# Analytical Solution to the Generalized Complex Duffing Equation

Authors: Alvaro H. Salas S; Gilder Cieza Altamirano; Lorenzo J. Martínez H
Journal: The Scientific World Journal
 (2022)

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## Abstract

Future scientific and technological evolution in many areas of applied mathematics and modern physics will necessarily depend on dealing with complex systems. Such systems are complex in both their composition and behavior, namely, dealing with complex dynamical systems using different types of Duffing equations, such as real Duffing equations and complex Duffing equations. In this paper, we derive an analytical solution to a complex Duffing equation. We extend the Krýlov–Bogoliúbov–Mitropólsky method for solving a coupled system of nonlinear oscillators and apply it to solve a generalized form of a complex Duffing equation.


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## Body

## 1. Introduction

Numerous scholars have effectively used the theory of linear oscillations to analyze and model oscillatory devices. However, nonlinear behavior can be found in a wide range of real applications. Thus, scholars from several fields of science explore nonlinear systems and try to model and investigate these complicated systems in order to find solutions and explanations to some mysterious problems, whether in the manufacture of small and large machines or electronic chips. Consequently, nonlinear oscillation is one of the most popular and widely researched fields due to its diverse applications in automobiles, sensing, microscale and nanoscale, 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 are used to model several nonlinear oscillations in different physical and engineering systems. The Duffing-type equation is one of the most famous and important equations that succeeded in explaining many different oscillations in different engineering, physical systems, and statistical mechanics.The Duffing equation is a nonlinear second-order differential equation that describes an oscillator with complex, sometimes chaotic behavior. The Duffing equation was originally the result of Georg Duffing’s systematic study of nonlinear oscillations. The behavior of the solution of the Duffing equation easily changes depending on the initial value and the polynomial coefficients, and it is difficult to predict its solution. To clarify the behavior of the solution, research based mainly on numerical analysis with high-precision calculations is conducted. Interest in the equation was later revived with the advent of chaos theory. Since then, the system has come to be regarded as one of the prototype systems in chaos theory, and related equations continue to find applications today, e.g., to describe the rolling of ships. The Duffing equation reads(1)x¨+δx˙+αx+βx3=γcosωt,where xt is the displacement at time t and the term γcosωt represents a sinusoidal driving force. The cubic term describes an asymmetry in the restoring force of a spring that softens or stiffens as it is stretched. One of the most remarkable results of dynamical systems theory is the ubiquitousness of chaotic behavior in nonlinear systems. Deterministic chaos has been observed both in mathematical models and in real physical systems. Although, from the point of view of the applications, chaotic behavior can have positive effects, improving, for example, mixing processes in chemical reactions, in other situations, such behavior can have harmful consequences, as is the case in different fields of engineering: aerodynamics, electronic circuits, and magnetic confinement of plasmas.Some recent works on complex chaos have focused on solving complex nonlinear differential equations, complex chaos control and synchronization, and so on. For example, Cveticanin developed an approximate analytic approach for solving strong nonlinear differential equations of the Duffing-type with a complex-valued function. Furthermore, excellent agreement is found between the analytic and numerical results.In [1], authors considered the following complex Duffing equation for modelling complex signal detection:(2)z¨+kz˙−z+εzz2=γexp−1t,where z=x+−1y is a complex function, k, ε, and γ≥0 are the real parameters, and the dots are the time derivatives. Its dynamical behavior was analyzed. Based on the proposed (2), they constructed a complex chaotic oscillator detection system to detect complex signals in noise. They investigated the influence of noise on the detection system and the detection performance of the system for complex signals.In their work [2], the authors considered a complex Duffing system subjected to nonstationary random excitation of the form(3)z¨+2ωξz˙+ω2z+ϵzz2=αFt,where z=zt is a complex function, α=1+−1, ω, ξ represent the natural frequency and damping coefficient, respectively, ϵ is the small perturbation parameter and nonlinearity strength, and Ft is a random function. This equation with Ft=0 has connection to the complex nonlinear Schrodinger equation which appears in many important fields of physics. Authors in [2] investigated the mean square response of a complex Duffing system subjected to nonstationary random excitation using the Wiener–Hermite expansion method combining the perturbation technique.In 2001, Mahmoud et al. [3] presented the following complex Duffing equation:(4)z¨−αz+εzz2=γexp−1ωt.Based on the work in [3], Li et al. [4] studied the problem of chaos control for a complex Duffing oscillation system. In general, few works are devoted to the complex Duffing equation.In this paper, we will consider the following complex Duffing equation:(5)z¨+2εz˙+αz+βzz2+γz¯3=f1t+−1f2t,z=zt.To our best knowledge, no work has been devoted to seeking analytical solutions to the complex Duffing equation. This is precisely the main objective of the present paper.


