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{{Use American English|date=January 2019}}
{{Use mdy dates|date=January 2019}}
{{Short description|Non-linear second order differential equation and its attractor}}
[[File:Duffing_oscillator.webm|thumb|Duffing oscillator plot, containing phase plot, trajectory, [[strange attractor]], Poincare section, and double well potential plot. The parameters are 
{{nowrap|<math>\alpha=-1</math>,}}
{{nowrap|<math>\beta=0.25</math>,}}
{{nowrap|<math>\delta=0.1</math>,}}
{{nowrap|<math>\gamma=2.5</math>,}} and
{{nowrap|<math>\omega=2</math>.}}]]

[[File:Forced Duffing equation Poincaré section.png|thumb|A [[Poincaré section]] of the forced Duffing equation suggesting chaotic behaviour {{nowrap|(<math>\alpha = 1</math>,}} {{nowrap|<math>\beta = 5</math>,}} {{nowrap|<math>\delta = 0.02</math>,}} {{nowrap|<math>\gamma = 8</math>,}} and {{nowrap|<math>\omega = 0.5</math>).}}]]

[[File:Duffing oscillator strange attractor with color.gif|thumb|The strange attractor of the Duffing oscillator, through 4 periods (<math>8\pi</math> time). Coloration shows how the points flow. 
{{nowrap|(<math>\alpha = -1</math>,}}
{{nowrap|<math>\beta = 1</math>,}}
{{nowrap|<math>\delta = 0.02</math>,}}
{{nowrap|<math>\gamma = 3</math>,}}
{{nowrap|<math>\omega = 1</math>.}}
The animation has time offset so driving force is {{nowrap|<math>\sin(\omega t)</math>}} rather than {{nowrap|<math>\cos(\omega t)</math>.)}}]]

The '''Duffing equation''' (or '''Duffing oscillator'''), named after [[Georg Duffing]] (1861–1944), is a [[non-linear]] second-order [[differential equation]] used to model certain [[Harmonic oscillator|damped and driven oscillator]]s. The equation is given by
<math display="block">\ddot{x} + \delta \dot{x} + \alpha x + \beta x^3 = \gamma \cos (\omega t),</math>
where the (unknown) function <math>x = x(t)</math> is the displacement at time {{mvar|t}}, <math>\dot{x}</math> is the first [[derivative]] of <math>x</math> with respect to time, i.e. [[velocity]], and <math>\ddot{x}</math> is the second time-derivative of <math>x,</math> i.e. [[acceleration]]. The numbers <math>\delta,</math> <math>\alpha,</math> <math>\beta,</math> <math>\gamma</math> and <math>\omega</math> are given constants.

The equation describes the motion of a damped oscillator with a more complex [[potential]] than in [[simple harmonic motion]] (which corresponds to the case <math>\beta=\delta=0</math>); in physical terms, it models, for example, an [[elastic pendulum]] whose spring's [[stiffness]] does not exactly obey [[Hooke's law]].

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

==Parameters==
The parameters in the above equation are:
*<math>\delta</math> controls the amount of [[Damping ratio|damping]],
*<math>\alpha</math> controls the linear [[stiffness]],
*<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]],
*<math>\gamma</math> is the [[amplitude]] of the periodic driving force; if <math>\gamma=0</math> the system is without a driving force, and
*<math>\omega</math> is the [[angular frequency]] of the periodic driving force.

The Duffing equation can be seen as describing the oscillations of a mass attached to a nonlinear [[spring (device)|spring]] and a linear damper. The restoring force provided by the nonlinear spring is then <math>\alpha x + \beta x^3.</math>

When <math>\alpha>0</math> and <math>\beta>0</math> the spring is called a ''hardening spring''. Conversely, for <math>\beta<0</math> it is a ''softening spring'' (still with <math>\alpha>0</math>). Consequently, the adjectives ''hardening'' and ''softening'' are used with respect to the Duffing equation in general, dependent on the values of <math>\beta</math> (and <math>\alpha</math>).<ref>{{cite book |title= Nonlinear Dynamics and Chaos |first1=J. M. T. |last1=Thompson |first2=H. B. |last2=Stewart |publisher= John Wiley & Sons |year=2002 |isbn=9780471876847 |page= 66}}</ref>

