The Anharmonic Harmonic Oscillator

Harmonic oscillators are one of the fundamental elements of physical theory.  They arise so often in so many different contexts that they can be viewed as a central paradigm that spans all aspects of physics.  Some famous physicists have been quoted to say that the entire universe is composed of simple harmonic oscillators (SHO).

Despite the physicist’s love affair with it, the SHO is pathological! First, it has infinite frequency degeneracy which makes it prone to the slightest perturbation that can tip it into chaos, in contrast to non-harmonic cyclic dynamics that actually protects us from the chaos of the cosmos (see my Blog on Chaos in the Solar System). Second, the SHO is nowhere to be found in the classical world.  Linear oscillators are purely harmonic, with a frequency that is independent of amplitude—but no such thing exists!  All oscillators must be limited, or they could take on infinite amplitude and infinite speed, which is nonsense.  Even the simplest of simple harmonic oscillators would be limited by nothing other than the speed of light.  Relativistic effects would modify the linearity, especially through time dilation effects, rendering the harmonic oscillator anharmonic.

Despite the physicist’s love affair with it, the SHO is pathological!

Therefore, for students of physics as well as practitioners, it is important to break the shackles imposed by the SHO and embrace the anharmonic harmonic oscillator as the foundation of physics. Here is a brief survey of several famous anharmonic oscillators in the history of physics, followed by the mathematical analysis of the relativistic anharmonic linear-spring oscillator.

Anharmonic Oscillators

Anharmonic oscillators have a long venerable history with many varieties.  Many of these have become central models in systems as varied as neural networks, synchronization, grandfather clocks, mechanical vibrations, business cycles, ecosystem populations and more.

Christiaan Huygens

Already by the mid 1600’s Christiaan Huygens (1629 – 1695) knew that the pendulum becomes slower when it has larger amplitudes.  The pendulum was one of the best candidates for constructing an accurate clock needed for astronomical observations and for the determination of longitude at sea.  Galileo (1564 – 1642) had devised the plans for a rudimentary pendulum clock that his son attempted to construct, but the first practical pendulum clock was invented and patented by Huygens in 1657.  However, Huygens’ modified verge escapement required his pendulum to swing with large amplitudes, which brought it into the regime of anharmonicity. The equations of the simple pendulum are truly simple, but the presence of the sinθ makes it the simplest anharmonic oscillator.

Therefore, Huygens searched for the mathematical form of a tautochrone curve for the pendulum (a curve that is traversed with equal times independently of amplitude) and in the process he invented the involutes and evolutes of a curve—precursors of the calculus.  The answer to the tautochrone question is a cycloid (see my Blog on Huygen’s Tautochrone Curve).

Hermann von Helmholtz

Hermann von Helmholtz (1821 – 1894) was possibly the greatest German physicist of his generation—an Einstein before Einstein—although he began as a medical doctor.  His study of muscle metabolism, drawing on the early thermodynamic work of Carnot, Clapeyron and Joule, led him to explore and to express the conservation of energy in its clearest form.  Because he postulated that all forms of physical processes—electricity, magnetism, heat, light and mechanics—contributed to the interconversion of energy, he sought to explore them all, bringing his research into the mainstream of physics.  His laboratory in Berlin became world famous, attracting to his laboratory the early American physicists Henry Rowland (founder and first president of the American Physical Society) and Albert Michelson (first American Nobel prize winner).

Even the simplest of simple harmonic oscillators would be limited by nothing other than the speed of light.  

Helmholtz also pursued a deep interest in the physics of sensory perception such as sound.  This research led to his invention of the Helmholtz oscillator which is a highly anharmonic relaxation oscillator in which a tuning fork was placed near an electromagnet that was powered by a mercury switch attached to the fork. As the tuning fork vibrated, the mercury came in and out of contact with it, turning on and off the magnet, which fed back on the tuning fork, and so on, enabling the device, once started, to continue oscillating without interruption. This device is called a tuning-fork resonator, and it became the first door-bell buzzers.  (These are not to be confused with Helmholtz resonances that are formed when blowing across the open neck of a beer bottle.)

