Some Novel Approaches for Analyzing the Unforced and Forced Duffing–Van der Pol Oscillators

Department of Mathematics and Statistics, Universidad Nacional de Colombia, FIZMAKO Research Group, Bogota, Colombia Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia Department of Physics, Faculty of Science, Port Said University, Port Said 42521, Egypt Research Center for Physics (RCP), Department of Physics, Faculty of Science and Arts, Al-Mikhwah, Al-Baha University, Al-Bahah, Saudi Arabia Department of Physics, College of Arts and Science in Wadi Al-Dawaser, Prince Sattam Bin Addulaziz University, Wadi-Dawaser 11991, Saudi Arabia Department of Physics, Faculty of Science, Ain Shams University, Cairo, Egypt


Introduction
e study of the dynamics of nonlinear oscillators is one of the topics of great importance for many researchers due to its many important applications in various areas of physics, applied mathematics, and practical engineering [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Di erential equations are one of the best and most successful models in modeling many nonlinear dynamical systems. For instance, Du ng-type equation is one of the most famous and successful equations that has been used for modeling and interpreting many nonlinear oscillations in many different dynamical systems such as electrical circuit, optical stability, the buckled beam, and di erent oscillations in a plasma [16][17][18][19]. In plasma physics, there are many evolution equations that can be reduced to Du ng-type equation, Helmholtz-type equation, Du ng-Helmhlotz equation, and Mathieu equation in order to investigate the various oscillations that occur within complicated plasma systems [20][21][22][23]. ere is another type of equation of motion that was used for modeling the nonlinear oscillations in biology, electronics, engineering, plasma physics, and chemistry which is called Van der Pol-Du ng (VdPD) (sometimes called Du ng-Van der Pol (DVdP)) equation and its family [24,25]. For example, a forced modi ed VdPD (mVdPD) equation was adopted for investigating the strong nonlinear oscillations in plasma [26]. Also, a mVdPD equation with asymmetric potential was used for modeling the nonlinear chemical dynamics [27]. Many numerical and analytical approaches were applied for solving the second-order nonlinear oscillator equations. For example, both HBM and MTSs techniques were devoted for analyzing a forced Van der Pol (VdP) generalized oscillator to obtain the amplitudes of the forced harmonic, superharmonic, and subharmonic oscillatory states [26]. Melnikov's method was used for analyzing a mVdPD equation to derive analytical criteria for the appearance of horseshoe chaos in chemical oscillations [27]. He et al. [16] used the Poincare´-Lindstedt technique (PLT) for solving and analyzing the Hybrid Rayleigh-van der pol-Duffing equation. Also, the homotopy analysis method (HAM) was used for analyzing DVdP oscillator [28]. Both methods of differentiable dynamics and Lie symmetry reduction method were devoted for analyzing the DVdPtype oscillator [29]. Moreover, DVdP oscillator was solved numerically via Adomian's decomposition method (ADM) [30]. Based on this approach, the equation of motion is converted to a system of first-order differential equations and then was solved to obtain a numerical approximation. Moreover, the authors made a comparison with Lindsted's method (LM) approximation. ey found that the obtained approximation using ADM is better than LM. However, in the two approaches, the approximations become convergence and more accurate only in the short time domain but these approximations become dis-convergence and not accurate for long time domain. Most methods in the literature lead to complicated formulas for the obtained approximations and the analysis of such solutions are much difficult or not convergence for a long time. However, the Krylov-Bogoliubov-Mitropolsky method (KBMM) was adopted for deriving the periodic steady-state solutions to the following DVdP driven oscillator [6].
where the overdot indicates the derivative with respect to "t".
Recently, Salas et al. [11] applied the ansatz method, HBM, PLT, and KBMM for analyzing the forced VdP oscillator and found that KBMM gives more accurate approximations. Motivated by the investigations in Ref. [6,11], we proceed to analyze the DVdP oscillator using a new effective and simplification technique based on KBMM. In our approach, we will prove that the new approach does not demand to solve any ordinary differential equations (odes). Moreover, we will prove that the new suggested approach gives highaccurate and convergence approximations in the whole time domain. Note that for ε � 0, i.v.p. (1) reduces to the forced Duffing oscillator whose general solution is well known [11]. Moreover, in this investigation, we try to improve He's frequency-amplitude formulation to be suitable for analyzing the DVdP oscillator. Also, the He's homotopy perturbation method (He's HPM) will apply for analyzing and investigating the DVdP oscillator. e rest of this paper is introduced in the following fashion: in Section 2, the new suggested approach is introduced. e analytical approximations to the unforced DVdP oscillator is reported in Section 3 using the new mentioned approach, the He's HPM, and improved He's frequency-amplitude formulation. Moreover, in Section 4, the new mentioned approach is devoted for getting an analytical approximation to the forced DVdP oscillator. e obtained results are summarized in Section 5.

