Duffing equation

The parameters in the above equation are:

• $\backslash delta$ controls the amount of damping,
• $\backslash alpha$ controls the linear stiffness,
• $\backslash beta$ controls the amount of non-linearity in the restoring force; if $\backslash beta=0,$ the Duffing equation describes a damped and driven simple harmonic oscillator,
• $\backslash gamma$ is the amplitude of the periodic driving force; if $\backslash gamma=0$ the system is without a driving force, and
• $\backslash omega$ 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 and a linear damper. The restoring force provided by the nonlinear spring is then $\backslash alpha\; x\; +\; \backslash beta\; x^3.$

When $\backslash alpha>0$ and $\backslash beta>0$ the spring is called a hardening spring. Conversely, for it is a softening spring (still with $\backslash alpha>0$). Consequently, the adjectives hardening and softening are used with respect to the Duffing equation in general, dependent on the values of $\backslash beta$ (and $\backslash alpha$).{{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}}

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 $x$ and time $t$ can be scaled as:{{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}} $\backslash tau\; =\; t\; \backslash sqrt\{\backslash alpha\}$ and $y\; =\; x\; \backslash alpha/\backslash gamma,$ assuming $\backslash alpha$ is positive (other scalings are possible for different ranges of the parameters, or for different emphasis in the problem studied). Then:{{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 }}

$$\backslash ddot\{y\}\; +\; 2\; \backslash eta\backslash ,\; \backslash dot\{y\}\; +\; y\; +\; \backslash varepsilon\backslash ,\; y^3\; =\; \backslash cos(\backslash sigma\backslash tau),$$

where

• $\backslash eta\; =\; \backslash frac\{\backslash delta\}\{2\backslash sqrt\{\backslash alpha\}\},$
• $\backslash varepsilon\; =\; \backslash frac\{\backslash beta\backslash gamma^2\}\{\backslash alpha^3\},$ and
• $\backslash sigma\; =\; \backslash frac\{\backslash omega\}\{\backslash sqrt\{\backslash alpha\}\}.$

The dots denote differentiation of $y(\backslash tau)$ with respect to $\backslash tau.$ This shows that the solutions to the forced and damped Duffing equation can be described in terms of the three parameters ($\backslash varepsilon$, $\backslash eta$, and $\backslash sigma$) and two initial conditions (i.e. for $y(t\_0)$ and $\backslash dot\{y\}(t\_0)$).

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 $x^3$ 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 complex but workable solution.
• Any of the various 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.{{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 }}{{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 }}

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

Multiplication of the undamped and unforced Duffing equation, $\backslash gamma\; =\; \backslash delta\; =\; 0,$ with $\backslash dot\{x\}$ gives:{{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 }}

\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}

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

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

$$\backslash begin\{align\}$$

&\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}

When both $\backslash alpha$ and $\backslash beta$ are positive, the solution is bounded:

$$|x|\; \backslash leq\; \backslash sqrt\{2H/\backslash alpha\}\; \backslash qquad\; \backslash text\{\; and\; \}\; \backslash qquad\; |\backslash dot\{x\}|\; \backslash leq\; \backslash sqrt\{2H\},$$

with the Hamiltonian {{math|H}} being positive. This bound on $x$ comes from dropping the term with $\backslash beta$. Including it gives a smaller but more complicated bound, by solving $(\backslash beta/4)x^4\; +\; (\backslash alpha/2)x^2\; -\; H\; =\; 0$, a quadratic equation for $x^2$.

Similarly, the damped oscillator converges globally, by Lyapunov function method{{scholarpedia|title=Duffing oscillator|urlname=Duffing_oscillator|curator=Takashi Kanamaru}}

\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}

since $\backslash delta\; \backslash ge\; 0$ 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 $\backslash alpha\; x\; +\; \backslash beta\; x^3\; =\; 0.$ If $\backslash alpha>0$ the stable equilibrium is at $x=0.$ If and $\backslash beta\; >\; 0$ the stable equilibria are at $x\; =\; +\backslash sqrt\{-\backslash alpha/\backslash beta\}$ and $x\; =\; -\backslash sqrt\{-\backslash alpha/\backslash beta\}.$

The forced Duffing oscillator with cubic nonlinearity is described by the following ordinary differential equation:

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

The frequency response of this oscillator describes the amplitude $z$ of steady state response of the equation (i.e. $x(t)$) at a given frequency of excitation $\backslash omega.$ For a linear oscillator with $\backslash beta=0,$ the frequency response is also linear. However, for a nonzero cubic coefficient $\backslash beta$, 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:{{harvnb|Jordan|Smith|2007|pages=223–233}}

$$\backslash left[\backslash left(\backslash omega^2\; -\; \backslash alpha\; -\; \backslash tfrac\{3\}\{4\}\; \backslash beta\; z^2\backslash right)^2\; +\; \backslash left(\backslash delta\backslash omega\backslash right)^2\backslash right]\; \backslash ,\; z^2\; =\; \backslash gamma^2.$$

For the parameters of the Duffing equation, the above algebraic equation gives the steady state oscillation amplitude $z$ 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:

$$x\; =\; a\backslash ,\; \backslash cos(\backslash omega\; t)\; +\; b\backslash ,\; \backslash sin(\backslash omega\; t)\; =\; z\backslash ,\; \backslash cos(\backslash omega\; t\; -\; \backslash phi),$$

with $z^2=a^2+b^2$ and $\backslash tan\backslash phi\; =\; \backslash frac\{b\}\{a\}.$

Application in the Duffing equation leads to:

\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}

Neglecting the superharmonics at $3\backslash omega,$ the two terms preceding $\backslash cos(\backslash omega\; t)$ and $\backslash sin(\backslash omega\; t)$ have to be zero. As a result,

\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}

Squaring both equations and adding leads to the amplitude frequency response:

$$\backslash left[\backslash left(\backslash omega^2-\backslash alpha-\backslash frac\{3\}\{4\}\backslash beta\; z^2\backslash right)^2\; +\; \backslash left(\backslash delta\backslash omega\backslash right)^2\backslash right]\backslash ,z^2=\backslash gamma^2,$$

as stated above.

}}

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 $z/\backslash gamma$ as a function of $\backslash omega/\backslash sqrt\{\backslash alpha\}$ for the Duffing equation, with $\backslash alpha\; =\; \backslash gamma=1$ and damping $\backslash delta\; =\; 0.1$. The dashed parts of the frequency response are unstable.

File:Duffing 3D surface plot.png|The same plot as a 3D diagram. Varying $\backslash beta$ is shown along a separate axis.

We may graphically solve for $z^2$ as the intersection of two curves in the $(z^2,\; y)$ plane:$$\backslash begin\{cases\}$$

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}For fixed $\backslash alpha,\; \backslash delta,\; \backslash gamma$, the second curve is a fixed hyperbola in the first quadrant. The first curve is a parabola with shape $y\; =\; \backslash tfrac\{9\}\{16\}\backslash beta^2\; (z^2)^2$, and apex at location $(\backslash tfrac\{4\}\{3\backslash beta\}(\backslash omega^2\; -\; \backslash alpha),\; \backslash delta^2\; \backslash omega^2)$. If we fix $\backslash beta$ and vary $\backslash omega$, then the apex of the parabola moves along the line $y\; =\; \backslash tfrac\{3\}\{4\}\backslash beta\backslash delta^2\; (z^2)+\; \backslash delta^2\backslash alpha$.

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

File:Duffing response jumps.svg

For certain ranges of the parameters in the Duffing equation, the frequency response may no longer be a single-valued function of forcing frequency $\backslash omega.$ For a hardening spring oscillator ($\backslash alpha>0$ and large enough positive $\backslash beta>\backslash beta\_\{c+\}\; >\; 0$) the frequency response overhangs to the high-frequency side, and to the low-frequency side for the softening spring oscillator ($\backslash alpha>0$ and ). 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 $\backslash omega$ is slowly increased (with other parameters fixed), the response amplitude $z$ drops at A suddenly to B,
• if the frequency $\backslash omega$ 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.

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.{{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|url-access=subscription }}

{{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 $\backslash gamma=0.20$

| image2 = Duffing gam0.28 ome1.2 del0.3.svg

| caption2 = period-2 oscillation at $\backslash gamma=0.28$

| image3 = Duffing gam0.29 ome1.2 del0.3.svg

| caption3 = period-4 oscillation at $\backslash gamma=0.29$

| image4 = Duffing gam0.37 ome1.2 del0.3.svg

| caption4 = period-5 oscillation at $\backslash gamma=0.37$

| image5 = Duffing gam0.50 ome1.2 del0.3.svg

| caption5 = chaos at $\backslash gamma=0.50$

| image6 = Duffing gam0.65 ome1.2 del0.3.svg

| caption6 = period-2 oscillation at $\backslash gamma=0.65$

}}

Some typical examples of the time series and phase portraits of the Duffing equation, showing the appearance of subharmonics through period-doubling bifurcation – as well chaotic behavior – are shown in the figures below. The forcing amplitude increases from $\backslash gamma\; =\; 0.20$ to {{nowrap|$\backslash gamma\; =\; 0.65$.}} The other parameters have the values: {{nowrap|$\backslash alpha\; =\; -1$,}} {{nowrap|$\backslash beta\; =\; +1$,}} $\backslash delta\; =\; 0.3$ and {{nowrap|$\backslash omega\; =\; 1.2$.}} The initial conditions are $x(0)\; =\; 1$ and $\backslash dot\{x\}(0)\; =\; 0.$ The red dots in the phase portraits are at times $t$ which are an integer multiple of the period {{nowrap|$T\; =\; 2\backslash pi\; /\; \backslash omega$.}}Based on the examples shown in {{harvnb|Jordan|Smith|2007|pages=453–462}}.

{{Reflist|2}}

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{{Refend}}

• [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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