Lord Rayleigh

Baron John Strutt, the Lord Rayleigh (1842 – 1919) like Helmholtz also was a generalist and had a strong interest in the physics of sound.  He was inspired by Helmholtz’ oscillator to consider general nonlinear anharmonic oscillators mathematically.  He was led to consider the effects of anharmonic terms added to the harmonic oscillator equation.  in a paper published in the Philosophical Magazine issue of 1883 with the title On Maintained Vibrations, he introduced an equation to describe the self-oscillation by adding an extra term to a simple harmonic oscillator. The extra term depended on the cube of the velocity, representing a balance between the gain of energy from a steady force and natural dissipation by friction.  Rayleigh suggested that this equation applied to a wide range of self-oscillating systems, such as violin strings, clarinet reeds, finger glasses, flutes, organ pipes, among others (see my Blog on Rayleigh’s Harp.)

Georg Duffing

The first systematic study of quadratic and cubic deviations from the harmonic potential was performed by the German engineer George Duffing (1861 – 1944) under the conditions of a harmonic drive. The Duffing equation incorporates inertia, damping, the linear spring and nonlinear deviations.

Fig. 1 The Duffing equation adds a nonlinear term to the spring force when alpha is positive, stiffening or weakening it for larger excursions when beta is positive or negative, respectively. And by making alpha negative and beta positive, it describes a damped driven double-well potential.

Duffing confirmed his theoretical predictions with careful experiments and established the lowest-order corrections to ideal masses on springs. His work was rediscovered in the 1960’s after Lorenz helped launch numerical chaos studies. Duffing’s driven potential becomes especially interesting when α is negative and β is positive, creating a double-well potential. The driven double-well is a classic chaotic system (see my blog on Duffing’s Oscillator).

Balthasar van der Pol

Autonomous oscillators are one of the building blocks of complex systems, providing the fundamental elements for biological oscillatorsneural networksbusiness cyclespopulation dynamics, viral epidemics, and even the rings of Saturn.  The most famous autonomous oscillator (after the pendulum clock) is named for a Dutch physicist, Balthasar van der Pol (1889 – 1959), who discovered the laws that govern how electrons oscillate in vacuum tubes, but the dynamical system that he developed has expanded to become the new paradigm of cyclic dynamical systems to replace the SHO (see my Blog on GrandFather Clocks.)

Fig. 2 The van der Pol equation is the standard simple harmonic oscillator with a gain term that saturates for large excursions leading to a limit cycle oscillator.

Turning from this general survey, let’s find out what happens when special relativity is added to the simplest SHO [1].

Relativistic Linear-Spring Oscillator

The theory of the relativistic one-dimensional linear-spring oscillator starts from a relativistic Lagrangian of a free particle (with no potential) yielding the generalized relativistic momentum

The Lagrangian that accomplishes this is [2]

where the invariant 4-velocity is

When the particle is in a potential, the Lagrangian becomes

The action integral that is minimized is

and the Lagrangian for integration of the action integral over proper time is

The relativistic modification in the potential energy term of the Lagrangian is not in the spring constant, but rather is purely a time dilation effect.  This is captured by the relativistic Lagrangian

where the dot is with respect to proper time τ. The classical potential energy term in the Lagrangian is multiplied by the relativistic factor γ, which is position dependent because of the non-constant speed of the oscillator mass.  The Euler-Lagrange equations are

where the subscripts in the variables are a = 0, 1 for the time and space dimensions, respectively.  The derivative of the time component of the 4-vector is

From the derivative of the Lagrangian with respect to speed, the following result is derived

where E is the constant total relativistic energy.  Therefore,

which provides an expression for the derivative of the coordinate time with respect to the proper time where

The position-dependent γ(x) factor is then

The Euler-Lagrange equation with a = 1 is

which gives

providing the flow equations for the (an)harmonic oscillator with respect to proper time

This flow represents a harmonic oscillator modified by the γ(x) factor, due to time dilation, multiplying the spring force term.  Therefore, at relativistic speeds, the oscillator is no longer harmonic even though the spring constant remains truly a constant.  The term in parentheses effectively softens the spring for larger displacement, and hence the frequency of oscillation becomes smaller. 

