Research Article
The Duffing Oscillator Equation and Its Applications in Physics
Alvaro Humberto Salas Salas ,
1
Jairo Ernesto Castillo Hern´ andez ,
2
and Lorenzo Julio Mart´ ınez Hern´ andez
3,4
1
Universidad Nacional de Colombia, Department of Mathematics and Statistics, Fizmako Research Group, Bogot´ a, Colombia
2
Universidad Distrital Francisco Jos´ e de Caldas, Fizmako Research Group, Bogot´ a, Colombia
3
Universidad Nacional de Colombia-Manizales-Caldas, Department of Mathematics and Statistics, Caldas, Colombia
4
Universidad de Caldas, Department of Mathematics and StatisticsManizales, Caldas, Colombia
Correspondence should be addressed to Alvaro Humberto Salas Salas; ahsalass@unal.edu.co
Received 21 March 2021; Accepted 23 August 2021; Published 30 November 2021
Academic Editor: Maria L. Gandarias
Copyright©2021AlvaroHumbertoSalasSalasetal.isisanopenaccessarticledistributedundertheCreativeCommonsAttribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, we solve the Duffing equation for given initial conditions. We introduce the concept of the discriminant for the
Duffingequationandwesolveitinthreecasesdependingonsignofthediscriminant.WealsoshowthewaytheDuffingequation
is applied in soliton theory.
1. Introduction
e nonlinear equation describing an oscillator with a cubic
nonlinearity is called the Duffing equation. Duffing [1], a
Germanengineer,wroteacomprehensivebookaboutthisin
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 ap-
proximate model of many physical systems or as a conve-
nient mathematical model to investigate new solution
methods[2–7].isequationexhibitsanenormousrangeof
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 [8–18].
2. 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)� 0maybeexpressedintermsofany
ofthetwelveJacobianellipticfunctions,asshowninTable1.
In this section, we will solve the initial value problem
u
″
(t)+ pu(t)+ qu
3
(t)� 0, u(0)� u
0
,u
′
(0)� _ u
0
,u
2
0
+ _ u
2
0
q ≠ 0. (1)
e number
Δ � p + qu
2
0
2
+ 2q _ u
2
0
(2)
Hindawi
Mathematical Problems in Engineering
Volume 2021, Article ID 9994967, 13 pages
https://doi.org/10.1155/2021/9994967