The Duffing Oscillator Equation and Its Applications in Physics

Universidad Nacional de Colombia, Department of Mathematics and Statistics, Fizmako Research Group, Bogotá, Colombia Universidad Distrital Francisco José de Caldas, Fizmako Research Group, Bogotá, Colombia Universidad Nacional de Colombia-Manizales-Caldas, Department of Mathematics and Statistics, Caldas, Colombia Universidad de Caldas, Department of Mathematics and StatisticsManizales, Caldas, Colombia


Introduction
e nonlinear equation describing an oscillator with a cubic nonlinearity is called the Duffing equation. Duffing [1], a German engineer, wrote a comprehensive book about this in 1918. Since then there has been a tremendous amount of work done on this equation, including the development of solution methods (both analytical and numerical) and the use of these methods to investigate the dynamic behavior of physical systems that are described by the various forms of the Duffing equation. Because of its apparent and enigmatic simplicity, and because so much is now known about the Duffing equation, it is used by many researchers as an approximate model of many physical systems or as a convenient mathematical model to investigate new solution methods [2][3][4][5][6][7]. is equation exhibits an enormous range of well-known behavior in nonlinear dynamical systems and is used by many educators and researchers to illustrate such behavior. Since the 1970s, it has become really popular with researchers into chaos, as it is possibly one of the simplest equations that describes chaotic behavior of a system. is equation is also useful in the study of soliton solutions to important physics models such as KdV equation, mKdV equation, sine-Gordon equation, Klein-Gordon equation, nonlinear Schrodinger equation, and shallow water wave equation [8][9][10][11][12][13][14][15][16][17][18].

Undamped and Unforced Duffing Equation
Let p, q, u 0 , and _ u 0 be real numbers. e general solution to the undamped and unforced Duffing equation u ″ (t) + pu(t) + qu 3 (t) � 0 may be expressed in terms of any of the twelve Jacobian elliptic functions, as shown in Table 1.

First
Case: Δ > 0. In the case, when _ u 0 � 0, we get Δ � (p + qu 2 0 ) 2 so that p + qu 2 0 ≠ 0 and the problem reduces to (3) Its solution is given by , m) (c 1 � a nonzero constant) be the Jacobi elliptic function cn with modulus m and parameter k defined by k 2 � m. We have y 3 (t) � 0, for any t, any c 1 ≠ 0.
(5) erefore, comparing (1) and (5) gives and we conclude that the analytic function is the general solution to the Duffing equation u ″ (t) + pu(t) + qu 3 (t) � 0 for arbitrary constants c 1 and c 2 . e values of these constants are determined from the initial conditions u(0) � u 0 and u ′ (0) � _ u 0 . We have and the value of c 1 results from solving the equation Squaring this last equation and taking into account relations (6) and the identities we arrive at the equation Solving equation (11) for c 1 gives To avoid the ambiguity with plus-minus signs, we define Making use of the addition formula the solution may also be written in the form where e solution is periodic and its main period equals In the case, when m > 1, we make use if the identities e main period will then be If m < 0, we transform the solution by means of the following identities: (20) Remember that cd � cn/dn, sd � sn/dn, and nd � 1/dn. For reference, Tables 2-4 give useful conversion formulas.
is given by is solution is periodic with main period . See Figure 1 for a comparison with Runge-Kutta numerical solution (dashed curve).
An equivalent expression without the imaginary unit is See Figure 2 for a comparison with Runge-Kutta numerical solution (dashed curve).

Second Case:
Since Δ < 0, necessarily q < 0. From the equality it is evident that δ > 0. We seek a solution to the i.v.p. (1) in the ansatz form where v � v(t) is the solution to some Duffing equations Mathematical Problems in Engineering

Mathematical Problems in Engineering
Equating to zero the coefficients of v 0 (t), v 1 (t), v 1 (t), and v 3 (t) in (32), we obtain an algebraic system. Solving it gives Observe that

us, the Duffing equation (29) has a positive discriminant. e solution to the i.v.p. (3) is then given by
where e values of a, β, and A are found from (33).

ird Case:
Δ � 0. When the discriminant vanishes, then q < 0 and the only solution to problem (1) which may be verified by direct computation.

New Trigonometric Jacobian Functions
Define the generalized cosine and sine functions as follows: , .

(38)
Our aim is to find some λ so that

Mathematical Problems in Engineering
We will choose λ so that e obtained approximations are good. is is seen from Tables 5 and 6.
We now will introduce new Jacobian "trigonometric functions" as follows: , , We extend the new functions (44)-(47) cn m (t) and sn m (t) for m > 1 and m < 0 and imaginary argument it using Tables 1-3 replacing the cn(t, m) with cn m (t) and cn m (t) with cn m (t) and so on.

Applications in Physics
Many partial differential equations arising in soliton theory may be reduced to odes or systems of odes by means of a traveling wave transformation. ese odes are generally nonlinear and some of them are Duffing type equations. Let us consider some important models of soliton theory.

e Klein-Gordon-Zakharov (KGZ) Equation in Plasmas.
e KGZ equation reads We transform the KGZ by means of the traveling wave substitution to obtain the system Mathematical Problems in Engineering We choose λ so that κ � (λω/k 2 ) and integrating the equation (51) twice taking null integration constants, we obtain and the problem reduces to solve a Duffing equation.

e Sine-Gordon Equation.
is is the equation is important model appears in differential geometry and relativistic field theory. It is denominated following its similar form to the Klein-Gordon equation. e equation, as well as several solution techniques, was known in the 19th century, but the equation grew greatly in importance when it was realized that it led to solutions ("kink" and "antikink") with the collision properties of solitons. e sine-Gordon equation is widely applied in physical and engineering applications, including the propagation of fluxons in Josephson junctions (a junction between two superconductors), the motion of rigid pendular attached to a stretched wire, and dislocations in crystals. It also arises in nonlinear optics. We apply the traveling wave transformation and the sine-Gordon equation converts into It may be easily verified that equation (55) holds for any solution u � u(ξ) of the Duffing equation
Equating the coefficients of y(ξ) 4 , y(ξ) 2 , and C + 2 DS + λR to zero gives an algebraic system. Solving it, we arrive at the expressions

e Nonlinear Schrodinger Equation.
e nonlinear Schrödinger equation is among the most prominent equations in nonlinear physics, especially in nonlinear optics. e nonlinear Schrödinger equation is of particular importance in the description of nonlinear effects in optical fibers. e nonlinear Schrödinger equation is a central model of nonlinear science, applying to hydrodynamics, plasma physics, molecular biology, and optics. It has been studied for more than 40 years, and it is employed in numerous fields well beyond plasma physics and nonlinear optics, where it originally appeared. e nonlinear Schrödinger equation (NLSE) is in the following form: where γ is a nonzero real constants and u � u(x, t) is a complex valued function of two real variables x, t. e Schrödinger equations occur in various areas of physics, including nonlinear optics, plasma physics, superconductivity, and quantum mechanics. e NLSE (67) exhibits soliton and periodic cnoidal wave solutions. Let Under this transformation, the NLSE (67) takes the form v ″ (ξ) − α 2 + β v(ξ) + 2cv 3 (ξ) � 0, which is a Duffing equation.

Data Availability
No data were used to support this study.

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