Analytical Solution to the Generalized Complex Duffing Equation

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.


Introduction
Numerous scholars have efectively 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. Tus, scholars from several felds of science explore nonlinear systems and try to model and investigate these complicated systems in order to fnd 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 felds due to its diverse applications in automobiles, sensing, microscale and nanoscale, fuid and solid interaction, nonlinear oscillations in plasma physics, bioengineering, and nonlinear oscillations in optics. Tere are many diferent and various equations of motion that are used to model several nonlinear oscillations in diferent physical and engineering systems. Te Dufng-type equation is one of the most famous and important equations that succeeded in explaining many diferent oscillations in diferent engineering, physical systems, and statistical mechanics.
Te Dufng equation is a nonlinear second-order differential equation that describes an oscillator with complex, sometimes chaotic behavior. Te Dufng equation was originally the result of Georg Dufng's systematic study of nonlinear oscillations. Te behavior of the solution of the Dufng equation easily changes depending on the initial value and the polynomial coefcients, and it is difcult to predict its solution. To clarify the behavior of the solution, research based mainly on numerical analysis with highprecision 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 fnd applications today, e.g., to describe the rolling of ships. Te Dufng equation reads where x(t) is the displacement at time t and the term c cos ωt represents a sinusoidal driving force. Te cubic term describes an asymmetry in the restoring force of a spring that softens or stifens 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 efects, improving, for example, mixing processes in chemical reactions, in other situations, such behavior can have harmful consequences, as is the case in diferent felds of engineering: aerodynamics, electronic circuits, and magnetic confnement of plasmas. Some recent works on complex chaos have focused on solving complex nonlinear diferential equations, complex chaos control and synchronization, and so on. For example, Cveticanin developed an approximate analytic approach for solving strong nonlinear diferential equations of the Dufng-type with a complex-valued function. Furthermore, excellent agreement is found between the analytic and numerical results.
In [1], authors considered the following complex Dufng equation for modelling complex signal detection: y is a complex function, k, ε, and c ≥ 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. Tey investigated the infuence 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 Dufng system subjected to nonstationary random excitation of the form , ω, ξ represent the natural frequency and damping coefcient, respectively, ϵ is the small perturbation parameter and nonlinearity strength, and F(t) is a random function. Tis equation with F(t) � 0 has connection to the complex nonlinear Schrodinger equation which appears in many important felds of physics. Authors in [2] investigated the mean square response of a complex Dufng 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 Dufng equation: Based on the work in [3], Li et al. [4] studied the problem of chaos control for a complex Dufng oscillation system. In general, few works are devoted to the complex Dufng equation.
In this paper, we will consider the following complex Dufng equation: To our best knowledge, no work has been devoted to seeking analytical solutions to the complex Dufng equation. Tis is precisely the main objective of the present paper.

Undamped and Unforced Complex Duffing Equation
Let us consider the i.v.p. Let Ten, Assume that x � x(t) and y � y(t) obey some Dufng equations: Ten, Equating to zero, the coefcients of x(t) and y(t) in (10) give Tus, On the other hand, the exact solution to the i.v.p.

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is expressed as where

Solution to the General Complex Duffing Equation by Means of the Krýlov-Bogoliúbov-Mitropólsky Method
Let us consider the i.v.p.
Here, α, β, and c are the real numbers. f 1 (t) and f 2 (t) are the real-valued functions, and z(t) � x(t) + �� � −1 √ y(t). Te system (16) may be written in the form Te initial conditions are Let us consider the following p-problem: Te solution is assumed to be in the ansatz form , (20) We choose the solutions in order to avoid the presence of the so-called secularity terms. Solving the odes gives Te approximate analytical solution is obtained by letting p � 1. It reads Te Scientifc World Journal a 3 (β + c)cos (3ψ) − 4ab 2 (β − 3c)cos (2Ψ)(2ψ sin(ψ) + cos(ψ)) + 32f 1 (t) , (22) Te constants c 0 , c 1 , d 0 , and d 1 are obtained from the initial conditions. Te obtained solution is valid for α > 0. Let α < 0 for the sake of simplicity; we will consider only the case when c � 0. Let us change α to −α. We are given that In the case when ε � 0 and f 1 (t) � f 2 (t) ≡ 0, direct calculations show that the following function will be the exact solution to € z − αz + εz|z| 2 � 0: Te constants c 0 , c 1 , d 0 , and d 1 are determined from the initial conditions: Let us solve the general case. Assume the solution in the ansatz form: Ten, € y + 2ε _ y + 2βrxy + βx 2 y + βy 3 � f 2 (t). (28) We may solve the above system using the KBM method. To this end, we consider the following p-problem: Proceeding in the same way as we did in the frst part, we obtain the following frst-order approximation:    Te Scientifc World Journal

Conclusions
Te nonlinear complex Dufng oscillators and many related oscillators, including the unforced undamped complex Dufng oscillator (CDO), the unforced damped CDO, and the forced damped CDO, have been analyzed using the ansatz method in order to fnd some approximations. For the unforced undamped CDO, the exact solution of the standard Dufng 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][6][7][8][9][10][11][12][13].

Data Availability
No data were used to support this study.

Conflicts of Interest
Te authors declare that they have no conficts of interest.