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Duffing equation
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<h2 id="parameters">Parameters</h2> <p>The parameters in the above equation are: </p> <ul><li> δ {\displaystyle \delta } controls the amount of <a href="/facts/Damping_ratio/UW3RsjEd">damping</a>,</li> <li> α {\displaystyle \alpha } controls the linear <a href="/facts/Stiffness/xlabLkfT">stiffness</a>,</li> <li> β {\displaystyle \beta } controls the amount of non-linearity in the restoring force; if β = 0 , {\displaystyle \beta =0,} the Duffing equation describes a damped and driven simple <a href="/facts/Harmonic_oscillator/7IQNToGt">harmonic oscillator</a>,</li> <li> γ {\displaystyle \gamma } is the <a href="/facts/Amplitude/aEckuXJM">amplitude</a> of the periodic driving force; if γ = 0 {\displaystyle \gamma =0} the system is without a driving force, and</li> <li> ω {\displaystyle \omega } is the <a href="/facts/Angular_frequency/VqLm3L3R">angular frequency</a> of the periodic driving force.</li></ul> <p>The Duffing equation can be seen as describing the oscillations of a mass attached to a nonlinear <a href="/facts/Spring_(device)/jknGXzTq">spring</a> and a linear damper. The restoring force provided by the nonlinear spring is then α x + β x 3 . {\displaystyle \alpha x+\beta x^{3}.} </p><p>When α > 0 {\displaystyle \alpha >0} and β > 0 {\displaystyle \beta >0} the spring is called a <i>hardening spring</i>. Conversely, for β < 0 {\displaystyle \beta <0} it is a <i>softening spring</i> (still with α > 0 {\displaystyle \alpha >0} ). Consequently, the adjectives <i>hardening</i> and <i>softening</i> are used with respect to the Duffing equation in general, dependent on the values of β {\displaystyle \beta } (and α {\displaystyle \alpha } ).<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p><p>The number of parameters in the Duffing equation can be reduced by two through scaling (in accord with the <a href="/facts/Buckingham_%25CF%2580_theorem/R3j8Dl2w">Buckingham π theorem</a>), e.g. the excursion x {\displaystyle x} and time t {\displaystyle t} can be scaled as:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> τ = t α {\displaystyle \tau =t{\sqrt {\alpha }}} and y = x α / γ , {\displaystyle y=x\alpha /\gamma ,} assuming α {\displaystyle \alpha } is positive (other scalings are possible for different ranges of the parameters, or for different emphasis in the problem studied). Then:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> y ¨ + 2 η y ˙ + y + ε y 3 = cos ( σ τ ) , {\displaystyle {\ddot {y}}+2\eta \,{\dot {y}}+y+\varepsilon \,y^{3}=\cos(\sigma \tau ),} where </p> <ul><li> η = δ 2 α , {\displaystyle \eta ={\frac {\delta }{2{\sqrt {\alpha }}}},} </li> <li> ε = β γ 2 α 3 , {\displaystyle \varepsilon ={\frac {\beta \gamma ^{2}}{\alpha ^{3}}},} and</li> <li> σ = ω α . {\displaystyle \sigma ={\frac {\omega }{\sqrt {\alpha }}}.} </li></ul> <p>The dots denote differentiation of y ( τ ) {\displaystyle y(\tau )} with respect to τ . {\displaystyle \tau .} This shows that the solutions to the forced and damped Duffing equation can be described in terms of the three parameters ( ε {\displaystyle \varepsilon } , η {\displaystyle \eta } , and σ {\displaystyle \sigma } ) and two <a href="/facts/Initial_conditions/sA4ohFP4">initial conditions</a> (i.e. for y ( t 0 ) {\displaystyle y(t_{0})} and y ˙ ( t 0 ) {\displaystyle {\dot {y}}(t_{0})} ). </p> <h2 id="methods-of-solution">Methods of solution</h2> <p>In general, the Duffing equation does not admit an exact symbolic solution. However, many approximate methods work well: </p> <ul><li>Expansion in a <a href="/facts/Fourier_series/s3fsxPAT">Fourier series</a> may provide an equation of motion to arbitrary precision.