In this investigation, some different approaches are implemented for analyzing a generalized forced damped complex Duffing oscillator, including the hybrid homotopy perturbation method (H-HPM), which is sometimes called the Krylov-Bogoliubov-Mitropolsky (KBM) method and the multiple scales method (MSM). All mentioned methods are applied to obtain some accurate and stable approximations to the proposed problem without decoupling the original problem. All obtained approximations are discussed graphically using different numerical values to the relevant parameters. Moreover, all obtained approximate solutions are compared with the 4th-order Runge-Kutta (RK4) numerical approximation. The maximum residual distance error (MRDE) is also estimated, in order to verify the high accuracy of the obtained analytic approximations.
Weaam Alhejaili, Alvaro H Salas, S A El-Tantawy. Analytical approximations to a generalized forced damped complex Duffing oscillator: multiple scales method and KBM approach[J]. Communications in Theoretical Physics, 2023, 75(2): 025002. DOI: 10.1088/1572-9494/aca9c0
1. Introduction
It is worth noting that all natural phenomena behave nonlinearly. Therefore, to accurately describe these phenomena, many researchers have modeled them based on many nonlinear differential equations [1, 2]. For instance, the averaging method was applied for studying some different examples of autonomous systems [3]: $\ddot{x}+\alpha x+\beta {x}^{3}+\varepsilon g\left(x,\dot{x}\right)=0$ for small ϵ and β. Also, the forced/unforced damped/undamped parametric pendulum oscillators [4]: $\ddot{\theta }+2\beta \dot{\theta }+\left({\omega }_{0}^{2}-{Q}_{0}\cos \left(\gamma t\right)\right)\sin \theta =f\left(t\right)$ and some other equations related to this oscillator have been analyzed and investigated using some different effective analytical and numerical techniques, such as the ansatz method [4], He's frequency-amplitude principle [4], He's homotopy perturbation method (HPM) [4], the Krylov–Bogoliúbov Mitropolsky (KBM) method [4], the 4th-order Runge Kutta (RK4), the hybrid Padé-finite difference method [4], the Chebyshev collocation method (CCM) [5], the Galerkin method [6], the ansatz method (AM) and He's frequency formulation [6]. Moreover, the AM and the HPT with the extended KBM were used in the study of the damped cubic nonlinearity Duffing–Mathieu-type oscillator [7]. Since many different types of oscillators are associated with several potential engineering and physical applications and many other complicated nonlinear phenomena, many researchers have devoted considerable effort to analyze these oscillators to arrive at an accurate interpretation that serves both engineering and physical applications. Consequently, nonlinear oscillation is one of the most popular and widely researched fields due to its diverse applications in automobiles, sensing, micro- and nano-scale, fluid and solid interaction, nonlinear oscillations in plasma physics, bioengineering, and nonlinear oscillations in optics. There are many different and various equations of motion that were used for modeling several nonlinear oscillations in different physical and engineering systems [8–11]. Most published papers about the nonlinear oscillatory equations focused on the one-dimensional oscillatory differential equations that succeeded in explaining many different oscillations in different engineering, physical systems (especially in plasma physics), and statistical mechanics such as the Duffing oscillatory equation (DOE), the damped DOE [12], the forced damping DOE [13], the fifth-order DOE [14], the damped Helmholtz oscillator equation [15], the damped/undamped Helmholtz-Duffing oscillatory equation [16], the Helmholtz-Fangzhu oscillator equation [17], the damped pendulum oscillator equation [18], and many others. On the other hand, there are some studies that have been conducted on the complex/coupled system of oscillatory differential equations [19–22]. A complex Duffing oscillator (CDO) is an example of nonlinear differential equations with complex variables [23, 24]. Also, this equation has many applications, including rotor dynamics, plasma waves and oscillations, vibrations of the nonlinear Jeffcott rotor [24], etc. In order to analyze and solve this type of differential equations (DEs), they separate into two components of the real and imaginary parts, which leads to a coupled system of DEs. For example, Cveticanin [23] studied the coupled system of third and fifth-order nonlinearity Duffing oscillators by using the hybrid elliptic Krylov-Bogolubov method (eKBM) with the power series method. Also, Cveticanin [24] used the eKBM for investigating the following complex Duffing-type oscillator,