The state-space diagram of the anharmonic oscillator is shown in Fig. 3 with respect to proper time (the time read on a clock co-moving with the oscillator mass).  At low energy, the oscillator is harmonic with a natural period of the SHO.  As the maximum speed exceeds β = 0.8, the period becomes longer and the trajectory less sinusoidal.  The position and speed for β = 0.9999 is shown in Fig. 4.  The mass travels near the speed of light as it passes the origin, producing significant time dilation at that instant.  The average time dilation through a single cycle is about a factor of three, despite the large instantaneous γ = 70 when the mass passes the origin.

Fig. 3 State-space diagram in relativistic units relative to proper time of a relativistic (an)harmonic oscillator with a constant spring constant for several relative speeds β. The anharmonicity becomes pronounced above β = 0.8.
Fig. 4 Position and speed in relativistic units relative to proper time of a relativistic (an)harmonic oscillator with a constant spring constant for β = 0.9999.  The period of oscillation in this simulation is nearly three times longer than the natural frequency at small amplitudes.

[1] W. Moreau, R. Easther, and R. Neutze, “RELATIVISTIC (AN)HARMONIC OSCILLATOR,” American Journal of Physics, Article vol. 62, no. 6, pp. 531-535, Jun (1994)

[2] D. D. Nolte, Introduction to Modern Dynamics: Chaos, Networks, Space and Time, 2nd. ed. (Oxford University Press, 2019)

Georg Duffing’s Equation

Although coal and steam launched the industrial revolution, gasoline and controlled explosions have sustained it for over a century.  After early precursors, the internal combustion engine that we recognize today came to life in 1876 from the German engineers Otto and Daimler with later variations by Benz and Diesel.  In the early 20th century, the gasoline engine was replacing coal and oil in virtually all mobile conveyances and had become a major industry attracting the top mechanical engineering talent.  One of those talents was the German engineer Georg Duffing (1861 – 1944) whose unlikely side interest in the quantum mechanics revolution brought him to Berlin to hear lectures by Max Planck, where he launched his own revolution in nonlinear oscillators.

The publication of this highly academic book by a nonacademic would establish Duffing as the originator of one of the most iconic oscillators in modern dynamics.

An Academic Non-Academic

Georg Duffing was born in 1861 in the German town of Waldshut on the border with Switzerland north of Zurich.  Within a year the family moved to Mannheim near Heidelberg where Georg received a good education in mathematics as well as music.  His mathematical interests attracted him to engineering, and he built a reputation that led to an invitation to work at Westinghouse in the United States in 1910.  When he returned to Germany he set himself up as a consultant and inventor with the freedom to move where he wished.  In early 1913 he wished to move to Berlin where Max Planck was lecturing on the new quantum mechanics at the University.  He was always searching for new knowledge, and sitting in on Planck’s lectures must have made him feel like he was witnessing the beginnings of a new era.            

At that time Duffing was interested in problems related to brakes, gears and engines.  In particular, he had become fascinated by vibrations that often were the limiting factors in engine performance.  He stripped the problem of engine vibration down to its simplest form, and he began a careful and systematic study of nonlinear oscillations.  While in Berlin, he had became acquainted with Prof. Meyer at the University who had a mechanical engineering laboratory.  Meyer let Duffing perform his experiments in the lab on the weekends, sometime accompanied by his eldest daughter.  By 1917 he had compiled a systematic investigation of various nonlinear effects in oscillators and had written a manuscript that collected all of this theoretical and experimental work.  He extended this into a small book that he published with Vieweg & Sohn in 1918 to be purchased for a price of 5 Deutsch Marks [1].   The publication of this highly academic book by a nonacademic would establish Duffing as the originator of one of the most iconic oscillators in modern dynamics.

Fig. 1 Cover of Duffing’s 1918 publication on nonlinear oscillators.

Duffing’s Nonlinear Oscillator

The mathematical and technical focus of Duffing’s book was low-order nonlinear corrections to the linear harmonic oscillator.  In one case, he considered a spring that either became stiffer or softer as it stretched.  This happens when a cubic term is added to the usual linear Hooke’s law.  In another case, he considered a spring that was stiffer in one direction than another, making the stiffness asymmetric.  This happens when a quadratic term is added.  These terms are shown in Fig. 2 from Duffing’s book.  The top equation is a free oscillation, and the bottom equation has a harmonic forcing function.  These were the central equations that Duffing explored, plus the addition of damping that he considered in a later chapter as shown in Fig. 3. The book lays out systematically, chapter by chapter, approximate and series solutions to the nonlinear equations, and in special cases described analytically exact solutions (such as for the nonlinear pendulum).

