The oscillator that answers at three times the question

Every minimum is a parabola near enough to the bottom, and that fact is the reason so much of physics is the theory of the harmonic oscillator. The interesting question is what the next term does. It is not that the answers get slightly less accurate. Three things start happening that the parabolic oscillator cannot do at all, and each of them is qualitative rather than quantitative.

The first is that the oscillator stops answering only at the frequency it was asked at.

The arithmetic of why is short enough to do here. If the displacement is close to $a\cos\omega t$ and the restoring force contains a term in $x^3$, then that term contains $a^3\cos^3\omega t$, and

$$\cos^3\theta = \tfrac34\cos\theta + \tfrac14\cos 3\theta.$$

A cube of a cosine is not a cosine. It is a cosine plus a cosine of three times the angle, and the second piece is a force at $3\omega$ acting on the oscillator whether or not anything is driving it there. The oscillator responds to it as it responds to any force, and a component at $3\omega$ appears in the motion. Nothing supplied it; the nonlinearity manufactured it out of the drive.

The exponent follows from the same line. The third-harmonic force is proportional to $a^3$, and $a$ is proportional to the drive while the response is weak, so the third harmonic grows as the cube of the drive. Halve the drive and the fundamental halves while the third harmonic falls by a factor of eight. That is why nonlinearity is always a specification and never a yes or no: every real oscillator has some, and the question is only at what level it becomes intolerable.

Which harmonics, and why #

Which harmonics appear is decided by a symmetry rather than by a size. If the potential is even in the displacement — the same on both sides of the minimum — then the equation of motion is unchanged when the displacement and the drive both change sign. A solution containing a second harmonic would violate that: reversing the drive would reverse the fundamental and leave the second harmonic alone, and the reversed motion would no longer solve the equation. So a symmetric well produces odd harmonics only, exactly, at every amplitude. The figure with no quadratic term shows the even components below one part in ten thousand of the fundamental, which is the integrator’s noise floor rather than a physical effect.

Break the symmetry and the even ones appear immediately. The quadratic term in the restoring force is the leading asymmetry of any real well — a chemical bond is easier to stretch than to compress, and so is a crystal — and it produces both a second harmonic and something that is not a harmonic at all: a shift in the average position, because the square of a cosine has a constant part. That constant is why heating a solid makes it expand and why a perfect spring would not. Thermal expansion and second-harmonic generation are the same coefficient seen twice, one in a steady average and one at twice the driving frequency.

The rule generalises usefully. A medium’s second-order nonlinearity vanishes wherever the medium has a centre of symmetry, which is why frequency doubling in optics requires a crystal without one, and why the same crystal cut a different way does nothing. The mechanics and the optics are the same statement about an even function.

Two tones, and the frequencies that are neither #

A single drive produces harmonics. Two drives at once produce something worse, and it is the reason nonlinearity is measured with two tones rather than one.

If the motion contains components at $\omega_1$ and $\omega_2$, the cubic term contains the cube of their sum, and expanding it gives terms at $2\omega_1-\omega_2$ and $2\omega_2-\omega_1$ as well as at the harmonics. Those are the awkward ones. A third harmonic of a drive at 1 kHz sits at 3 kHz, a long way off, and can be removed by a filter. But two drives at 1.00 and 1.05 kHz produce components at 0.95 and 1.10 kHz — inside the band, a few per cent away from the wanted signals, and impossible to filter out without removing the signal too.

The size of those components grows as the cube of the drive, exactly as the third harmonic does, because they come from the same term. That gives the standard way of quoting a nonlinearity: extrapolate the fundamental, which rises with slope one, and the third-order product, which rises with slope three, and quote the drive at which the extrapolated lines cross. It is a fictitious operating point — nothing works that hard — and it is a complete specification of the cubic coefficient in one number.

The same arithmetic in a mechanical setting is the reason a loudspeaker cone or a stiffening suspension produces sum and difference tones that were not in the recording, and the reason a nonlinear medium mixes light: two frequencies entering a crystal leave it accompanied by their sums and differences, generated by exactly the term drawn in the first figure of this essay.

