the system oscillates around the equilibrium point  = 0.

Exact Solution
When  1 > 0,  5 < 0, and 0 <  0 < (− 1 / 5 ) 1/4 For case (d) it is necessary to obtain the equation for the period again since if  5 < 0, the root √ 5 /3 is not a real number and so ( 7) cannot be used.In order to obtain the exact solution when  1 > 0,  5 < 0, and 0 <  0 < (− 1 / 5 ) 1/4 in which case the system oscillates around the equilibrium position  = 0 and the periodic solution  satisfies  ∈ [− 0 ,  0 ], we proceed as follows.From ( 6) we can write 6.1.Calculation of the Exact Period.We can consider the same four trajectories that we analysed at the beginning of Section 3.2.Doing the change of variable  2 = , it follows that for trajectory 1 → 2 (0 ≤  ≤ /4 and  0 ≥  ≥ 0) we have which can be written as where  1 and  2 are defined by the following equations: It is easy to verify that it satisfies for  >  ≥  >  >  [20, 6 th edition, Formula 3.147, Integral 5, page 272]: where (, ) is the incomplete elliptic integral of the first kind defined in (13) and  and  are given by the following equations: We have  =  1 ,  =  2 0 ,  = 0, and  =  2 .Then the value of the integral in (59) is ) . ( As can be seen in Figure 1(d), the period of the oscillation is four times the time taken by the oscillator to go from  = 0 and  =  2 0 .Therefore, where   is given by 0 ,  2 , and  4 are defined in (19) and () is the complete elliptic integral of the first kind defined in (16).

Discussion
In this section we briefly discuss the derived solutions presented in this paper compared to the exact one derived by Elías-Zúñiga [11], providing a discussion in which both solutions are compared for all the cases discussed in the manuscript.As it was pointed out in the introduction, Elías-Zúñiga does not solve the nonlinear differential equation exactly as we have done here, but he assumes an expression for the solution.He proposed a solution for (1) with the initial conditions given in (2) which can be written as follows: 2 () = 1  + cn 2 (, ) .
Elías-Zúñiga also concluded that the corresponding exact period of oscillation  of this nonlinear oscillator is given by [11, Eq. ( 15), page 2576] and pointed out that, depending on the system parameter values of  1 ,  5 , and  0 , we can have real, complex, or imaginary values for , , , and .Now we are going to compare the exact period and solutions presented in this paper with Elías-Zúñiga's paper for all the cases discussed in the manuscript.

Exact Solution
When  1 ≥ 0,  5 > 0, and  0 > 0. We have obtained the exact period given in (17) and we have written the exact periodic solution of the quintic oscillator in the compact form given in (40).For this case, Elías-Zúñiga exact period is given in (72) and the exact solution has to be written as a piecewise function as follows:

Figure 8 (
b) shows the displacement  as a function of time  when  1 = −1,  5 = 3, and