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

Salas, Alvaro H.; Albalawi, Wedad; El-Tantawy, S. A.; El-Sherif, L. S.

doi:https://doi.org/10.1155/2022/2174192

Journal of Mathematics

On this page

Abstract Introduction Conclusion Appendix Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Special Issue

Analytical Methods to Model Nature

View this Special Issue

Research Article | Open Access

Volume 2022 | Article ID 2174192 | https://doi.org/10.1155/2022/2174192

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

Alvaro H. Salas,¹Wedad Albalawi,²S. A. El-Tantawy,^3,4and L. S. El-Sherif^5,6

Academic Editor: Melike Kaplan

Received15 Jan 2022

Revised27 May 2022

Accepted21 Jun 2022

Published31 Jul 2022

Abstract

In the current investigation, both unforced and forced Duffing–Van der Pol oscillator (DVdPV) oscillators with a strong nonlinearity and external periodic excitations are analyzed and investigated analytically and numerically using some new and improved approaches. The new approach is constructed based on Krylov–Bogoliubov–Metroolsky method (KBMM). One of the most important features of this approach is that we do not need to solve a system of differential equations, but only solve a system of algebraic equations. Moreover, the ease and faster of applying this method gives high-accurate results and this approach is better than many approaches in the literature. This approach is applied for analyzing (un)forced DVdP oscillators. Also, some improvements are made to He’s frequency-amplitude formulation in order to solve unforced DVdP oscillator to obtain high-accurate results. Furthermore, the He’s homotopy perturbation method (He’s HPM) is employed for analyzing unforced DVdP oscillator. The comparison between all mentioned approaches is carried out. The application of our approach is not limited to (un)forced DVdPV oscillators only but can be applied to analyze many higher-order nonlinearity oscillators for any odd power and it gives more accurate results than other approaches. Both used methods and obtained approximations will help many researchers in general and plasma physicists in particular in the interpretation of their results.

1. Introduction

The 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 [1–17]. Differential equations are one of the best and most successful models in modeling many nonlinear dynamical systems. For instance, Duffing-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 different oscillations in a plasma [16–19]. In plasma physics, there are many evolution equations that can be reduced to Duffing-type equation, Helmholtz-type equation, Duffing–Helmhlotz equation, and Mathieu equation in order to investigate the various oscillations that occur within complicated plasma systems [20–23]. There 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–Duffing (VdPD) (sometimes called Duffing–Van der Pol (DVdP)) equation and its family [24, 25]. For example, a forced modified 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 DVdP-type 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. They 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 high-accurate and convergence approximations in the whole time domain. Note that for , 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.

The rest of this paper is introduced in the following fashion: in Section 2, the new suggested approach is introduced. The 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. The obtained results are summarized in Section 5.

2. New Approach Based on KBM for Solving Strongly Nonlinear Oscillators

Let us consider the following general form to the second-order i.v.p.:where the expression is an odd polynomial of .

To introduce a -parameter solution to i.v.p. (2), we rewrite this problem in the following new form:where indicates the solution to i.v.p. (3), we can call a -parameter solution. Then, the solution to the original i.v.p. (2) can be obtained for .

Assuming the solution to i.v.p. (3) is given by the following ansatz form:where the functions and are assumed to vary with time according to the following ordinary differential equations (odes):

We plug the ansatz (4) with the relations (5) and (6) into given in equation (3) and equating the coefficients of () , cos , and sin (, ) to zero, then we can get a system of algebraic equations. By solving this system, we can determine the unknown coefficients , , , and .

Remark 1. We can obtain another method by replacing (6) withIn the below section, we will use this method for analyzing both the unforced DVdP oscillator, i.e., i.v.p. (1) for and the forced DVdP oscillator (1).

3. Analytical Approximations to the Unforced Duffing–Van Der Pol Oscillator

Here, we can analyze the unforced DVdP oscillator, i.e., i.v.p. (1) for using three different approaches, namely:(i)Our new approach mentioned in the above section(ii)The He’s HPM(iii)Improved He’s frequency-amplitude formulation (He’s FAF)

3.1. Our New Approach

By using in solution (4), then the solution to the following unforced problem (3)can be introduced in following form:

Accordingly, we getwith

On the other handwithwhere the values of coefficients are defined in Appendix (i).