The resonance that leans over #

The second thing the extra term does is move the resonance, and it moves it by an amount that depends on the amplitude.

The mechanism is the amplitude dependence of the period, which for a pendulum is the fact that a wide swing takes longer than a narrow one. A stiffening spring is the other sign: it swings faster the further it goes. Either way the oscillator’s own frequency is a function of how hard it is being driven, so the peak of the response cannot sit at a fixed place. It sits wherever the drive frequency and the amplitude are consistent with each other, and the locus of those points is a curve leaning away from the vertical.

At small drive the lean is a small distortion of a symmetric peak. Past a threshold it is a change of kind: the curve folds back on itself, so a vertical line at a fixed drive frequency crosses it three times. Two of those crossings are stable and the middle one is not, which is the same arithmetic as a fluid isotherm crossing three volumes at one pressure with the middle root unstable for the same structural reason.

Nothing about the drive changed to produce the fold. Only its size.

Two answers, and which one depends on the past #

The consequence of a folded response curve is hysteresis, and hysteresis is a strong statement. The steady amplitude of a linear oscillator is a function of the drive frequency: give the frequency and the amplitude follows. Here it is not a function of anything the drive is doing now. Two identical oscillators, driven identically at the same instant, can be at amplitudes differing by a factor of ten because one of them was driven at a lower frequency a minute ago.

The jumps themselves are worth looking at, because they are not gradual. Climbing the upper branch, the state reaches the fold, and past it there is no upper solution at all — so the amplitude collapses onto the lower branch over a few dozen cycles, which for a mechanical oscillator is a fraction of a second and sounds like a click. Coming down, the same thing happens in reverse at a different frequency. The pair of frequencies bracketing the hysteresis is a measurable quantity, and its width is a direct measurement of the nonlinear coefficient — which is how the coefficient is measured in micromechanical resonators, where nothing about the material is known in advance.

This is also the practical reason a nonlinear resonator is hard to use as a frequency reference. A linear one has a peak whose centre is a property of the device; a nonlinear one has a peak whose position depends on how hard it is driven, and a device driven hard enough to give a good signal is driven hard enough to lean. The design compromise between signal and linearity is a compromise between two things that are the same knob.

Where the term comes from #

There is nothing special about the cubic term. It is the third term of a Taylor expansion of whatever potential the oscillator actually sits in, and its coefficient is a third derivative at the minimum. What makes it worth singling out is that it is the first term that does anything qualitatively new: the constant is nothing, the linear term vanishes at a minimum, the quadratic term is the parabola, and the cubic is the first correction with a name.

The size of the coefficient decides only when the effects appear, not whether. A pendulum’s expansion has no quadratic term by symmetry and a negative cubic one, so it softens; a chemical bond has both, so it does everything at once; a crystal has a lattice whose anharmonicity is what carries heat by making phonons scatter off one another and what makes it expand when warmed. The equation drawn here is the same equation in all three cases with a different third derivative.

It is worth putting a number on when the term stops being ignorable, because the answer is not intuitive. The frequency shift is of order the cubic coefficient times the square of the amplitude divided by the linear stiffness, so a one per cent shift needs an amplitude of about a tenth of the distance over which the well’s shape changes appreciably. For a pendulum that distance is a radian, so a one per cent shift arrives at a swing of six degrees — which is why a pendulum clock’s escapement is designed to hold the amplitude constant rather than merely to keep it small. For a crystal lattice the distance is a bond length, so the shift is negligible for sound and dominant for anything at the amplitude of a shock.

The other number worth carrying is how sharply the effects turn on. The frequency shift goes as the square of the amplitude, the third harmonic as the cube, and the width of the hysteresis loop as the amplitude to the fourth. So a device operated at twice its previous amplitude has four times the frequency pull, eight times the distortion and sixteen times the hysteresis, and the first thing to fail is whichever the application cares about most.

And the expansion has a range. Everything above is the leading nonlinear behaviour, valid while the excursion is small enough that the fourth term does not matter, and a real oscillator driven far past that does not merely acquire more harmonics. It period-doubles: the response starts repeating every two drive cycles, then every four, and the way that cascade proceeds is the same for every system that does it. The leaning peak and the fold are the last things a perturbative description can describe.