Fig. 2 Duffing’s equations without damping for free oscillation and driven oscillation with quadratic (producing an asymmetric potential) and cubic (producing stiffening or softening) corrections to the spring force.
Fig. 3 Inclusion of damping in the case with cubic corrections to the spring force.

Duffing was a practical engineer as well as a mathematical one, and he built experimental systems to test his solutions.  An engineering drawing of his experimental test apparatus is shown in Fig. 4. The small test pendulum is at S in the figure. The large pendulum at B is the drive pendulum, chosen to be much heavier than the test pendulum so that it can deliver a steady harmonic force through spring F1 to the test system. The cubic nonlinearity of the test system was controlled through the choice of the length of the test pendulum, and the quadratic nonlinearity (the asymmetry) was controlled by allowing the equilibrium angle to be shifted from vertical. The relative strength of the quadratic and cubic terms was adjusted by changing the position of the mass at G. Duffing derived expressions for all the coefficients of the equations in Fig. 1 in terms of experimentally-controlled variables. Using this apparatus, Duffing verified to good accuracy his solutions for various special cases.

Fig. 4 Duffing’s experimental system he used to explore and verify his equations and solutions.

           Duffing’s book is a masterpiece of careful systematic investigation, beginning in general terms, and then breaking the problem down into its special cases, finding solutions for each one with accurate experimental verifications. These attributes established the importance of this little booklet in the history of science and technology, but because it was written in German, most of the early citations were by German scientists.  The first use of Duffing’s name associated to the nonlinear oscillator problem occurred in 1928 [2], as was the first reference to him in a work in English in a book by Timoshenko [3].  The first use of the phrase “Duffing Equation” specifically to describe an oscillator with a linear and cubic restoring force was in 1942 in a series of lectures presented at Brown University [4], and this nomenclature had become established by the end of that decade [5].  Although Duffing had spent considerable attention in his book to the quadratic term for an asymmetric oscillator, the term “Duffing Equation” now refers to the stiffening and softening problem rather than to the asymmetric problem.

Fig. 5 The Duffing equation is generally expressed as a harmonic oscillator (first three terms plus the harmonic drive) modified by a cubic nonlinearity and driven harmonically.

Duffing Rediscovered

Nonlinear oscillations remained mainly in the realm of engineering for nearly half a century, until a broad spectrum of physical scientists began to discover deep secrets hiding behind the simple equations.  In 1963 Edward Lorenz (1917 – 2008) of MIT published a paper that showed how simple nonlinearities in three equations describing the atmosphere could produce a deterministic behavior that appeared to be completely chaotic.  News of this paper spread as researchers in many seemingly unrelated fields began to see similar signatures in chemical reactions, turbulence, electric circuits and mechanical oscillators.  By 1972 when Lorenz was invited to give a talk on the “Butterfly Effect” the science of chaos was emerging as new frontier in physics, and in 1975 it was given its name “chaos theory” by James Yorke (1941 – ).  By 1976 it had become one of the hottest new areas of science. 

        Through the period of the emergence of chaos theory, the Duffing oscillator was known to be one of the archetypical nonlinear oscillators.  A particularly attractive aspect of the general Duffing equations is the possibility of studying a “double-well” potential.  This happens when the “alpha” in the equation in Fig. 5 is negative and the “beta” is positive.  The double-well potential has a long history in physics, both classical and modern, because it represents a “two-state” system that exhibits bistability, bifurcations, and hysteresis.  For a fixed “beta” the potential energy as a function of “alpha” is shown in Fig. 6.  The bifurcation cascades of the double-well Duffing equation was investigated by Phillip Holmes (1945 – ) in 1976 [6], and the properties of the strange attractor were demonstrated in 1978 [7] by Yoshisuke Ueda (1936 – ).  Holmes, and others, continued to do detailed work on the chaotic properties of the Duffing oscillator, helping to make it one of the most iconic systems of chaos theory.