</li> <li>The x 3 {\displaystyle x^{3}} term, also called the <i>Duffing term</i>, can be approximated as small and the system treated as a <a href="/facts/Perturbation_theory/JWrNgbVm">perturbed</a> simple harmonic oscillator.</li> <li>The <a href="/facts/Frobenius_method/Jj0LHFkC">Frobenius method</a> yields a complex but workable solution.</li> <li>Any of the various <a href="/facts/Numerical_analysis/QT9cYa9z">numeric methods</a> such as <a href="/facts/Euler%2527s_method/7yF5B4z9">Euler's method</a> and <a href="/facts/Runge%25E2%2580%2593Kutta_methods/YRjbyHpY">Runge–Kutta methods</a> can be used.</li> <li>The <a href="/facts/Homotopy_analysis_method/kGVyYLYX">homotopy analysis method</a> (HAM) has also been reported for obtaining approximate solutions of the Duffing equation, also for strong nonlinearity.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li></ul> <p>In the special case of the <a href="/facts/Undamped/UW3RsjEd">undamped</a> ( δ = 0 {\displaystyle \delta =0} ) and undriven ( γ = 0 {\displaystyle \gamma =0} ) Duffing equation, an exact solution can be obtained using <a href="/facts/Jacobi%2527s_elliptic_functions/BLk6Rxew">Jacobi's elliptic functions</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> <h2 id="boundedness-of-the-solution-for-the-unforced-oscillator">Boundedness of the solution for the unforced oscillator</h2> <h3>Undamped oscillator</h3> <p>Multiplication of the undamped and unforced Duffing equation, γ = δ = 0 , {\displaystyle \gamma =\delta =0,} with x ˙ {\displaystyle {\dot {x}}} gives:<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> x ˙ ( x ¨ + α x + β x 3 ) = 0 ⟹ d d t [ 1 2 ( x ˙ ) 2 + 1 2 α x 2 + 1 4 β x 4 ] = 0 ⟹ 1 2 ( x ˙ ) 2 + 1 2 α x 2 + 1 4 β x 4 = H , {\displaystyle {\begin{aligned}&{\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{aligned}}} with <i>H</i> a constant. The value of <i>H</i> is determined by the initial conditions x ( 0 ) {\displaystyle x(0)} and x ˙ ( 0 ) . {\displaystyle {\dot {x}}(0).} </p><p>The substitution y = x ˙ {\displaystyle y={\dot {x}}} in <i>H</i> shows that the system is <a href="/facts/Hamiltonian_system/1jx7lA7t">Hamiltonian</a>: x ˙ = + ∂ H ∂ y , y ˙ = − ∂ H ∂ x ⟹ H = 1 2 y 2 + 1 2 α x 2 + 1 4 β x 4 . {\displaystyle {\begin{aligned}&{\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{aligned}}} </p><p>When both α {\displaystyle \alpha } and β {\displaystyle \beta } are positive, the solution is bounded:<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> | x | ≤ 2 H / α and | x ˙ | ≤ 2 H , {\displaystyle |x|\leq {\sqrt {2H/\alpha }}\qquad {\text{ and }}\qquad |{\dot {x}}|\leq {\sqrt {2H}},} with the Hamiltonian <i>H</i> being positive. This bound on x {\displaystyle x} comes from dropping the term with β {\displaystyle \beta } . Including it gives a smaller but more complicated bound, by solving ( β / 4 ) x 4 + ( α / 2 ) x 2 − H = 0 {\displaystyle (\beta /4)x^{4}+(\alpha /2)x^{2}-H=0} , a quadratic equation for x 2 {\displaystyle x^{2}} . </p> <h3>Damped oscillator</h3> <p>Similarly, the damped oscillator converges globally, by <a href="/facts/Lyapunov_function/RB590Utb">Lyapunov function</a> method<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> x ˙ ( x ¨ + δ x ˙ + α x + β x 3 ) = 0 ⟹ d d t [ 1 2 ( x ˙ ) 2 + 1 2 α x 2 + 1 4 β x 4 ] = − δ ( x ˙ ) 2 ⟹ d H d t = − δ ( x ˙ ) 2 ≤ 0 , {\displaystyle {\begin{aligned}&{\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}\leq 0,\end{aligned}}} since δ ≥ 0 {\displaystyle \delta \geq 0} for damping. Without forcing the damped Duffing oscillator will end up at (one of) its <i>stable</i> <a href="/facts/Equilibrium_point/DgbGMPkg">equilibrium point</a>(s). The equilibrium points, stable and unstable, are at α x + β x 3 = 0. {\displaystyle \alpha x+\beta x^{3}=0.