Who noticed, and in what order #

The three effects were found in a different order from the one they are presented in here, and the order says something about which of them is easiest to see.

Helmholtz was arguing about combination tones in the 1850s — the frequencies a listener hears that are in neither of two sounded notes — and attributed them to a nonlinear response somewhere in the chain. That is the two-tone effect, and it was noticed first because the ear is an extremely sensitive detector of a frequency that should not be there.

Rayleigh treated the amplitude dependence of the frequency in the 1870s and had the hard and soft spring, the leaning peak, and the observation that such a system can have two responses. Duffing’s monograph of 1918 gave the equation its name and, more usefully, the systematic approximation for solving it: assume a single cosine, keep the terms at the driving frequency, and balance. Everything in the response curves here is that method, and its virtue is that it turns a differential equation nobody can solve into a cubic nobody minds solving.

The jump itself was demonstrated experimentally in the 1920s and 1930s in electrical circuits with iron-cored inductors, where the iron’s magnetic saturation supplies a very strong stiffening nonlinearity. That is worth noting because it is the same phenomenon in a system with no moving parts at all: the magnetisation of iron falls behind the field that drives it and the resulting inductance depends on the current, which is a nonlinear restoring term in the circuit equation with the same cubic leading behaviour.

Where the model stops #

The response curves come from harmonic balance, which keeps one frequency. The amplitude is assumed to be a single cosine at the drive frequency, and everything at $3\omega$ is thrown away when balancing. That is consistent — the third harmonic is small in exactly the regime where the assumption holds — and it is the reason the figure marks points obtained by integrating the full equation, which does not make the assumption. They agree away from the fold and would not agree far past it.

The oscillator is one degree of freedom. A real spring, beam or crystal has many, and the cubic term couples them: energy fed in at one mode can come out at another, which is a mechanism the one-dimensional equation cannot represent at all. In a stretched string it produces the coupling between the transverse and longitudinal motions that gives a plucked string its slight sharpening, and in a crystal it is the whole of phonon–phonon scattering.

Damping is linear here and often is not. The equation uses a force proportional to velocity, which is right for a viscous fluid and wrong for internal friction in a solid, where the loss per cycle is more nearly independent of frequency. A nonlinear damping term produces its own amplitude-dependent effects, including a saturation of the response that looks like a leaning peak and is not one.

And the hysteresis is drawn for a slow sweep. How slow is not a detail: the jump happens when the state reaches the fold, and near the fold the approach to the steady state slows down without limit, so a fast sweep overshoots and jumps late. The frequencies reported are the quasi-static ones, and a measurement made by sweeping quickly gives a wider hysteresis loop than the equation’s.

What the pictures cannot show #

The bar charts draw the size of each Fourier component and not its phase. The phases are not incidental: the third harmonic’s phase relative to the fundamental is what decides whether the combined waveform is flattened or peaked, and two motions with identical component sizes and different phases look nothing alike. What a nonlinearity does to a waveform is a statement about phases, and the spectrum has thrown them away.

The response curves draw a steady amplitude and cannot show the unstable branch’s role. The middle crossing is not merely unstable; it is the boundary between the basins of the two stable states, so it decides which state a knocked oscillator ends up in. A figure showing that would be a figure in the plane of displacement and velocity rather than of frequency and amplitude, and it is where the answer to “which state” actually lives.

Where the ladder goes next #

This ladder began with every minimum being a parabola, continued with two oscillators that will not stop swapping their energy and with the thermal consequence of the term beyond the parabola. This rung asks what that term does to a driven oscillator. The rungs after it: parametric driving, where the stiffness itself is modulated and the response grows without any force at the resonant frequency at all, which a child on a swing already knows; mode coupling, where the cubic term moves energy between degrees of freedom; and the transition to chaos, where the perturbative description stops entirely.

The habit worth carrying away is about what an approximation throws out. The parabola does not lose accuracy gradually; it loses entire phenomena at once. Harmonic generation, an amplitude-dependent frequency and two answers at one drive are not small corrections to a linear response, and no amount of care with a linear model will produce any of them.