The number of parameters in the Duffing equation can be reduced by two through scaling (in accord with the [[Buckingham π theorem]]), e.g. the excursion <math>x</math> and time <math>t</math> can be scaled as:<ref>{{cite book |first1=R. |last1=Lifshitz |first2=M. C. |last2=Cross |chapter=Nonlinear mechanics of nanomechanical and micromechanical resonators |title=Reviews of Nonlinear Dynamics and Complexity |editor-last=Schuster |editor-first=H. G. |isbn=9783527407293 |lccn=2008459659 |year=2008 |publisher=Wiley |pages=8–9}}</ref> <math>\tau = t \sqrt{\alpha}</math> and <math>y = x \alpha/\gamma,</math> assuming <math>\alpha</math> is positive (other scalings are possible for different ranges of the parameters, or for different emphasis in the problem studied). Then:<ref name=BKCW2008>{{cite journal |title= On the jump-up and jump-down frequencies of the Duffing oscillator |journal= Journal of Sound and Vibration |volume= 318 |number= 4–5 |pages= 1250–1261 |year= 2008 |doi= 10.1016/j.jsv.2008.04.032 |first1= M. J. |last1= Brennan |first2= I. |last2= Kovacic |first3= A. |last3= Carrella |first4= T. P. |last4= Waters|bibcode= 2008JSV...318.1250B }}</ref>
<math display="block">\ddot{y} + 2 \eta\, \dot{y} + y + \varepsilon\, y^3 = \cos(\sigma\tau),</math>
where
*<math>\eta = \frac{\delta}{2\sqrt{\alpha}},</math>
*<math>\varepsilon = \frac{\beta\gamma^2}{\alpha^3},</math> and
*<math>\sigma = \frac{\omega}{\sqrt{\alpha}}.</math>

The dots denote differentiation of <math>y(\tau)</math> with respect to <math>\tau.</math> This shows that the solutions to the forced and damped Duffing equation can be described in terms of the three parameters (<math>\varepsilon</math>, <math>\eta</math>, and <math>\sigma</math>) and two [[initial conditions]] (i.e. for <math>y(t_0)</math> and <math>\dot{y}(t_0)</math>).

==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]] may provide an equation of motion to arbitrary precision.
*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.
*The [[Frobenius method]] yields a complex but workable solution.
*Any of the various [[numerical analysis|numeric methods]] such as [[Euler's method]] and [[Runge–Kutta methods]] can be used.
*The [[homotopy analysis method]] (HAM) has also been reported for obtaining approximate solutions of the Duffing equation, also for strong nonlinearity.<ref name="KovacicBrennan2011">{{Citation | title = The Duffing Equation: Nonlinear Oscillators and their Behaviour | editor1-first= I. | editor1-last= Kovacic | editor2-first= M. J. | editor2-last = Brennan | publisher= Wiley | year= 2011 | isbn= 978-0-470-71549-9 | pages = 123–127 }}</ref><ref name=TYP2016>{{cite journal |last1=Tajaddodianfar |first1=F. |first2=M. R. H. |last2=Yazdi |last3=Pishkenari |first3=H. N. |title=Nonlinear dynamics of MEMS/NEMS resonators: analytical solution by the homotopy analysis method|journal=Microsystem Technologies | date=2016 | volume=23 |issue=6 |pages=1913–1926 |doi=10.1007/s00542-016-2947-7|s2cid=113216381 }}</ref>

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]].<ref>{{Citation |first=R. H. |last=Rand |title=Lecture notes on nonlinear vibrations |publisher=Cornell University |version=53 |pages=13–17 |year=2012 |url=https://ecommons.cornell.edu/bitstream/handle/1813/28989/NonlinearVibrations_ver53.pdf?sequence=2&isAllowed=y}}</ref>

==Boundedness of the solution for the unforced oscillator==
===Undamped oscillator===
Multiplication of the undamped and unforced Duffing equation, <math>\gamma = \delta = 0,</math> with <math>\dot{x}</math> gives:<ref name="BO99">{{Citation | publisher= Springer | isbn= 9780387989310 | last1= Bender | first1= C. M. | author1-link= Carl M. Bender | first2= S. A. | last2= Orszag | author2-link= Steven Orszag | title= Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory | year= 1999 | page = 546 | bibcode= 1999amms.book.....B }}</ref>
<math display="block">
\begin{align}
  & \dot{x} \left( \ddot{x} + \alpha x + \beta x^3 \right) = 0
  \\[1ex]
  \Longrightarrow {} &
  \frac{\mathrm{d}}{\mathrm{d}t} \left[ \frac 1 2 \left( \dot{x} \right)^2 + \frac 1 2 \alpha x^2 + \frac 1 4 \beta x^4 \right] = 0
  \\[1ex]
  \Longrightarrow {} &
  \frac 1 2 \left( \dot{x} \right)^2 + \frac 1 2 \alpha x^2 + \frac 1 4 \beta x^4 = H,
\end{align}
</math>
with {{math|''H''}} a constant. The value of {{math|''H''}} is determined by the initial conditions <math>x(0)</math> and <math>\dot{x}(0).</math>