New Approach Based on KBM for Solving Strongly Nonlinear Oscillators
Let us consider the following general form to the secondorder i.v.p.: where the expression F(x, 0) is an odd polynomial of x.

2
Journal of Mathematics Remark 1. We can obtain another method by replacing (6) with In the below section, we will use this method for analyzing both the unforced DVdP oscillator, i.e., i.v.p. (1) for f � 0 and the forced DVdP oscillator (1).

Analytical Approximations to the Unforced Duffing-Van Der Pol Oscillator
Here, we can analyze the unforced DVdP oscillator, i.e., i.v.p.
Equating the coefficients of p and p 2 to zero in also, equating to zero the coefficients of a 2j+1 , cos((2i + 1)ψ), and sin((2i + 1)ψ), where (i, j � 1, 2, 3, . . . , N), i.e., S i � 0, we get an algebraic system. e solution of this system yields Journal of Mathematics From the above values in equation (11), we have Accordingly, the odes for determining the functions (a, ψ) read For p � 1, the value of _ a given in equation (18) reduces to e amplitude for the limit cycle is obtained from the condition _ a| p�1 � 0: Solving equation (20) gives Observe that this is called the cycle amplitude for the VdP oscillator. We can use the following Chebyshev approximation in order to facilitate the solution to the ode system (18): with κ � 1 16 By solving equation (23), we get Also, the expression for determining ψ can be obtained from the second equation in (18) for p � 1 whose solution reads where the values of W i (i � 1, 2, 3, 4) are defined in Appendix (ii). e constants c 0 and c 1 are determined from the initial conditions (ICs) x(0) � x 0 and x ′ (0) � _ x 0 . In all above expressions, for p � 1, the approximation to the following i.v.p. can be obtained:

He's Homotopy Perturbation
Method. Moreover, the approximate solution to the DVdP i.v.p. (1) using He's HPM is obtained. Briefly, He's HPM can be used for a series of nonlinear oscillators differential equations which many classical perturbation methods failed to solve them or to give some accurate solutions. is method suggests the solution in the following ansatz: with Substitute equations (28) and (29) into i.v.p. (1), and by collecting the coefficients of same powers of p, we finally obtain some reduced equations. We have where τ � tω.
Equating to zero the coefficients of p j and solving the resulting odes gives where A represents the amplitude of the oscillator. Secularity terms in the last expression are not allowed so that the coefficients of cos(t) and sin(t) must be equal to zero which lead to
He considered Duffing oscillator where A represents the amplitude of the oscillator. Based on He's principle, we have where ω denotes the frequency of oscillator. Now, by considering the following DVdP oscillator: in this case, the function f(x) reads which leads to Since the amplitude now depends on time, we will reason heuristically to determine it . (43) Note that κ ⟶ 1 as β ⟶ 0. en, the improved He's solution becomes with , where e constants c 0 and c 1 are determined from the ICs x(0) � x 0 and x ′ (0) � _ x 0 . e amplitude for the limit cycle reads As a numerical example, we can use the same model and data that were given in Ref. [36], which lead to the following unforced DVdP i.v.p. (27): Solution (10) and RK numerical approximation to i.v.p. (48) are graphically mapped as shown in Figure 1. Moreover, the approximation (34) using He's HMP and the approximation (44) using the improved He's FAF are compared with the obtained analytical approximation (10) and RK numerical approximation as illustrated in Figure 1. In addition, the maximum distance error in the whole time domain (0 ≤ t ≤ 50) with respect to RK numerical approximation is estimated 6 Journal of Mathematics It is clear that the analytical approximation (10) and RK numerical approximation are very compatible with each other. Also, they are more accurate than He's FAF and He's HPM approximations.