where $z\equiv z\left(t\right)=x+$ iy is a complex function and $\bar{z}$ is the conjugate of z, whereas x ≡ x(t) and y ≡ y(t) indicate the real and imaginary parts, respectively, $z\bar{z}=| z{| }^{2}$ and $G\left(z,\dot{z},{cc}\right)$ represent a complex defection, “cc” indicates the complex conjugate function, and i $=\sqrt{-1}$. Cveticanin [24] derived an analytic approximation in the form of the Jacobi elliptic functions for the unforced case. Also, Cveticanin [24] tested the accuracy of the obtained approximations by comparing them with the RK numerical approximation and found that there is a good agreement between them. Moreover, some coupled nonlinear differential equations related to an enzymatic reaction have been solved using the HPM [25]. In [2], the complex Duffing system with damping term has been studied using the Wiener-Hermite expansion. Furthermore, a complex Duffing system with a nonstationary random excitation has been investigated using the Wiener–Hermite expansion method [26]. In [27], the authors analyzed the dynamical behavior of the following new form to a complex Duffing equation (CDE): $\ddot{z}+k\dot{z}-z+\beta z{\left|z\right|}^{2}\,=\gamma \exp \left({\rm{i}}t\right)$. The authors [27] used the later equation for detecting complex signals in noise. However, in our investigation, we will consider the following generalized forced damped CDO [23],
where $\unicode{x003DC}\left(t\right)={f}_{1}(t)+$ if2(t) represents the excited periodic force in which both real and imaginary components $\left({f}_{1}(t),{f}_{2}(t)\right)$ may take the following values: ${f}_{1}(t)={\gamma }_{1}\cos \left({\omega }_{1}t\right)$ and ${f}_{2}(t)={\gamma }_{2}\cos \left({\omega }_{2}t\right)$ or any other time-dependent function. In equation (2), the coefficients $\left(\varepsilon ,\alpha ,\beta ,\gamma \right)$ are real constants. More details about the derivation and potential applications of equation (2) can be found in [23]. However, in [23], the author did not investigate the forced case. Therefore, the main objective of this study is to analyze and discuss both unforced ($\unicode{x003DC}\left(t\right)=0$) and forced $(\unicode{x003DC}\left(t\right)\ne 0)$ general damped CDO (2) and find some approximation for them using two different approaches, including the hybrid HPM (H-HPM), sometimes also called the KBM method [28–30], and the multiple scales method (MSM) [31, 32].
The rest of this paper consists of the following parts: in section 2, the mathematical methods for analyzing the generalized forced damped CDO are introduced. In this section, two different approaches including both the H-HMP and MSM are implemented to find some approximations to the generalized forced damped CDO. In section 3, all of the obtained approximations are discussed in detail. Some numerical examples are considered for studying the impact of different parameters on the profile of the oscillator. Finally, the most significant findings we make are outlined in section 4.
2. Mathematical methods for analyzing the generalized forced damped CDO
Let us write the general forced damped complex Duffing (2) in the following initial value problem (i.v.p.):
Here, the coefficients $\left(\varepsilon ,\alpha ,\beta ,\gamma \right)$ are real constants. For analyzing this problem, we first should reduce it to the standard two forced damped Duffing oscillators. Using the relation z = x + iy in the i.v.p. (3), we have
Now, we try to analyze and solve the coupled system of equations (4)–(6) directly without decoupling using some effective and accurate analytical approaches including both the H-HPM/KBM method and the MSM. The proposed methods are illustrated in detail below.
2.1. Hybrid homotopy perturbation method (H-HMP)
Let us construct the homotopy for the i.v.p. (3) as follows:
Equating to zero the coefficients of p, $\cos \left(\psi \left(t\right)\right)$, $\sin \left(\psi \left(t\right)\right)$, $\cos \left({\rm{\Psi }}\left(t\right)\right),$ and $\sin \left({\rm{\Psi }}\left(t\right)\right)$, in the relations of $\left(\tilde{R},\tilde{S}\right)$, the following system of algebraic-differential equations is obtained:
where the constants $\left({a}_{0},{a}_{1},{b}_{0},{b}_{1}\right)$ are obtained from the ICs (6).
2.2. Multiple scales method (MSM)
To apply the MSM [31, 32] for analyzing the general forced damped complex Duffing i.v.p. (3), we first construct the p − problem to the system of equations (4) and (5) for α > 0 as follows:
where $a\left({T}_{1}\right)$, $b\left({T}_{1}\right)$, $\varphi \left({T}_{1}\right)$, and $\phi \left({T}_{1}\right)$ are undetermined time-dependent functions.
Solving system (25) yields the values of $\left({u}_{1},{v}_{1}\right)$ as follows:
Now, to avoid the secularity and to determine the values of $a\left({T}_{1}\right)$, $b\left({T}_{1}\right)$, φ(t), and φ(t), we should solve the ode system S1,2 = 0 and W1,2 = 0, which leads to
Accordingly, system (27) reduces to u1 = S0 and v1 = W0. Inserting equations (26)–(28) into system (19), and for p → 1, we finally get the first-order approximations to $\left(x,y,z\right)$ as follows:
where the constants c0, d0, c1 and d1 are found from the initial conditions.