The substitution <math>y=\dot{x}</math> in ''H'' shows that the system is [[Hamiltonian system|Hamiltonian]]:
<math display="block">\begin{align}
&\dot{x} = + \frac{\partial H}{\partial y}, \qquad \dot{y} = - \frac{\partial H}{\partial x} \\[1ex]
\Longrightarrow {} & H = \tfrac 1 2 y^2 + \tfrac 1 2 \alpha x^2 + \tfrac 1 4 \beta x^4.
\end{align}</math>

When both <math>\alpha</math> and <math>\beta</math> are positive, the solution is bounded:<ref name="BO99" />
<math display="block"> |x| \leq \sqrt{2H/\alpha} \qquad \text{ and } \qquad |\dot{x}| \leq \sqrt{2H},</math>
with the Hamiltonian {{math|''H''}} being positive.

===Damped oscillator===
Similarly, the damped oscillator converges globally, by [[Lyapunov function]] method<ref>{{scholarpedia|title=Duffing oscillator|urlname=Duffing_oscillator|curator=Takashi Kanamaru}}</ref>
<math display="block">
\begin{align}
  & \dot{x} \left( \ddot{x} + \delta \dot{x} + \alpha x + \beta x^3 \right) = 0
  \\[1ex]
  \Longrightarrow{}&
  \frac{\mathrm{d}}{\mathrm{d}t} \left[ \frac 1 2 \left( \dot{x} \right)^2 + \frac 1 2 \alpha x^2 + \frac 1 4 \beta x^4 \right]
    = -\delta\, \left(\dot{x}\right)^2
  \\[1ex]
  \Longrightarrow{}&
  \frac{\mathrm{d}H}{\mathrm{d}t} = -\delta\, \left(\dot{x}\right)^2 \le 0,
\end{align}
</math>
since <math>\delta \ge 0</math> for damping. Without forcing the damped Duffing oscillator will end up at (one of) its ''stable'' [[equilibrium point]](s). The equilibrium points, stable and unstable, are at <math>\alpha x + \beta x^3 = 0.</math> If <math>\alpha>0</math> the stable equilibrium is at <math>x=0.</math> If <math>\alpha < 0</math> and <math>\beta > 0</math> the stable equilibria are at <math display="inline">x = +\sqrt{-\alpha/\beta}</math> and <math display="inline">x = -\sqrt{-\alpha/\beta}.</math>

==Frequency response==
The forced Duffing oscillator with cubic nonlinearity is described by the following ordinary differential equation:
<math display="block">\ddot{x} + \delta \dot{x} + \alpha x + \beta x^3 = \gamma \cos (\omega t).</math>

The [[frequency response]] of this oscillator describes the [[amplitude]] <math>z</math> of steady state response of the equation (i.e. <math>x(t)</math>) at a given [[frequency]] of excitation <math>\omega.</math> For a linear oscillator with <math>\beta=0,</math> the frequency response is also linear. However, for a nonzero cubic coefficient <math>\beta</math>, the frequency response becomes nonlinear. Depending on the type of nonlinearity, the Duffing oscillator can show hardening, softening or mixed hardening–softening frequency response. Anyway, using the [[homotopy analysis method]] or [[harmonic balance]], one can derive a frequency response equation in the following form:<ref name=JordanSmith>{{harvnb|Jordan|Smith|2007|pages=223–233}}</ref><ref name=TYP2016/>
<math display="block">\left[\left(\omega^2 - \alpha - \tfrac{3}{4} \beta z^2\right)^2 + \left(\delta\omega\right)^2\right] \, z^2 = \gamma^2.</math>

For the parameters of the Duffing equation, the above algebraic equation gives the [[steady state]] oscillation amplitude <math>z</math> at a given excitation frequency.

{{math proof
| title = Derivation of the frequency response
| proof = Using the method of harmonic balance, an approximate solution to the Duffing equation is sought of the form:<ref name=JordanSmith/>
<math display="block">x = a\, \cos(\omega t) + b\, \sin(\omega t) = z\, \cos(\omega t - \phi),</math>
with <math>z^2=a^2+b^2</math> and <math>\tan\phi = \frac{b}{a}.</math>