Analytical Approximation to the Forced Duffing-Van Der Pol Oscillator
Let us consider the following forced DVdP i.v.p.: Assume that the solution to i.v.p. (50) is given by the following ansatz: where y ≡ y(t) is a solution to the unforced DVdP oscillator Putting solution (51) into (50), we have where h.o.t. represents higher-order terms. By neglecting y(t) and y ′ (t) from system (54) at (R, S) (0, 0), then the constants d 0 and d 1 can be determined from the system From this system, we get where the values of Y i (i 0, 1, 2, 3) and Z i (i 0, 1, 2, 3) are de ned in Appendix (iii). We choose the least in magnitude real roots to cubics (56) and (57). As a numerical example, the two approximations (51) and (34) according to ICs (35) for i.v.p. (50) are displayed in Figure 2 for (ε, ω 0 , β, F, Ω) (0.1, 1, 0.01, 1, 5.2). Also, the maximum distance error for the two approximations is estimated as follows: On the other side, for arbitrary ICs, the analytical approximation (51) versus the RK numerical approximation is presented in Figure 3 for (ω 0 , β, F, Ω, x 0 , _ x 0 ) (1, 0.01, 1, 5.2, 0, 0.183) and di erent values to ε. Also, the maximum distance error at the same values of the physical  Figure 3 is estimated as follows: We can conclude that in all cases, both two analytical approximations (10) and (51) for the unforced and forced DVdP oscillators are more accurate and convergence as compared to He's FAF and He's HPM.

Conclusion
Both higher-order nonlinearity unforced Du ng-Van der Pol (DVdP) oscillator and forced DVdP oscillator having linear and cubic nonlinear terms have been analyzed using some e ectiveness and more accurate approaches. e new approach was constructed based on the Krylov-Bogoliubov-Metroolsky method (KBMM). e new approach was discussed in detail for the two issues. In our analysis, we only stopped at the rst approximation because it is su cient in all cases. Also, this new approach can be used for analyzing many strongly nonlinear oscillators. Moreover, the new approach does not demand to solve any ordinary di erential equations (odes) because we only simply equate the coe cients of the trigonometric functions a(t) and p to zero in order to get a simple system of algebraic equations. en, this system becomes very easy to solve it to determine the undetermined coecients. Also, this new method is also characterized by being direct and fast. Moreover, it is characterized by high-accuracy if it is compared with other methods in the literature. Also, we improved He's frequency-amplitude formulation technique in order to solve unforced DVdP oscillator to obtain high-accurate results. Furthermore, the unforced DVdP oscillator was analyzed via He's homotopy perturbation method. e maximum distance error in the whole time domain with respect to Runge-Kutta numerical approach has been estimated. It was found that our new approach is better than all method approaches. Moreover, the new approach can be devoted for analyzing many strong nonlinearity oscillators with any odd power. Also, the new approach can be applied for arbitrary initial conditions. Future work: the ansatz that has been used in this paper is called the KBM rst-Variant. is approach cannot recover He's amplitude formula. On the other side, we will use another new ansatz which maybe called the KBM second-Variant, in this case, He's amplitude formula can be recovered.

Data Availability
All data generated or analyzed during this study are included in this published article (more details can be requested from El-Tantawy).

Conflicts of Interest
e authors declare that they have no conflicts of interest.