} If α > 0 {\displaystyle \alpha >0} the stable equilibrium is at x = 0. {\displaystyle x=0.} If α < 0 {\displaystyle \alpha <0} and β > 0 {\displaystyle \beta >0} the stable equilibria are at x = + − α / β {\textstyle x=+{\sqrt {-\alpha /\beta }}} and x = − − α / β . {\textstyle x=-{\sqrt {-\alpha /\beta }}.} </p> <h2 id="frequency-response">Frequency response</h2> <p>The forced Duffing oscillator with cubic nonlinearity is described by the following ordinary differential equation: x ¨ + δ x ˙ + α x + β x 3 = γ cos ( ω t ) . {\displaystyle {\ddot {x}}+\delta {\dot {x}}+\alpha x+\beta x^{3}=\gamma \cos(\omega t).} </p><p>The <a href="/facts/Frequency_response/zsHmnCJ4">frequency response</a> of this oscillator describes the <a href="/facts/Amplitude/aEckuXJM">amplitude</a> z {\displaystyle z} of steady state response of the equation (i.e. x ( t ) {\displaystyle x(t)} ) at a given <a href="/facts/Frequency/QUSbPaW8">frequency</a> of excitation ω . {\displaystyle \omega .} For a linear oscillator with β = 0 , {\displaystyle \beta =0,} the frequency response is also linear. However, for a nonzero cubic coefficient β {\displaystyle \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 <a href="/facts/Homotopy_analysis_method/kGVyYLYX">homotopy analysis method</a> or <a href="/facts/Harmonic_balance/w6ICbkHY">harmonic balance</a>, one can derive a frequency response equation in the following form:<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> [ ( ω 2 − α − 3 4 β z 2 ) 2 + ( δ ω ) 2 ] z 2 = γ 2 . {\displaystyle \left[\left(\omega ^{2}-\alpha -{\tfrac {3}{4}}\beta z^{2}\right)^{2}+\left(\delta \omega \right)^{2}\right]\,z^{2}=\gamma ^{2}.} </p><p>For the parameters of the Duffing equation, the above algebraic equation gives the <a href="/facts/Steady_state/qgrL4BNs">steady state</a> oscillation amplitude z {\displaystyle z} at a given excitation frequency. </p> Derivation of the frequency response <p>Using the method of harmonic balance, an approximate solution to the Duffing equation is sought of the form:<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> x = a cos ( ω t ) + b sin ( ω t ) = z cos ( ω t − ϕ ) , {\displaystyle x=a\,\cos(\omega t)+b\,\sin(\omega t)=z\,\cos(\omega t-\phi ),} with z 2 = a 2 + b 2 {\displaystyle z^{2}=a^{2}+b^{2}} and tan ϕ = b a . {\displaystyle \tan \phi ={\frac {b}{a}}.} </p><p>Application in the Duffing equation leads to: ( − ω 2 a + ω δ b + α a + 3 4 β a 3 + 3 4 β a b 2 − γ ) cos ( ω t ) + ( − ω 2 b − ω δ a + 3 4 β b 3 + α b + 3 4 β a 2 b ) sin ( ω t ) + ( 1 4 β a 3 − 3 4 β a b 2 ) cos ( 3 ω t ) + ( 3 4 β a 2 b − 1 4 β b 3 ) sin ( 3 ω t ) = 0. {\displaystyle {\begin{aligned}&\left(-\omega ^{2}\,a+\omega \,\delta \,b+\alpha \,a+{\tfrac {3}{4}}\,\beta \,a^{3}+{\tfrac {3}{4}}\,\beta \,a\,b^{2}-\gamma \right)\,\cos \left(\omega \,t\right)\\&+\left(-\omega ^{2}\,b-\omega \,\delta \,a+{\tfrac {3}{4}}\,\beta \,b^{3}+\alpha \,b+{\tfrac {3}{4}}\,\beta \,a^{2}\,b\right)\,\sin \left(\omega \,t\right)\\&+\left({\tfrac {1}{4}}\,\beta \,a^{3}-{\tfrac {3}{4}}\,\beta \,a\,b^{2}\right)\,\cos \left(3\omega t\right)+\left({\tfrac {3}{4}}\,\beta \,a^{2}\,b-{\tfrac {1}{4}}\,\beta \,b^{3}\right)\,\sin \left(3\omega t\right)=0.