It is shown that the approximations (30)–(32) do not recover only the solution of the forced damped CDO (2), but can also recover the solution of the unforced damped CDO, i.e., equation (2) for $\unicode{x003DD}\left(t\right)=0$ or γ1 = γ2 = 0, as follows:
First, let us check the behavior of the obtained analytical approximations to the i.v.p. (3) for the unforced case, i.e., for γ1 = γ2 = 0, by using the following numerical example
The approximations to real and imaginary parts $\left(x,y\right)$ using both the KBM method (approximations (16) and (17)) and the MSM (approximations (33) and (34)) are compared with the RK4 approximations as illustrated in figure 1. Moreover, the absolute approximations ∣z∣ = ∣x + iy∣ using both the KBM method and MSM are compared with the absolute approximation using the RK4 approach as shown in figure 2. In addition, the maximum residual distance error (MRDE) LMRDE to the real, imaginary, and absolute approximations, according to the following relations
Figure 1. The analytical approximations (17), (31), and (32) using H-HPM and MSM as well as the RK4 numerical approximations to the i.v.p. (3) for (a), (c) the real part (4) and (b), (d) the imaginary part (5) are considered using data from example (1).
Figure 2. The analytical approximations (a) (16) using H-HPM and (b) (30) using MSM are compared with the RK4 numerical approximations to the i.v.p. (3) for the absolute value ∣z∣ using data from example (1).
Table 1. The MRDE LMRDE for all proposed methods as compared to the RK4 approach using data from example (1).
Method
${\left.{L}_{{MRDE}}\right|}_{x}$
${\left.{L}_{{MRDE}}\right|}_{y}$
${\left.{L}_{{MRDE}}\right|}_{| z| }$
H-HPM
0.001 724 24
0.001 724 24
0.002 438 44
MSM
0.001 803
0.001 803
0.002 549 82
3.2. Example (2)
Now, we can investigate the characteristics of the obtained analytical approximations of the i.v.p. (3) in the existence of excited force, i.e., γ1 ≠ 0 and γ2 ≠ 0 by using the following numerical example:
Both real and imaginary parts approximations $\left(x,y\right)$ using the H-HPM (approximations (16) and (17)) and the MSM (approximations (30)–(32)) are compared with the numerical approximations using the RK4 approach as illustrated in figure 3. In addition, the absolute approximations ∣z∣ using the proposed methods (H-HPM and MSM) are compared with the absolute numerical approximations using the RK4 approach as shown in figure 4. Moreover, the MRDE LMRDE for the real, imaginary, and absolute approximations is estimated as shown in table 2.
Figure 3. The analytical approximations (17), (31), and (32) using H-HPM and MSM as well as the RK4 numerical approximations to the i.v.p. (3) for (a), (c) the real part (4) and (b), (d) the imaginary part (5) are considered using data from example (2).
Figure 4. The analytical approximations (a) (16) using H-HPM and (b) (30) using MSM are compared with the RK4 numerical approximations to the i.v.p. (3) for the absolute value ∣z∣ using data from example (2).
Table 2. The MRDE LMRDE for all proposed methods as compared to the RK4 approach using data from example (2).
Method
${\left.{L}_{{MRDE}}\right|}_{x}$
${\left.{L}_{{MRDE}}\right|}_{y}$
${\left.{L}_{{MRDE}}\right|}_{| z| }$
H-HPM
0.004 544 2
0.000 923 124
0.004 579 19
MSM
0.004 152 09
0.000 800 344
0.004 026 37
3.3. Example (3)
Here, we consider the following new values to the relevant parameters with non-zero ICs:
Figure 5 represents the real and imaginary parts approximations $\left(x,y\right)$ using the KBM method (approximations (16) and (17)) and the MSM (approximations (30)–(32)) as well as the RK4 numerical approximations. Also, the absolute approximations ∣z∣ using all mentioned approaches are presented in figure 6. Furthermore, the MRDE LMRDE for the real, imaginary, and absolute approximations to all proposed methods is estimated as illustrated in table 3. It is clear from tables 1–3 that the accuracies of all obtained analytical approximations using both the KBM method and MSM are highly compatible with each other. Moreover, all obtained approximations are characterized by high accuracy and are more stable for a long time, as illustrated in figures 1–6. Furthermore, one can see that the MSM accuracy is sometimes better than those of H-HPM. Despite this, there is great harmony and agreement between both the analytical and numerical approximations, which enhances the high accuracy of all obtained analytic approximations.