Application in the Duffing equation leads to:
<math display="block">
\begin{align}
  & \left( -\omega^2\, a + \omega\,\delta\,b + \alpha\,a + \tfrac34\,\beta\,a^3 + \tfrac34\,\beta\,a\,b^2 - \gamma \right)\,
      \cos \left( \omega\,t \right)
  \\
  & + \left( -\omega^2\, b - \omega\,\delta\,a + \tfrac34\,\beta\,b^3 + \alpha\,b + \tfrac34\,\beta\,a^2\,b \right)\,
      \sin \left( \omega\,t \right)
  \\
  & + \left( \tfrac14\,\beta\,a^3 - \tfrac34\,\beta\,a\,b^2 \right)\, \cos \left( 3 \omega t \right)
    + \left( \tfrac34\,\beta\,a^2\,b - \tfrac14\,\beta\,b^3 \right)\, \sin \left( 3 \omega t \right)
  = 0.
\end{align}
</math>

Neglecting the [[superharmonic]]s at <math>3\omega,</math> the two terms preceding <math>\cos(\omega t)</math> and <math>\sin(\omega t)</math> have to be zero. As a result,
<math display="block">
\begin{align}
  & -\omega^2\, a + \omega\,\delta\,b + \alpha\,a + \tfrac34\,\beta\,a^3 + \tfrac34\,\beta\,a\,b^2 = \gamma
  \qquad \text{and}
  \\
  & -\omega^2\, b - \omega\,\delta\,a + \tfrac34\,\beta\,b^3 + \alpha\,b + \tfrac34\,\beta\,a^2\,b = 0.
\end{align}
</math>

Squaring both equations and adding leads to the amplitude frequency response:
<math display="block">\left[\left(\omega^2-\alpha-\frac{3}{4}\beta z^2\right)^2 + \left(\delta\omega\right)^2\right]\,z^2=\gamma^2,</math>
as stated above.
}}
<gallery mode="nolines" widths="300" heights="300">
File:Duffing frequency response.svg|alt=Frequency response  as a function of  for the Duffing equation, with  and damping  The dashed parts of the frequency response are unstable.[3]|Frequency response <math>z/\gamma</math> as a function of <math>\omega/\sqrt{\alpha}</math> for the Duffing equation, with <math>\alpha = \gamma=1</math> and damping <math>\delta = 0.1</math>. The dashed parts of the frequency response are unstable.<ref name="BKCW2008" />
File:Duffing 3D surface plot.png|The same plot as a 3D diagram. Varying <math>\beta</math> is shown along a separate axis.
</gallery>

=== Graphically solving for frequency response ===

We may graphically solve for <math>z^2</math> as the intersection of two curves in the <math>(z^2, y)</math> plane:<math display="block">\begin{cases}
y = \left(\omega^2-\alpha-\frac{3}{4} \beta z^2\right)^{2}+\left(\delta\omega\right)^2 \\[1ex]
y = \dfrac{\gamma^2}{z^2}
\end{cases}</math>For fixed <math>\alpha, \delta, \gamma</math>, the second curve is a fixed hyperbola in the first quadrant. The first curve is a parabola with shape <math display="inline">y = \tfrac{9}{16}\beta^2 (z^2)^2</math>, and apex at location <math display="inline">(\tfrac{4}{3\beta}(\omega^2 - \alpha), \delta^2 \omega^2)</math>. If we fix <math>\beta</math> and vary <math>\omega</math>, then the apex of the parabola moves along the line <math display="inline">y = \tfrac{3}{4}\beta\delta^2 (z^2)+ \delta^2\alpha</math>.

Graphically, then, we see that if <math>\beta</math> is a large positive number, then as <math>\omega</math> varies, the parabola intersects the hyperbola at one point, then three points, then one point again. Similarly we can analyze the case when <math>\beta</math> is a large negative number.

===Jumps===
[[File:Duffing response jumps.svg|thumb|right|Jumps in the frequency response. The parameters are: <math>\alpha = \gamma = 1,</math>, <math>\beta = 0.04</math>, and <math>\delta = 0.1</math>.<ref name=JordanSmith/>]]
For certain ranges of the parameters in the Duffing equation, the frequency response may no longer be a [[single-valued function]] of forcing frequency <math>\omega.</math> For a hardening spring oscillator (<math>\alpha>0</math> and large enough positive <math>\beta>\beta_{c+} > 0</math>) the frequency response overhangs to the high-frequency side, and to the low-frequency side for the softening spring oscillator (<math>\alpha>0</math> and <math>\beta<\beta_{c-} < 0</math>). The lower overhanging side is unstable – i.e. the dashed-line parts in the figures of the frequency response – and cannot be realized for a sustained time. Consequently, the jump phenomenon shows up: 
* when the angular frequency <math>\omega</math> is slowly increased (with other parameters fixed), the response [[amplitude]] <math>z</math> drops at A suddenly to B,
* if the frequency <math>\omega</math> is slowly decreased, then at C the amplitude jumps up to D, thereafter following the upper branch of the frequency response. 
The jumps A–B and C–D do not coincide, so the system shows [[hysteresis]] depending on the frequency sweep direction.<ref name=JordanSmith/>

