# Duffing equation

A Poincaré section of the forced Duffing equation suggesting chaotic behaviour

The Duffing equation (or Duffing oscillator), named after Georg Duffing, is a non-linear second-order differential equation used to model certain damped and driven oscillators. The equation is given by

${\displaystyle {\ddot {x}}+\delta {\dot {x}}+\alpha x+\beta x^{3}=\gamma \cos(\omega t)\,}$

where the (unknown) function x=x(t) is the displacement at time t, ${\displaystyle {\dot {x}}}$ is the first derivative of x with respect to time, i.e. velocity, and ${\displaystyle {\ddot {x}}}$ is the second time-derivative of x, i.e. acceleration. The numbers ${\displaystyle \delta }$, ${\displaystyle \alpha }$, ${\displaystyle \beta }$, ${\displaystyle \gamma }$ and ${\displaystyle \omega }$ are given constants.

The equation describes the motion of a damped oscillator with a more complicated potential than in simple harmonic motion (which corresponds to the case β=δ=0); in physical terms, it models, for example, a spring pendulum whose spring's stiffness does not exactly obey Hooke's law.

The Duffing equation is an example of a dynamical system that exhibits chaotic behavior. Moreover, the Duffing system presents in the frequency response the jump resonance phenomenon that is a sort of frequency hysteresis behaviour.

## Parameters

• ${\displaystyle \delta }$ controls the size of the damping.
• ${\displaystyle \alpha }$ controls the size of the stiffness.
• ${\displaystyle \beta }$ controls the amount of non-linearity in the restoring force. If ${\displaystyle \beta =0}$, the Duffing equation describes a damped and driven simple harmonic oscillator.
• ${\displaystyle \gamma }$ controls the amplitude of the periodic driving force. If ${\displaystyle \gamma =0}$ we have a system without driving force.
• ${\displaystyle \omega }$ controls the frequency of the periodic driving force.

## Methods of solution

In general, the Duffing equation does not admit an exact symbolic solution. However, many approximate methods work well:

• Expansion in a Fourier series will provide an equation of motion to arbitrary precision.
• The ${\displaystyle x^{3}}$ term, also called the Duffing term, can be approximated as small and the system treated as a perturbed simple harmonic oscillator.
• The Frobenius method yields a complicated but workable solution.
• Any of the various numeric methods such as Euler's method and Runge-Kutta can be used.
• The Homotopy analysis method (HAM) has also been reported for analytical solution of the Duffing equation.[1]

In the special case of the undamped (${\displaystyle \delta =0}$) and undriven (${\displaystyle \gamma =0}$) Duffing equation, an exact solution can be obtained using Jacobi's elliptic functions.

## Boundedness of the solution for the undamped and unforced oscillator

Multiplication of the undamped and unforced Duffing equation, ${\displaystyle \gamma =\delta =0,}$ with ${\displaystyle {\dot {x}}}$ gives:[2]

{\displaystyle {\begin{aligned}&{\dot {x}}\left({\ddot {x}}+\alpha x+\beta x^{3}\right)=0\\&\Rightarrow {\frac {\text{d}}{{\text{d}}t}}\left[{\tfrac {1}{2}}\left({\dot {x}}\right)^{2}+{\tfrac {1}{2}}\alpha x^{2}+{\tfrac {1}{4}}\beta x^{4}\right]=0\\&\Rightarrow {\tfrac {1}{2}}\left({\dot {x}}\right)^{2}+{\tfrac {1}{2}}\alpha x^{2}+{\tfrac {1}{4}}\beta x^{4}=H,\end{aligned}}}

with H a constant. The value of H is determined by the initial conditions ${\displaystyle x(0)}$ and ${\displaystyle {\dot {x}}(0).}$

The substitution ${\displaystyle y={\dot {x}}}$ in H shows that the system is Hamiltonian:

${\displaystyle {\dot {x}}=+{\frac {\partial H}{\partial y}},}$   ${\displaystyle {\dot {y}}=-{\frac {\partial H}{\partial x}}}$   with   ${\displaystyle \quad H={\tfrac {1}{2}}y^{2}+{\tfrac {1}{2}}\alpha x^{2}+{\tfrac {1}{4}}\beta x^{4}.}$

When both ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are positive, the solution is bounded:[2]

${\displaystyle |x|\leq {\sqrt {2H/\alpha }}}$   and   ${\displaystyle |{\dot {x}}|\leq {\sqrt {2H}},}$

with the Hamiltonian H being positive.

## References

### Inline

1. ^ Tajaddodianfar, Farid (2016). "Nonlinear dynamics of MEMS/NEMS resonators: analytical solution by the homotopy analysis method". Microsystem Technologies. doi:10.1007/s00542-016-2947-7.
2. ^ a b Bender & Orszag (1999, p. 546)

### Other

• Bender, C.M.; Orszag, S.A. (1999), Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer, pp. 545–551, ISBN 9780387989310
• Addison, P.S. (1997), Fractals and Chaos: An illustrated course, CRC Press, pp. 147–148, ISBN 9780849384431