The Duffing Oscillator: Hardening Springs, Jumps and Chaos

A perfect spring obeys Hooke’s law: double the stretch, double the force. Real springs, beams and pendulums do not, once the swing gets large. In 1918 Georg Duffing (1861–1944) studied forced oscillations whose natural frequency changes with amplitude Duffing equation, Wikipedia^[1] Kovacic and Brennan 2011^[2]. The equation now named after him is now a widely used model of damped, driven nonlinear oscillators, and it hides resonance jumps, hysteresis and chaos Duffing equation, Wikipedia^[1]. Its strange attractor is printed on our Duffing's Strange Attractor poster.

What is the Duffing equation?

For a mass on a nonlinear spring with linear damping, driven by a periodic force, the equation of motion per unit mass is Duffing equation, Wikipedia^[1]

ẍ + δẋ + αx + βx³ = γ cos(ωt)

Here x is the displacement in metres, ẋ the velocity and ẍ the acceleration. δ (s⁻¹) sets the damping, α (s⁻²) the linear stiffness and β (m⁻² s⁻²) the cubic nonlinearity. γ (m s⁻²) is the forcing amplitude and ω (rad/s) the driving angular frequency Duffing equation, Wikipedia^[1]. With β = 0 it is the ordinary damped, driven harmonic oscillator Duffing equation, Wikipedia^[1]. The restoring force αx + βx³ comes from the potential

V(x) = ½αx² + ¼βx⁴

What are hardening and softening springs?

With α > 0 and β > 0 the spring gets stiffer as it stretches: a hardening spring. With β < 0 it gets weaker: a softening spring Duffing equation, Wikipedia^[1]. A pendulum is a softening example, because the restoring torque grows like sin θ ≈ θ − θ³/6, more slowly than θ itself. That is why big swings take longer, a correction that matters for the precise clocks and pendulums in our Foucault pendulum guide.

With α < 0 and β > 0 the potential has two wells separated by a hump at x = 0. A mass can rattle in one well or be thrown back and forth between them. This double-well case is where the most dramatic chaos lives Holmes 1979^[3].

Why does the Duffing resonance jump?

Drive a linear oscillator and its amplitude peaks smoothly at one frequency. In the Duffing oscillator the effective natural frequency depends on the amplitude z, so the resonance peak bends over. A harmonic-balance approximation gives the steady-state amplitude z (in metres) at each driving frequency Duffing equation, Wikipedia^[1]:

[(ω² − α − ¾βz²)² + (δω)²] z² = γ²

Without damping or driving this reduces to the backbone curve

ω² ≈ α + ¾βz²

which tilts to the right for a hardening spring and to the left for a softening one. Where the tilted peak overhangs, three amplitudes solve the equation at the same frequency. The middle one is unstable Duffing equation, Wikipedia^[1]. Sweep the frequency slowly upward and the amplitude rides the upper branch until it falls off the edge and drops suddenly. Sweep it downward and it jumps up at a different frequency. The two jumps do not coincide, so the response depends on the direction of the sweep: frequency hysteresis Duffing equation, Wikipedia^[1].

How does the Duffing oscillator become chaotic?

The harmonic-balance picture assumes that the response is dominated by the driving frequency. With strong enough forcing, higher harmonics matter and the motion can become chaotic, most commonly through successive period doublings Duffing equation, Wikipedia^[1]. In a standard double-well example (α = −1, β = 1, δ = 0.3, ω = 1.2, in scaled units), the motion repeats every drive cycle at γ = 0.20, every two cycles at γ = 0.28, every four at γ = 0.29, and is chaotic at γ = 0.50 Duffing equation, Wikipedia^[1].

Period doubling is a universal route to chaos. Feigenbaum showed that, for one-dimensional maps with a single quadratic maximum, the parameter intervals between successive doublings shrink by the same ratio, about 4.669, now called the Feigenbaum constant, so every system of that kind bifurcates at the same rate Feigenbaum constants, Wikipedia^[4] Feigenbaum 1978^[5].

What is the Duffing strange attractor?

A good way to see the chaos is to strobe the motion once per drive period and plot position against velocity. A periodic orbit leaves a few dots. A chaotic one leaves a folded, layered set of points that never repeats: a strange attractor Duffing equation, Wikipedia^[1]. In 1979 Philip Holmes analysed the double-well Duffing oscillator and found that, for a wide range of moderate forcing, extremely complicated non-periodic motions arise. He linked them to homoclinic orbits and used Melnikov’s method to prove they exist Holmes 1979^[3]. The same year Yoshisuke Ueda reported “randomly transitional phenomena” in a system governed by Duffing’s equation Ueda 1979^[6], and Moon and Holmes described a magnetoelastic strange attractor in a laboratory experiment Moon and Holmes 1979^[7].

Quick FAQ

Is the Duffing oscillator just a curiosity?

No. It is a model for any oscillator whose stiffness depends on amplitude, from beams to pendulums, and a whole book is devoted to its behaviour Kovacic and Brennan 2011^[2]. It is still Newton’s second law, the subject of Newton, Galileo and Kepler’s laws.

Does chaos mean the equation is random?

No. The equation is deterministic. Chaos means tiny differences in the starting state grow until prediction fails Holmes 1979^[3]. For ordinary, linear resonance, see our Oscillayne resonance tee, and more prints are in our physics posters.

References

1. Wikipedia, “Duffing equation”. https://en.wikipedia.org/wiki/Duffing_equation
2. I. Kovacic and M. J. Brennan (eds.) (2011), “The Duffing Equation: Nonlinear Oscillators and their Behaviour,” Wiley (DOI 10.1002/9780470977859; metadata verified via Crossref). https://doi.org/10.1002/9780470977859
3. P. Holmes (1979), “A nonlinear oscillator with a strange attractor,” Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 292, 419–448 (DOI 10.1098/rsta.1979.0068; metadata verified via Crossref). https://doi.org/10.1098/rsta.1979.0068
4. Wikipedia, “Feigenbaum constants”. https://en.wikipedia.org/wiki/Feigenbaum_constants
5. Mitchell J. Feigenbaum (1978), “Quantitative universality for a class of nonlinear transformations,” Journal of Statistical Physics 19, 25–52 (DOI 10.1007/BF01020332; metadata verified via Crossref). https://doi.org/10.1007/BF01020332
6. Yoshisuke Ueda (1979), “Randomly transitional phenomena in the system governed by Duffing's equation,” Journal of Statistical Physics 20, 181–196 (DOI 10.1007/BF01011512; metadata verified via Crossref). https://doi.org/10.1007/BF01011512
7. F.C. Moon and P.J. Holmes (1979), “A magnetoelastic strange attractor,” Journal of Sound and Vibration 65, 275–296 (DOI 10.1016/0022-460X(79)90520-0; metadata verified via Crossref). https://doi.org/10.1016/0022-460X(79)90520-0

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.