=== Transition to chaos ===
The above analysis assumed that the base frequency response dominates (necessary for performing harmonic balance), and higher frequency responses are negligible. This assumption fails to hold when the forcing is sufficiently strong. Higher order harmonics cannot be neglected, and the dynamics become chaotic. There are different possible transitions to chaos, most commonly by successive period doubling.<ref>{{Cite journal |last=Ueda |first=Yoshisuke |date=1991-01-01 |title=Survey of regular and chaotic phenomena in the forced Duffing oscillator |url=https://dx.doi.org/10.1016/0960-0779%2891%2990032-5 |journal=Chaos, Solitons & Fractals |language=en |volume=1 |issue=3 |pages=199–231 |doi=10.1016/0960-0779(91)90032-5 |bibcode=1991CSF.....1..199U |issn=0960-0779}}</ref>

==Examples==
{{Multiple image
| direction=vertical
| align=right
| width=300
| header=Time traces and phase portraits
| image1 = Duffing gam0.20 ome1.2 del0.3.svg
| caption1 = period-1 oscillation at <math>\gamma=0.20</math>
| image2 = Duffing gam0.28 ome1.2 del0.3.svg
| caption2 = period-2 oscillation at <math>\gamma=0.28</math>
| image3 = Duffing gam0.29 ome1.2 del0.3.svg
| caption3 = period-4 oscillation at <math>\gamma=0.29</math>
| image4 = Duffing gam0.37 ome1.2 del0.3.svg
| caption4 = period-5 oscillation at <math>\gamma=0.37</math>
| image5 = Duffing gam0.50 ome1.2 del0.3.svg
| caption5 = chaos at <math>\gamma=0.50</math>
| image6 = Duffing gam0.65 ome1.2 del0.3.svg
| caption6 = period-2 oscillation at <math>\gamma=0.65</math>
}}
Some typical examples of the [[time series]] and [[phase portrait]]s of the Duffing equation, showing the appearance of [[subharmonic]]s through [[period-doubling bifurcation]] – as well [[chaos theory|chaotic behavior]] – are shown in the figures below. The forcing amplitude increases from <math>\gamma = 0.20</math> to {{nowrap|<math>\gamma = 0.65</math>.}} The other parameters have the values: {{nowrap|<math>\alpha = -1</math>,}} {{nowrap|<math>\beta = +1</math>,}} <math>\delta = 0.3</math> and {{nowrap|<math>\omega = 1.2</math>.}} The initial conditions are <math>x(0) = 1</math> and <math>\dot{x}(0) = 0.</math> The red dots in the phase portraits are at times <math>t</math> which are an [[integer]] multiple of the [[frequency|period]] {{nowrap|<math>T = 2\pi / \omega</math>.}}<ref>Based on the examples shown in {{harvnb|Jordan|Smith|2007|pages=453–462}}.</ref>

==References==

===Citations===
{{Reflist|2}}

===Bibliography===
{{refbegin}}
*{{Citation
| first=G. 
| last=Duffing
| title=Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung
| publisher=Vieweg
| volume=Heft 41/42
| location=Braunschweig
| year=1918
| pages=vi+134&nbsp;pp
| no-pp=yes
| language=de
| trans-title=Forced oscillations with variable natural frequency and their technical relevance
| oclc=12003652
}}
*{{Citation
| publisher= CRC Press
| isbn= 9780849384431
| last= Addison
| first= P. S.
| title= Fractals and Chaos: An illustrated course
| year= 1997
| pages= 147–148
}}
*{{Citation
| title= Nonlinear ordinary differential equations – An introduction for scientists and engineers
| edition= 4th
| first1= D. W.
| last1= Jordan
| first2= P.
| last2= Smith
| year= 2007
| publisher= Oxford University Press
| isbn= 978-0-19-920824-1
}}
{{Refend}}

==External links==
*[http://scholarpedia.org/article/Duffing_oscillator Duffing oscillator on Scholarpedia]
*[http://mathworld.wolfram.com/DuffingDifferentialEquation.html MathWorld page]
* {{cite journal  
| last1=Pchelintsev | first1=A. N.
| last2=Ahmad | first2=S.
| title=Solution of the Duffing equation by the power series method
| journal=Transactions of TSTU 
| year = 2020 | volume = 26 | issue = 1 | pages=118–123
| url=http://vestnik.tstu.ru/rus/t_26/pdf/26_1_013.pdf 
}}

{{Chaos theory}}

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