Fig. 6 Potential energy of the Duffing Oscillator. The position variable is x, and changing alpha is along the other axis. For positive beta and alpha the potential is a quartic. For positive beta and negative alpha the potential is a double well.

Python Code for the Duffing Oscillator:

This Python code uses the simple ODE solver on the driven-damped Duffing double-well oscillator to display the configuration-space trajectories and the Poincaré map of the strange attractor.

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
Created on Wed May 21 06:03:32 2018
@author: nolte
D. D. Nolte, Introduction to Modern Dynamics: Chaos, Networks, Space and Time, 2nd ed. (Oxford,2019)
import numpy as np
import matplotlib as mpl
from mpl_toolkits.mplot3d import Axes3D
from scipy import integrate
from matplotlib import pyplot as plt
from matplotlib import cm
import time
import os


# model_case 1 = Pendulum
# model_case 2 = Double Well
print(' ')

alpha = -1       # -1
beta = 1         # 1
delta = 0.3       # 0.3
gam = 0.15    # 0.15
w = 1
def flow_deriv(x_y_z,tspan):
    x, y, z = x_y_z
    a = y
    b = delta*np.cos(w*tspan) - alpha*x - beta*x**3 - gam*y
    c = w
T = 2*np.pi/w

px1 = np.random.rand(1)
xp1 = np.random.rand(1)
w1 = 0

x_y_z = [xp1, px1, w1]

# Settle-down Solve for the trajectories
t = np.linspace(0, 2000, 40000)
x_t = integrate.odeint(flow_deriv, x_y_z, t)
x0 = x_t[39999,0:3]

tspan = np.linspace(1,20000,400000)
x_t = integrate.odeint(flow_deriv, x0, tspan)
siztmp = np.shape(x_t)
siz = siztmp[0]

y1 = x_t[:,0]
y2 = x_t[:,1]
y3 = x_t[:,2]
lines = plt.plot(y1[1:2000],y2[1:2000],'ko',ms=1)
plt.setp(lines, linewidth=0.5)

for cloop in range(0,3):

#phase = np.random.rand(1)*np.pi;
    phase = np.pi*cloop/3

    repnum = 5000
    px = np.zeros(shape=(2*repnum,))
    xvar = np.zeros(shape=(2*repnum,))
    cnt = -1
    testwt = np.mod(tspan-phase,T)-0.5*T;
    last = testwt[1]
    for loop in range(2,siz):
        if (last < 0)and(testwt[loop] > 0):
            cnt = cnt+1
            del1 = -testwt[loop-1]/(testwt[loop] - testwt[loop-1])
            px[cnt] = (y2[loop]-y2[loop-1])*del1 + y2[loop-1]
            xvar[cnt] = (y1[loop]-y1[loop-1])*del1 + y1[loop-1]
            last = testwt[loop]
            last = testwt[loop]
    if cloop == 0:
        lines = plt.plot(xvar,px,'bo',ms=1)
    elif cloop == 1:
        lines = plt.plot(xvar,px,'go',ms=1)
        lines = plt.plot(xvar,px,'ro',ms=1)

Fig. 7 Strange attractor of the double-well Duffing equation for three selected phases.

[1] G. Duffing, Erzwungene Schwingungen bei veranderlicher Eigenfrequenz und ihre technische Bedeutung, Vieweg & Sohn, Braunschweig, 1918.

[2] Lachmann, K. “Duffing’s vibration problem.” Mathematische Annalen 99: 479-492. (1928)

[3] S. Timoshenko, Vibration Problems in Engineering, D. Van Nostrand Company, Inc.,New York, 1928.

[4] K.O. Friedrichs, P. Le Corbeiller, N. Levinson, J.J. Stoker, Lectures on Non-Linear Mechanics delivered at Brown University, New York, 1942.

[5] Kovacic, I. and M. J. Brennan, Eds. The Duffing Equation: Nonlinear Oscillators and their Behavior. Chichester, United Kingdom, Wiley. (2011)

[6] Holmes, P. J. and D. A. Rand. “Bifurcations of Duffings Equation – Application of Catastrophe Theory.” Journal of Sound and Vibration 44(2): 237-253. (1976)

[7] Ueda, Y. “Randomly Transitional Phenomena in the System Governed by Duffings Equation.” Journal of Statistical Physics 20(2): 181-196. (1979)