Duffing equation

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

{\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, ${\dot {x}}$ is the first derivative of x with respect to time, i.e. velocity, and ${\ddot {x}}$ is the second time-derivative of x, i.e. acceleration. The numbers $\delta$ , $\alpha$ , $\beta$ , $\gamma$ and $\omega$ 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

$\delta$ controls the size of the damping.
$\alpha$ controls the size of the stiffness.
$\beta$ controls the amount of non-linearity in the restoring force. If $\beta =0$ , the Duffing equation describes a damped and driven simple harmonic oscillator.
$\gamma$ controls the amplitude of the periodic driving force. If $\gamma =0$ we have a system without driving force.
$\omega$ 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 $x^{3}$ 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.

In the special case of the undamped ( $\delta =0$ ) and undriven ( $\gamma =0$ ) 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, $\gamma =\delta =0,$ with ${\dot {x}}$ gives:^[1]

{\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 $x(0)$ and ${\dot {x}}(0).$

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

{\dot {x}}=+{\frac {\partial H}{\partial y}},

{\dot {y}}=-{\frac {\partial H}{\partial x}}

with

\quad H={\tfrac {1}{2}}y^{2}+{\tfrac {1}{2}}\alpha x^{2}+{\tfrac {1}{4}}\beta x^{4}.

When both $\alpha$ and $\beta$ are positive, the solution is bounded:^[1]

|x|\leq {\sqrt {2H/\alpha }}

and

|{\dot {x}}|\leq {\sqrt {2H}},

with the Hamiltonian H being positive.

References

Inline

^ ^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

External links

[BO99-1] Bender & Orszag (1999, p. 546)

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