Equating the coefficients of and to zero inalso, equating to zero the coefficients of , , and where (, ), i.e., , we get an algebraic system. The solution of this system yields

From the above values in equation (11), we have

Accordingly, the odes for determining the functions read

For , the value of given in equation (18) reduces to

The amplitude for the limit cycle is obtained from the condition :

Solving equation (20) gives

Observe thatthis 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

By solving equation (23), we get

Also, the expression for determining can be obtained from the second equation in (18) for whose solution readswhere the values of are defined in Appendix (ii). The constants and are determined from the initial conditions (ICs) and .

In all above expressions, for , the approximation to the following i.v.p. can be obtained:

3.2. 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. This 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 , we finally obtain some reduced equations. We havewhere .

Equating to zero the coefficients of and solving the resulting odes giveswhere represents the amplitude of the oscillator.

Secularity terms in the last expression are not allowed so that the coefficients of and must be equal to zero which lead to

Solving the last system gives

Then, the following solution to and for is obtained:where and . This solution is obtained under the initial conditions (ICs)

Solution (34) recovers the unforced case for .

3.3. Improved He’s Frequency-Amplitude Formulation

To demonstrate the general idea of He’s frequency-amplitude formulation [31–35], let us consider the following oscillator:where indicates the nonlinear restoring force. The following conditions are hold: and .

He considered Duffing oscillatorwhere represents the amplitude of the oscillator. Based on He’s principle, we havewhere denotes the frequency of oscillator.

Now, by considering the following DVdP oscillator:in this case, the function readswhich leads to

Since the amplitude now depends on time, we will reason heuristically to determine itwithNote that as .

Then, the improved He’s solution becomeswithwhere

The constants and are determined from the ICs and . The 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 with respect to RK numerical approximation is estimated

(a)

(b)

(c)

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.

4. 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 is a solution to the unforced DVdP oscillator

Putting solution (51) into (50), we havewithwhere h.o.t. represents higher-order terms. By neglecting and from system (54) at , then the constants and can be determined from the system

From this system, we getwhere the values of and are defined 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 . Also, the maximum distance error for the two approximations is estimated as follows:

(a)

(b)

On the other side, for arbitrary ICs, the analytical approximation (51) versus the RK numerical approximation is presented in Figure 3 for and different values to . Also, the maximum distance error at the same values of the physical parameters and ICs mentioned in Figure 3 is estimated as follows:

(a)

(b)

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.

5. Conclusion

Both higher-order nonlinearity unforced Duffing–Van der Pol (DVdP) oscillator and forced DVdP oscillator having linear and cubic nonlinear terms have been analyzed using some effectiveness and more accurate approaches. The new approach was constructed based on the Krylov–Bogoliubov–Metroolsky method (KBMM). The new approach was discussed in detail for the two issues. In our analysis, we only stopped at the first approximation because it is sufficient 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 differential equations (odes) because we only simply equate the coefficients of the trigonometric functions and to zero in order to get a simple system of algebraic equations. Then, this system becomes very easy to solve it to determine the undetermined coefficients. 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. The 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 first-Variant. This 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.

Appendix

(i)The coefficients of solution (13)(ii)The values of for equation (26)(iii)The values of and for equations (56) and (57)

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

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this study and approved the final manuscript.

Acknowledgments

The authors expressed their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R157), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

