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| | Greetings! I am Marvella and I really feel comfortable when people use the full title. Playing baseball is the pastime he will by no means stop performing. California is where I've always been residing and I adore each working day living here. Bookkeeping is her working day job now.<br><br>Also visit my page ... [http://www.ninfeta.tv/blog/99493 http://www.ninfeta.tv/blog/99493] |
| [[File:Forced Duffing equation Poincaré section.png|thumb|A [[Poincaré section]] of the forced Duffing equation suggesting chaotic behaviour]]
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| The '''Duffing equation''', named after [[Georg Duffing]], is a [[non-linear]] second-order [[differential equation]] used to model certain [[damping|damped]] and driven [[oscillator]]s. The equation is given by
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| :<math>\ddot{x} + \delta \dot{x} + \alpha x + \beta x^3 = \gamma \cos (\omega t)\,</math>
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| where the (unknown) function ''x''=''x''(''t'') is the displacement at time ''t'', <math>\dot{x}</math> is the first [[derivative]] of ''x'' with respect to time, i.e. [[velocity]], and <math>\ddot{x}</math> is the second time-derivative of ''x'', i.e. [[acceleration]]. The numbers <math>\delta</math>, <math>\alpha</math>, <math>\beta</math>, <math>\gamma</math> and <math>\omega</math> are given constants.
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| 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]].
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| The Duffing equation is an example of a dynamical system that exhibits [[chaos theory|chaotic behavior]].
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| Moreover the Duffing system presents in the frequency response the jump resonance phenomenon that is a sort of frequency hysteresis behaviour.
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| ==Parameters==
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| * <math>\delta</math> controls the size of the [[damping]].
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| * <math>\alpha</math> controls the size of the [[stiffness]].
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| * <math>\beta</math> controls the amount of non-linearity in the restoring force. If <math>\beta=0</math>, the Duffing equation describes a damped and driven simple harmonic oscillator.
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| * <math>\gamma</math> controls the [[amplitude]] of the periodic driving force. If <math>\gamma=0</math> we have a system without driving force.
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| * <math>\omega</math> controls the [[frequency]] of the periodic driving force.
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| ==Methods of solution==
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| [[File:Duffing oscillator limit cycle.gif|thumb|right|300px|Duffing oscillator limit cycle γ>0]]
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| [[File:Duffing oscillator limit cycle phase animation.gif|thumb|right|300px|Duffing oscillator limit cycle phase animation γ>0]]
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| [[File:Duffing oscillator chaos.gif|thumb|300px|right|Duffing oscillator chaos oscillation γ<0]]
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| [[File:Duffing oscillator attractors animation.gif|thumb|right|300px|Duffing oscillator attractors animation γ<0]]
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| In general, the Duffing equation does not admit an exact symbolic solution. However, many approximate methods work well:
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| * Expansion in a [[Fourier series]] will provide an equation of motion to arbitrary precision.
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| * The <math>x^3</math> term, also called the ''Duffing term'', can be approximated as small and the system treated as a [[perturbation theory|perturbed]] simple harmonic oscillator.
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| * The [[Frobenius method]] yields a complicated but workable solution.
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| * Any of the various [[numerical analysis|numeric methods]] such as [[Euler's method]] and [[Runge-Kutta]] can be used.
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| In the special case of the undamped (<math>\delta = 0</math>) and undriven (<math>\gamma = 0</math>) Duffing equation, an exact solution can be obtained using [[Jacobi's elliptic functions]].
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| ==References==
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| {{reflist}}
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| == External links ==
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| * [http://scholarpedia.org/article/Duffing_oscillator Duffing oscillator on Scholarpedia]
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| * [http://mathworld.wolfram.com/DuffingDifferentialEquation.html MathWorld page]
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| {{Chaos theory}}
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| [[Category:Ordinary differential equations]]
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| {{mathapplied-stub}}
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Greetings! I am Marvella and I really feel comfortable when people use the full title. Playing baseball is the pastime he will by no means stop performing. California is where I've always been residing and I adore each working day living here. Bookkeeping is her working day job now.
Also visit my page ... http://www.ninfeta.tv/blog/99493