Figure 5. The analytical approximations (17), (31), and (32) using H-HPM and MSM as well as the RK4 numerical approximations to the i.v.p. (3) for (a), (c) the real part (4) and (b), (d) the imaginary part (5) are considered using data from example (3).
Figure 6. The analytical approximations (a) (16) using H-HPM and (b) (30) using MSM are compared with the RK4 numerical approximations to the i.v.p. (3) for the absolute value ∣z∣ using data from example (3).
Table 3. The MRDE LMRDE for all proposed methods as compared to RK4 approach using data of example (3).
Method
${\left.{L}_{{MRDE}}\right|}_{x}$
${\left.{L}_{{MRDE}}\right|}_{y}$
${\left.{L}_{{MRDE}}\right|}_{| z| }$
H-HPM
0.001 423 05
0.001 485 84
0.001 612 87
MSM
0.001 504 99
0.001 182 36
0.001 480 71
4. Conclusion
The nonlinear complex Duffing oscillators including both unforced and forced damped complex Duffing oscillators (CDO), have been analyzed using some different approaches. Two different approaches have been applied for deriving some effective, accurate, and stable approximations. In the first approach, the hybrid homotopy perturbation method (H-HPM)/Krylov-Bogoliubov-Mitropolsky (KBM) method has been carried out for deriving an approximation in terms of trigonometric functions. In the second approach, the proposed problem has been analyzed via the multiple scales method (MSM). In both the H-HPM/KBM method and MSM, only the first-order approximations have been derived because the first-order approximations are sufficient to obtain high accuracy and more stable solutions.
It was clear from the numerical results as well as the graphical analysis that all proposed methods give good results compared to the numerical approximations using the 4th-order Runge Kutta (RK4) method. Moreover, it was noted that all of the obtained approximations are characterized by high accuracy and are more stable for a long time. Note that in this investigation, we applied all mentioned approaches for solving and analyzing the nonlinear complex Duffing oscillators problem without decoupling.
Author contributions
All authors contributed equally and approved the final manuscript.
Conflicts of interest
The authors declare that they have no conflicts of interest.
Data Availability
All data generated or analyzed during this study are included in this published article (More details can be requested from El-Tantawy).
Funding
The authors extend their appreciation to the Deputyship for Research & Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number RI-44-0143.
NayfehA HZavodneyL D1988 Experimental observation of amplitude and phase modulated responses of two internally coupled oscillators to a harmonic excitation J. Appl. Mech.55 706 710
Alyousef HaifaASalas AlvaroHAlharthiM REl-TantawyS A2022 Galerkin method, ansatz method, and He's frequency formulation for modeling the forced damped parametric driven pendulum oscillators J. Low Freq. Noise V. A.41 1426 1445
Alyousef HaifaASalas AlvaroHAlkhateeb SadahAEl-TantawyS A2022 Some novel analytical approximations to the (un)damped duffing-mathieu oscillators J. Math.2022 2715767
Aljahdaly NoufeHEl-TantawyS A2021 On the multistage differential transformation method for analyzing damping Duffing oscillator and its applications to plasma physics Mathematics9 432
El-TantawyS ASalas AlvaroHAlharthiM R2021 On the analytical solutions of the forced damping duffing equation in the form of weierstrass elliptic function and its applications Math. Probl. Eng.2021 6678102
Salas AlvaroHEl-TantawyS AJairoECastilloH2021 On the approximate and analytical solutions to the fifth-order Duffing oscillator and its physical applications Waves Random Complex Media (10.1080/17455030.2021.1949072)
15
El-TantawyS ASalas AlvaroHAlharthiM R2021 A new approach for modelling the damped Helmholtz oscillator: applications to plasma physics and electronic circuits Commun. Theor. Phys.73 035501
SalasSAlvaroHEl-TantawyS AAlharthiM R2021 Novel solutions to the (un)damped Helmholtz-Duffing oscillator and its application to plasma physics: Moving boundary method Phys. Scr.96 104003
AlbalawiWSalasA HEl-TantawyS AYoussef Amr AbdA-R2021 Approximate analytical and numerical solutions to the damped pendulum oscillator: Newton-Raphson and moving boundary methods J. Taibah Univ. Sci.15 479 485
VaradharajanGRajendranL2011 Analytical solution of coupled non-linear second order reaction differential equations in enzyme kinetics Natural Science03 459 465
SalasA HAlbalawiWEl-TantawyS AEl-SherifL S2022 Some novel approaches for analyzing the unforced and forced duffing-van der pol oscillators J. Math.2022 2174192
SalasA HAlbalawiWAlharthiM REl-TantawyS A2022 Some novel solutions to a quadratically damped pendulum oscillator: analytical and numerical approximations Complexity2022 7803798