References

J. Soukup and J. Volek, Vibration of Mechanic Systems-Vehicles, UJEP, Usti nad Labem, Czechia, 2008.
J. R. Ellis, Road Vehicle Dynamics, SAE International, Warrendale, PA, USA, 1989.
A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, John Wiley, New York, NY, USA, 1973.
W. Albalawi, A. H. Salas, S. A. El-Tantawy, and A. A. A. R. Youssef, “Approximate analytical and numerical solutions to the damped pendulum oscillator: Newton–Raphson and moving boundary methods,” Journal of Taibah University for Science, vol. 15, no. 1, pp. 479–485, 2021.
View at: Publisher Site | Google Scholar
G. M. Moatimid, “Stability analysis of a parametric duffing oscillator,” Journal of Engineering Mechanics, vol. 146, no. 5, Article ID 05020001, 2020.
View at: Publisher Site | Google Scholar
J. Kyzioł and A. Okniński, “The duffing–van der Pol equation: metamorphoses of resonance curves,” Nonlinear Dynamics and Systems Theory, vol. 15, no. 1, pp. 25–31, 2015.
View at: Google Scholar
A. Pikovsky, M. Rosenblum, and J. S. Kurths, A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, UK, 2001.
A. H. Salas and S. A. El-Tantawy, “On the approximate solutions to a damped harmonic oscillator with higher-order nonlinearities and its application to plasma physics: semi-analytical solution and moving boundary method,” European Physical Journal A: Hadrons and Nuclei, vol. 135, no. 10, p. 833, 2020.
View at: Publisher Site | Google Scholar
A. H. Salas and S. A. El-Tantawy, Analytical Solutions of Some Strong Nonlinear Oscillators, IntechOpen, London, UK, 2021.
Y. O. El-Dib, “Multiple scales homotopy perturbation method for nonlinear oscillators,” Nonlinear Sci Lett A, vol. 8, pp. 352–364, 2017.
View at: Google Scholar
A. Salas, J. M. H. Lorenzo, and L. O. R. David, “Analytical and Numerical Study to a Forced Van der,” Pol Oscillator” Mathematical Problems in Engineering, vol. 2022, Article ID 9736427, 9 pages, 2022.
View at: Publisher Site | Google Scholar
Ji-H. He, N. Anjum, and P. Skrzypacz, “A variational principle for a nonlinear oscillator arising in the microelectromechanical system,” Journal of Applied and Computational Mechanics, vol. 7, no. 1, pp. 78–83, 2021.
View at: Google Scholar
Ji-H. He, Q. Yang, C.-H. He, and Y. Khan, “A simple frequency formulation for the tangent oscillator,” Axioms, vol. 10, no. 4, p. 320, 2021.
View at: Publisher Site | Google Scholar
Ji-H. He and Y. O. El-Dib, “Homotopy perturbation method with three expansions for Helmholtz-Fangzhu oscillator,” International Journal of Modern Physics B, vol. 35, no. 24, Article ID 2150244, 2021.
View at: Publisher Site | Google Scholar
Ji-H. He, Y. O. El-Dib, and A. A. Mady, “Homotopy perturbation method for the fractal toda oscillator,” Fractal Fract, vol. 5, no. 3, p. 93, 2021.
View at: Publisher Site | Google Scholar
C.-H. He, D. Tian, G. M. Moatimid, H. F. Salman, and M. H. Zekry, “Hybrid Rayleigh–van der pol–duffing oscillator: stability analysis and controller,” Journal of Low Frequency Noise, Vibration and Active Control, vol. 41, no. 1, pp. 244–268, 2022.
View at: Publisher Site | Google Scholar
Y. Ueda, “Randomly transitional phenomena in the system governed by Duffing’s equation,” Journal of Statistical Physics, vol. 20, no. 2, pp. 181–196, 1979.
View at: Publisher Site | Google Scholar
H. G. Enjieu Kadji, J. B. Chabi Orou, and P. Woafo, “Regular and chaotic behaviors of plasma oscillations modeled by a modified Duffing equation,” Physica Scripta, vol. 77, no. 2, Article ID 025503, 2008.
View at: Publisher Site | Google Scholar
S. A. El-Tantawy, A. H. Salas, and M. R. Alharthi, “On the analytical solutions of the forced damping duffing equation in the form of weierstrass elliptic function and its applications,” Mathematical Problems in Engineering, vol. 2021, pp. 1–9, 2021.
View at: Publisher Site | Google Scholar
N. H. Aljahdaly and S. A. El-Tantawy, “On the multistage differential transformation method for analyzing damping Duffing oscillator and its applications to plasma physics,” Mathematics, vol. 9, no. 4, p. 432, 2021.