## 2. Undamped and Unforced Complex Duffing Equation

Let us consider the i.v.p.(6)z¨+αz+βzz2+γz¯3=0,z0=x0+−1y0andz′0=x˙0+−1y˙0.Let(7)zt=xt+−1yt.Then,(8)x″t+xtα+β−3γyt2+β+γxt3=0,y″t+ytα+β−3γxt2+β+γyt3=0.Assume thatx=xt and y=yt obey some Duffing equations:(9)x¨+px+qx3=0,y¨+ry+sy3=0.Then,(10)xtα−p+β−3γyt2+xt3β+γ−q=0,α−ryt+yt3β+γ−s+β−3γxt2yt=0.Equating to zero, the coefficients ofxt and yt in (10) give(11)p=α,q=4β3,r=α,s=4β3andγ=β3.Thus,(12)x¨+αx+4β3x3=0,x0=x0,x′0=x˙0,y¨+αy+4β3y3=0,y0=y0,y′0=y˙0.On the other hand, the exact solution to the i.v.p.(13)u¨+Au+Bu3=0,u0=u0andu′0=u˙0is expressed as(14)ut=u0cnωt,m+u˙0ωsnωt,mdnωt,m/1+bsn2ωt,m,where(15)b=B1−2mu022A−m,ω=A1−2m,m=121±AA+Bu022+2Bu˙02.


## 3. Solution to the General Complex Duffing Equation by Means of the Krýlov–Bogoliúbov–Mitropólsky Method

Let us consider the i.v.p.(16)z¨+2εz˙+αz+βzz2+γz¯3=f1t+−1f2t,z0=z0andz′0=z˙0.Here,α, β, and γ are the real numbers. f1t and f2t are the real-valued functions, and zt=xt+−1yt. The system (16) may be written in the form(17)x¨+αx+2εx˙+β−3γxy2+β+γx3−f1t=0,y¨+αy+2εy˙+β−3γx2y+β+γy3−f2t=0.The initial conditions are(18)x0=x0,y0=y0,x′0=x˙0,y′0=y˙0.Let us consider the followingp-problem:(19)x¨+αx+p2εx˙+β−3γxy2+β+γx3−f1t=0,y¨+αy+p2εy˙+β−3γx2y+β+γy3−f2t=0.The solution is assumed to be in the ansatz form(20)xt=atcosψt+∑n=1Npnvnat,bt,ψt,Ψt,yt=btcosΨt+∑n=1Npnwnat,bt,ψt,Ψt,a′t=∑n=1NpnAnat,ψ′t=α+∑n=1Npnφnat,b′t=∑n=1NpnBnbt,Ψ′t=α+∑n=1Npnϕnbt.We choose the solutions in order to avoid the presence of the so-called secularity terms. Solving the odes gives(21)φ1a=3β+γ+2b2β−3γ8αa2,ϕ1a=5β−3γ8αa2,A1a=−aε,B1b=−bε,v1a,ψ,Ψ=132αa3β+γcos3ψ−4ab2β−3γcos2Ψ2ψsinψ+cosψ+32f1t,w1b,ψ,Ψ=132α−4a2bβ−3γcos2ψ2ΨsinΨ+cosΨ+b3β+γcos3Ψ+32f2t,a˙=−aεp,b˙=−bεp,ψ˙=α+p8α3a2β+3a2γ+2b2β−6b2γ,Ψ˙=α+p8α2a2β−6a2γ+3b2β+3b2γ.The approximate analytical solution is obtained by lettingp=1. It reads(22)xt=acosψ+132αa3β+γcos3ψ−4ab2β−3γcos2Ψ2ψsinψ+cosψ+32f1t,yt=bcosΨ+132α−4a2bβ−3γcos2ψ2ΨsinΨ+cosΨ+b3β+γcos3Ψ+32f2t.The expressions fora,b,ψ,andΨare(23)a=at=c0exp−εt,b=bt=d0exp−εt,ψt=18αexp−εtsinhεt3c02β+γ+2d02β−3γε+8αt+c1,Ψt=18αexp−εtsinhεt2c02β−3γ+3d02β+γε+8αt+d1.The constantsc0, c1, d0, and d1 are obtained from the initial conditions.The obtained solution is valid forα>0. Let α<0 for the sake of simplicity; we will consider only the case when γ=0. Let us change α to −α. We are given that(24)z¨+2εz˙−αz+βzz2=f1t+−1f2t,z0=z0andz′0=z˙0.In the case whenε=0 and f1t=f2t≡0, direct calculations show that the following function will be the exact solution to z¨−αz+εzz2=0:(25)zt=c0dnεc02+d022t+c1|2α−εc02−εd02εc02+d02+−1d0dnεc02+d022t+d1|2α−εc02−εd02εc02+d02.The constantsc0, c1, d0, and d1 are determined from the initial conditions:(26)z0=x0+−1y0andz′0=x˙0+−1y˙0.Let us solve the general case. Assume the solution in the ansatz form:(27)zt=r+xt+−1yt,r2=αβ.Then,(28)x¨+2αx+2εx˙+3βrx2+βry2+βx3+βxy2=f1t,y¨+2εy˙+2βrxy+βx2y+βy3=f2t.We may solve the above system using the KBM method. To this end, we consider the followingp-problem:(29)x¨+2αx+p2εx˙+3βrx2+βry2+βx3+βxy2−f1t=0,y¨+y+p−y+2εy˙+2βrxy+βx2y+βy3−f2t=0.Proceeding in the same way as we did in the first part, we obtain the following first-order approximation:(30)xt=e−3εt64αβc03cos3ψ−4βc0d02cos2ϕ2ψsinψ+cosψ+16βreεtc02cos2ψ−3−2d02cos2ϕ+64αc0e2εtcosψ+12αf1t,yt=132d0e−3εt−8βc0ϕsinϕ+cosϕc0cos2ψ+4reεtcosψ+βd02cos3ϕ+32e2εtcosϕ+f2t.Here,(31)ψ=ψt=e−2εt32αε4ε8αc1e2εt+2tβd02+8αe2εt+32βc02e2εt−1,ϕ=ϕt=1164βc02te−2εt+βd123−3e−2εtε+16d1+8t.The constantsc0, c1, d0, and d1 are determined from the initial conditions:(32)z0=x0+−1y0andz′0=x˙0+−1y˙0.


## 4. Applications

Let us check the accuracy of the obtained results in concrete examples.Example 1.
Let(33)z″t+3zt+0.04z′t+ztzt2+0.2zt∗3=Ft,Ft≔=0.1cn0.1t|0.9+0.+0.1isn0.1t|0.9.z0=0∧z′0=0.
See Figures1–3.Figure 1
Real part compared with the Runge–Kutta numerical solution.Figure 2
Imaginary part compared with the Runge–Kutta numerical solution.Figure 3
Absolute value compared with the Runge–Kutta numerical solution.Example 2.
Let(34)z.+2z+0.2z˙+zz2+0.2z¯3=0.1cos0.2t−icos0.1t∧z0=0∧z′0=0.
The approximate analytical solution reads(35)zapproxt=xt+iyt,where(36)x=e−0.3tcos2Ψ6.23E−6ψsinψ+3.15E−6cosψ−2.4E−6cos3ψ−0.0501254e−0.1tcosψ+0.05cos0.2ty=e−0.3tcos2ψ−6.3E−6ΨsinΨ−3.15E−6cosΨ+2.−4E−6cos3Ψ+0.0501254e−0.1tcosΨ−0.05cos0.1tψ=1.41421t−0.00488578e−0.2t−0.0656304.Ψ=1.41421t−0.00488578e−0.2t−0.0656304.
See Figures4–6.Figure 4
Real part compared with the Runge–Kutta numerical solution.Figure 5
Imaginary part compared with the Runge–Kutta numerical solution.Figure 6
Absolute valuez compared with the Runge–Kutta numerical solution.


## 5. Conclusions

The nonlinear complex Duffing oscillators and many related oscillators, including the unforced undamped complex Duffing oscillator (CDO), the unforced damped CDO, and the forced damped CDO, have been analyzed using the ansatz method in order to find some approximations. For the unforced undamped CDO, the exact solution of the standard Duffing oscillator (DO) with the ansatz method was used for deriving an analytical approximation in terms of the Jacobi elliptic function. Also, the unforced damped CDO has been analyzed using the ansatz method, and with the help of the approximation of the unforced damped DO, an approximation in the form of a trigonometric form was obtained. Moreover, the forced damped CDO has been examined via the Krýlov–Bogoliúbov–Mitropólsky method (KBM), and a new analytical approximation in the form of a trigonometric formula has been derived. We demonstrated the way we may use the KBM in order to solve coupled systems of nonlinear oscillators. Other works related to nonlinear oscillators may be found in [5–13].



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