\end{aligned}}} </p><p>Neglecting the <a href="/facts/Superharmonic/WHchDTSz">superharmonics</a> at 3 ω , {\displaystyle 3\omega ,} the two terms preceding cos ( ω t ) {\displaystyle \cos(\omega t)} and sin ( ω t ) {\displaystyle \sin(\omega t)} have to be zero. As a result, − ω 2 a + ω δ b + α a + 3 4 β a 3 + 3 4 β a b 2 = γ and − ω 2 b − ω δ a + 3 4 β b 3 + α b + 3 4 β a 2 b = 0. {\displaystyle {\begin{aligned}&-\omega ^{2}\,a+\omega \,\delta \,b+\alpha \,a+{\tfrac {3}{4}}\,\beta \,a^{3}+{\tfrac {3}{4}}\,\beta \,a\,b^{2}=\gamma \qquad {\text{and}}\\&-\omega ^{2}\,b-\omega \,\delta \,a+{\tfrac {3}{4}}\,\beta \,b^{3}+\alpha \,b+{\tfrac {3}{4}}\,\beta \,a^{2}\,b=0.\end{aligned}}} </p><p>Squaring both equations and adding leads to the amplitude frequency response: [ ( ω 2 − α − 3 4 β z 2 ) 2 + ( δ ω ) 2 ] z 2 = γ 2 , {\displaystyle \left[\left(\omega ^{2}-\alpha -{\frac {3}{4}}\beta z^{2}\right)^{2}+\left(\delta \omega \right)^{2}\right]\,z^{2}=\gamma ^{2},} as stated above. </p> <h3>Graphically solving for frequency response</h3> <p>We may graphically solve for z 2 {\displaystyle z^{2}} as the intersection of two curves in the ( z 2 , y ) {\displaystyle (z^{2},y)} plane: { y = ( ω 2 − α − 3 4 β z 2 ) 2 + ( δ ω ) 2 y = γ 2 z 2 {\displaystyle {\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 α , δ , γ {\displaystyle \alpha ,\delta ,\gamma } , the second curve is a fixed hyperbola in the first quadrant. The first curve is a parabola with shape y = 9 16 β 2 ( z 2 ) 2 {\textstyle y={\tfrac {9}{16}}\beta ^{2}(z^{2})^{2}} , and apex at location ( 4 3 β ( ω 2 − α ) , δ 2 ω 2 ) {\textstyle ({\tfrac {4}{3\beta }}(\omega ^{2}-\alpha ),\delta ^{2}\omega ^{2})} . If we fix β {\displaystyle \beta } and vary ω {\displaystyle \omega } , then the apex of the parabola moves along the line y = 3 4 β δ 2 ( z 2 ) + δ 2 α {\textstyle y={\tfrac {3}{4}}\beta \delta ^{2}(z^{2})+\delta ^{2}\alpha } . </p><p>Graphically, then, we see that if β {\displaystyle \beta } is a large positive number, then as ω {\displaystyle \omega } varies, the parabola intersects the hyperbola at one point, then three points, then one point again. Similarly we can analyze the case when β {\displaystyle \beta } is a large negative number. </p> <h3>Jumps</h3> <p>For certain ranges of the parameters in the Duffing equation, the frequency response may no longer be a <a href="/facts/Single-valued_function/WH5ua6BL">single-valued function</a> of forcing frequency ω . {\displaystyle \omega .} For a hardening spring oscillator ( α > 0 {\displaystyle \alpha >0} and large enough positive β > β c + > 0 {\displaystyle \beta >\beta _{c+}>0} ) the frequency response overhangs to the high-frequency side, and to the low-frequency side for the softening spring oscillator ( α > 0 {\displaystyle \alpha >0} and β < β c − < 0 {\displaystyle \beta <\beta _{c-}<0} ). 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: </p> <ul><li>when the angular frequency ω {\displaystyle \omega } is slowly increased (with other parameters fixed), the response <a href="/facts/Amplitude/aEckuXJM">amplitude</a> z {\displaystyle z} drops at A suddenly to B,</li> <li>if the frequency ω {\displaystyle \omega } is slowly decreased, then at C the amplitude jumps up to D, thereafter following the upper branch of the frequency response.</li></ul> <p>The jumps A–B and C–D do not coincide, so the system shows <a href="/facts/Hysteresis/9M2pwR62">hysteresis</a> depending on the frequency sweep direction.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> </p> <h3>Transition to chaos</h3> <p>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.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p> <h2 id="examples">Examples</h2> <p>Some typical examples of the <a href="/facts/Time_series/fSXPR817">time series</a> and <a href="/facts/Phase_portrait/H3h1eG8p">phase portraits</a> of the Duffing equation, showing the appearance of <a href="/facts/Subharmonic/gYbxy8mu">subharmonics</a> through <a href="/facts/Period-doubling_bifurcation/UchwCLYh">period-doubling bifurcation</a> – as well <a href="/facts/Chaos_theory/LTC17vsF">chaotic behavior</a> – are shown in the figures below. The forcing amplitude increases from γ = 0.20 {\displaystyle \gamma =0.20} to γ = 0.65 {\displaystyle \gamma =0.65} . The other parameters have the values: α = − 1 {\displaystyle \alpha =-1} , β = + 1 {\displaystyle \beta =+1} , δ = 0.3 {\displaystyle \delta =0.3} and ω = 1.2 {\displaystyle \omega =1.2} . The initial conditions are x ( 0 ) = 1 {\displaystyle x(0)=1} and x ˙ ( 0 ) = 0. {\displaystyle {\dot {x}}(0)=0.} The red dots in the phase portraits are at times t {\displaystyle t} which are an <a href="/facts/Integer/PUwwrawx">integer</a> multiple of the <a href="/facts/Frequency/QUSbPaW8">period</a> T = 2 π / ω {\displaystyle T=2\pi /\omega } .<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> </p> <h3>Citations</h3> <h3>Bibliography</h3> <ul><li>Duffing, G. (1918), <i>Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung</i> [<i>Forced oscillations with variable natural frequency and their technical relevance</i>] (in German), vol. Heft 41/42, Braunschweig: Vieweg, vi+134 pp, <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/12003652">12003652</a></li> <li>Addison, P. S. (1997), <i>Fractals and Chaos: An illustrated course</i>, CRC Press, pp. 147–148, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780849384431</li> <li>Jordan, D. W.; Smith, P. (2007), <i>Nonlinear ordinary differential equations – An introduction for scientists and engineers</i> (4th ed.), Oxford University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-920824-1</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://scholarpedia.org/article/Duffing_oscillator">Duffing oscillator on Scholarpedia</a></li> <li><a href="http://mathworld.wolfram.com/DuffingDifferentialEquation.html">MathWorld page</a></li> <li>Pchelintsev, A. N.; Ahmad, S. (2020). <a href="http://vestnik.tstu.ru/rus/t_26/pdf/26_1_013.pdf">"Solution of the Duffing equation by the power series method"</a> (PDF). <i>Transactions of TSTU</i>. 26 (1): 118–123.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Thompson, J. M. T.; Stewart, H. B. (2002). Nonlinear Dynamics and Chaos. John Wiley & Sons. p. 66. ISBN 9780471876847. <a href="9780471876847" target="_blank">9780471876847</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Lifshitz, R.; Cross, M. C. (2008). "Nonlinear mechanics of nanomechanical and micromechanical resonators". In Schuster, H. G. (ed.). Reviews of Nonlinear Dynamics and Complexity. Wiley. pp. 8–9. ISBN 9783527407293. LCCN 2008459659. <a href="9783527407293" target="_blank">9783527407293</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Brennan, M. J.; Kovacic, I.; Carrella, A.; Waters, T. P. (2008). "On the jump-up and jump-down frequencies of the Duffing oscillator". Journal of Sound and Vibration. 318 (4–5): 1250–1261. Bibcode:2008JSV...318.1250B. doi:10.1016/j.jsv.2008.04.032. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Kovacic, I.; Brennan, M. J., eds. (2011), The Duffing Equation: Nonlinear Oscillators and their Behaviour, Wiley, pp. 123–127, ISBN 978-0-470-71549-9 <a href="978-0-470-71549-9" target="_blank">978-0-470-71549-9</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Tajaddodianfar, F.; Yazdi, M. R. H.; Pishkenari, H. N. (2016). "Nonlinear dynamics of MEMS/NEMS resonators: analytical solution by the homotopy analysis method". Microsystem Technologies. 23 (6): 1913–1926. doi:10.1007/s00542-016-2947-7. S2CID 113216381. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Rand, R. H. (2012), Lecture notes on nonlinear vibrations (PDF), 53, Cornell University, pp. 13–17 <a href="https://ecommons.cornell.edu/bitstream/handle/1813/28989/NonlinearVibrations_ver53.pdf?sequence=2&isAllowed=y" target="_blank">https://ecommons.cornell.edu/bitstream/handle/1813/28989/NonlinearVibrations_ver53.pdf?sequence=2&isAllowed=y</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Bender, C. M.; Orszag, S. A. (1999), Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer, p. 546, Bibcode:1999amms.book.....B, ISBN 9780387989310 <a href="9780387989310" target="_blank">9780387989310</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Bender, C. M.; Orszag, S. A. (1999), Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer, p. 546, Bibcode:1999amms.book.....B, ISBN 9780387989310 <a href="9780387989310" target="_blank">9780387989310</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Takashi Kanamaru (ed.). "Duffing oscillator". Scholarpedia. <a href="http://www.scholarpedia.org/article/Duffing_oscillator" target="_blank">http://www.scholarpedia.org/article/Duffing_oscillator</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Jordan & Smith 2007, pp. 223–233 - Jordan, D. W.; Smith, P. (2007), Nonlinear ordinary differential equations – An introduction for scientists and engineers (4th ed.), Oxford University Press, ISBN 978-0-19-920824-1 <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Tajaddodianfar, F.; Yazdi, M. R. H.; Pishkenari, H. N. (2016). "Nonlinear dynamics of MEMS/NEMS resonators: analytical solution by the homotopy analysis method". Microsystem Technologies. 23 (6): 1913–1926. doi:10.1007/s00542-016-2947-7. S2CID 113216381. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Jordan & Smith 2007, pp. 223–233 - Jordan, D. W.; Smith, P. (2007), Nonlinear ordinary differential equations – An introduction for scientists and engineers (4th ed.), Oxford University Press, ISBN 978-0-19-920824-1 <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Jordan & Smith 2007, pp. 223–233 - Jordan, D. W.; Smith, P. (2007), Nonlinear ordinary differential equations – An introduction for scientists and engineers (4th ed.), Oxford University Press, ISBN 978-0-19-920824-1 <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Ueda, Yoshisuke (January 1, 1991). "Survey of regular and chaotic phenomena in the forced Duffing oscillator". Chaos, Solitons & Fractals. 1 (3): 199–231. Bibcode:1991CSF.....1..199U. doi:10.1016/0960-0779(91)90032-5. ISSN 0960-0779. <a href="https://dx.doi.org/10.1016/0960-0779%2891%2990032-5" target="_blank">https://dx.doi.org/10.1016/0960-0779%2891%2990032-5</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Based on the examples shown in Jordan & Smith 2007, pp. 453–462. - Jordan, D. W.; Smith, P. (2007), Nonlinear ordinary differential equations – An introduction for scientists and engineers (4th ed.), Oxford University Press, ISBN 978-0-19-920824-1 <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> </ol>