View at: Publisher Site | Google Scholar
H. S. Alvaro, S. A. El-Tantawy, and H. J. E. Castillo, “On the approximate and analytical solutions to the fifth-order duffing oscillator and its physical applications,” Waves Random Complex Media, 2021, in Press.
View at: Publisher Site | Google Scholar
A. H. Salas S, S. A. El-Tantawy, M. R. Alharthi, and M. R. Alharthi, “Novel solutions to the (un)damped Helmholtz-Duffing oscillator and its application to plasma physics: moving boundary method,” Physica Scripta, vol. 96, no. 10, Article ID 104003, 2021.
View at: Publisher Site | Google Scholar
S. A. El-Tantawy, A. H. Salas, and M. R. Alharthi, “A new approach for modelling the damped Helmholtz oscillator: applications to plasma physics and electronic circuits,” Communications in Theoretical Physics, vol. 73, no. 3, Article ID 035501, 2021.
View at: Publisher Site | Google Scholar
W. Szemplińska-Stupnicka and J. Rudowski, “The coexistence of periodic, almost-periodic and chaotic attractors in the Van der Pol – duffing oscillator,” Journal of Sound and Vibration, vol. 199, no. 2, pp. 165–175, 1997.
View at: Publisher Site | Google Scholar
A. Chudzik, P. Perlikowski, A. Stefański, and T. Kapitaniak, “Multistability and rare attractors in van der Pol-Duffing oscillator,” International Journal of Bifurcation and Chaos, vol. 21, no. 7, pp. 1907–1912, 2011.
View at: Publisher Site | Google Scholar
C. H. Miwadinou, L. A. Hinvi, A. V. Monwanou, and J. B. Chabi Orou, “Nonlinear dynamics of plasma oscillations modeled by a forced modified Van der Pol-Duffing oscillator,” 2013, https://arxiv.org/abs/1308.6132.
View at: Google Scholar
C. H. Miwadinou, A. V. Monwanou, J. Yovogan, L. A. Hinvi, P. R. Nwagoum Tuwa, and J. B. Chabi Orou, “Modeling nonlinear dissipative chemical dynamics by a forced modified Van der Pol-Duffing oscillator with asymmetric potential: chaotic behaviors predictions,” Chinese Journal of Physics, vol. 56, no. 3, pp. 1089–1104, 2018.
View at: Publisher Site | Google Scholar
A. Kimiaeifar, A. R. Saidi, G. H. Bagheri, M. Rahimpour, and D Domairry, “Analytical solution for van der Pol–duffing oscillators,” Chaos, Solitons & Fractals, vol. 42, no. 5, pp. 2660–2666, 2009.
View at: Publisher Site | Google Scholar
Z. Feng, “Duffing-van der Pol-type oscillator systems,” Discrete & Continuous Dynamical Systems—S, vol. 7, no. 6, pp. 1231–1257, 2014.
View at: Publisher Site | Google Scholar
Gh. Asadi Cordshooli1 and A. R. Vahidi, “Solutions of duffing-van der Pol equation using decomposition method,” Advanced Studies in Theoretical Physics, vol. 5, no. 3, pp. 121–129, 2011.
View at: Google Scholar
J. H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141–1199, 2006.
View at: Publisher Site | Google Scholar
J. H. He, “An improved amplitude-frequency formulation for nonlinear oscillators,” International Journal of Nonlinear Sciences and Numerical Stimulation, vol. 9, no. 2, pp. 211-212, 2008.
View at: Publisher Site | Google Scholar
J. H. He, “Comment on He’s frequency formulation for nonlinear oscillators,” European Journal of Physics, vol. 29, no. 4, pp. 19–22, 2008.
View at: Publisher Site | Google Scholar
J. H. He, “An approximate amplitude-frequency relationship for a nonlinear oscillator with discontinuity,” Nonlinear Science Letters A Mathematics, vol. 7, pp. 77–85, 2016.
View at: Google Scholar
J. H. He, “Amplitude-frequency relationship for conservative nonlinear oscillators with odd nonlinearities,” International Journal of Applied and Computational Mathematics, vol. 3, no. 2, pp. 1557–1560, 2017.
View at: Publisher Site | Google Scholar
A. F. N. Rasedee, M. H. A. Sathar, H. M. Ijam, K. I. Othman, N. Ishak, and S. R. Hamzah, “A numerical solution for Duffing-Van Der Pol oscillators using a backward difference formulation,” AIP Conference Proceedings, vol. 2016, Article ID 020120, 2018.
View at: Google Scholar

Copyright

Copyright © 2022 Alvaro H. Salas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies