Christian, JM

Middleton-Spencer, HAJ

English

University of Salford Institutional Repository

chalkdustThe magnetic pendulumA tabletop demonstration of chaosJames Christian and Holly Middleton-SpencerY have to go a long way to beat the magnetic pendulum for demonstrating the deep andprofound nature of physics. One is pictured on the right. On the face of it, a swingingbob seems simple enough to understand. But as is so oen the case, simplicity is maskingcomplexity and the motion possesses an almost magical quality. Here, we will take a glimpse intojust how unpredictable the predictable can really be.The magnetic pendulum: it com-prises a bob with a small magnetsuspended by a string above a baseplane that contains similar magnetsarranged with opposite poles to en-sure aractionPull the pendulum back some distance in any direction,let it go, and prepare to be mesmerised by the way it dartsback and forth in an erratic and seemingly random way.The bob is drawn simultaneously to all three base-planemagnets until, finally, it tends towards a precarious stateof rest above one of them. Playing like this, it does nottake long to become convinced of two immutable facts.Firstly, one can never know beforehand over which mag-net the bob will stop. Secondly, and perhaps more subtly,the motion is not reproducible. No maer how hard onetries to replicate the initial displacement, the bob neverfollows the same winding path twice and the magnet thatultimately ‘wins’ seems to be governed by chance. Howcan we possibly find such randomness in a tabletop toy?9 autumn 2018chalkdustOf course, the bob is not moving randomly at all and we are at best starting it off each time fromonly roughly the same initial conditions. Its motion is prescribed by purely deterministic equa-tions, a combination of classical mechanics and electromagnetics, and at that level there is norandomness. Were we able to release the bob from exactly the same starting position each time,the subsequent motions would all be identical and they would always stop at the same place. Nouncertainty. No unpredictability. No endless fascination!Since our first pull can never be repeated with infinite precision, what we are seeing is sensitivedependence on initial conditions or, more colloquially, the buerfly effect. Any change in input—any, no maer how imperceptibly small—can have a dramatic impact on output. That intriguingphenomenon turns out to be far more widespread and pervasive than one might first imagine.Moreover, it provides our working definition of chaos and crystallises what we mean by saying “asystem is chaotic”. In this article, we will start to explore how the magnetic pendulum embodieschaos in the scientific (rather than the everyday) sense.A phenomenological modelIt is not too difficult to devise a model that exhibits all the key qualitative features of the magneticpendulum. Our approach is a phenomenological one, meaning that we are aiming to capture theessence of the motion using intuitive physical ideas rather than focusing on all the mathematicalminutiae. A relatively easy way forward is to consider looking down on the pendulum from aplan view (see figure below). The bob’s trajectory in the three-dimensional space is projecteddownwards onto the horizontal base plane, and the origin of the (x, y) coordinates is fixed at thecentre of an equilateral triangle. The magnets are subsequently located at vertices X1, X2, and X3,all of which lie along the circumference of a circle with a radius taken to be the unit length.Le: Schematic diagram of the magnetic pendulumRight: Projecting the position of the bob (white square atposition x) onto the horizontal (x, y) plane. The base-planemagnets (red squares 1, 2, and 3) are positioned at the verticesof an equilateral triangle.The position vector of the bob maybe represented by x(t) at time t.To account for gravity, it is suffi-cient for our purposes to consider arestoring force F grav ∝ −x whoseinfluence, due to the minus sign, al-ways acts to pull the bob towardsthe origin x = 0 (the constantof proportionality is set equal to 1,just to keep things simple). Dissi-pation might be introduced by wayof the standard velocity-dependentdamping force familiar from text-book physics. Here, we use F losses =−bu, where u ≡ dx/dt. For someconstant b > 0, the effect of F losses is to drain kinetic energy from the motion through air resis-tance at low speeds. Finally, we must look to include the aractive forces due to the base-planemagnets. It is tempting to reach immediately for the inverse-square rule familiar from Coulomb’schalkdustmagazine.com 10chalkdustlaw of electrostatics and Newton’s law of universal gravitation. We instead opt for a 1/distance4rule as this form tends to describe the forces exchanged by magnetic dipoles.SinceNewton’s second law ofmotion equatesmass× acceleration to the combined forces of gravity,magnetism, and damping, we can write down a governing equation of the formd2xdt 2+ bdxdt+ x =3∑n=1Xn − x(|Xn − x|2 + h2)5/2. (1)Pythagoras’s theorem has been deployed here, and so each contribution in the summation onthe right-hand side of (1) corresponds to a ‘distance/distance5 = 1/distance4’ type of term. Anadditional parameter, h2, has also appeared. Its role is to suppress unphysical (that is, infinite!)accelerations that would otherwise result whenever x approaches Xn. We then interpret h as beingrelated to the average height of the bob above the base plane.The equilibrium points are defined to be those positions x = xeq that are unchanging in time. Sincethe velocity and acceleration of the bob must be zero at those points (hence satisfying dxeq/dt = 0and d2xeq/dt 2 = 0, respectively), it follows from (1) thatxeq =3∑n=1Xn − xeq(|Xn − xeq|2 + h2)5/2 . (2)Aer playing with the pendulum and noting the positions where the bob tends to stop, we mightreasonably expect to find maybe three or at most four solutions (with xeq = 0 being the origin). Itis worth mentioning that the nontrivial roots of (2) do not occur at xeq = Xn, as one might initiallysuspect. Instead, they lie at the same angular positions as Xn (as symmetry demands) but at aradial distance |xeq| that is slightly less than unit length. At these positions, the competing pullsfrom gravity and magnetism are perfectly balanced.Equation (1) does a surprisingly good job at mimicking the unpredictability so readily seen inexperimental demonstrations; the le-hand side is just the damped harmonic oscillator problemfrommechanics while the right-hand side sums over the pairwisemagnetic-dipole interactions. Onthe one hand, any urges to seek analytical solutions should be kept in check. Even this stripped-down toy model confronts us with a formidable mathematical beast living in a four-dimensionalrealm whose axes are (x, y, ux, uy). On the other hand, computing a numerical solution can berelatively quick and easy for given initial conditions, say x(0) = x0 and u(0) = 0.Basins of attractionPhysically, we anticipate that the bob will almost inevitably come to rest at one of the non-trivialequilibrium points xeq [cf. (2)] as t → ∞. Those special points can be thought of as positions inthe (x, y) plane that aract the trajectory, and accordingly they are oen referred to as fixed-pointaractors. The idea now is to use a computer to carry out a systematic set of simulations, recordingwhich magnet ‘wins’ (interpreted as the output) as we vary the starting point x0 (taken to be theinput). By associating the outcome of each computation with a colour (eg red for magnet 1, white11 f 2018chalkdustfor magnet 2, and black for magnet 3), we can overlay the output on top of the input (x0, y0) planeto produce a kind of abstract map. The set of all initial conditions lying in the red region is thebasin of araction for magnet 1—that is, any x0 lying on a red point will always end up at magnet1 (and similar for the other colours and magnets), though the colour itself gives no informationabout the path taken by the bob to arrive at that point.Figure 1. Basins of araction for the magnetic pendulum with b = 0.1 and h2 = 1/4. In thefirst pane, the grey squares denote the position of the base-plane magnets and the doed greyline is a circle with radius equal to the unit length. The second and third panes show successivemagnifications.Figure 2. An illustration of FSS in the magnetic pendulum. Two initial conditions that are arbi-trarily close together can give rise to subsequent trajectories that will start to move away fromone another aer a finite amount of time. The red path ends at magnet 1, while the black pathends at magnet 3.Using this recipe, we discover a rather striking paern (see fig. 1). Regions around the originappear relatively simple. There are large single-colour lobes which indicate that variations in x0tend to have lile impact on which magnet wins. Further out beyond the unit circle, there is muchgreater complexity and all three colours are intertwined in a beautifully complicated way. In thoseregions, the pendulum is extremely sensitive to changes in x0. Successive zooming-in suggests theintertwining survives down to smaller and smaller length-scales. That feature—proportional levelsof paern detail persisting under arbitrary magnifications—is a defining characteristic of a fractal.chalkdustmagazine.com 12chalkdustA less obvious but equally fascinating property relates to the nature of the boundary separatingtwo differently-coloured regions. We typically cannot cross from a region of red to an adjacentregion of black without touching the white (similar is true for other permutations of colours). Thissituation is reminiscent of the delightfully strange Wada property from the field of topology.Final state sensitivityFigure 3. Variation of the basins of arac-tion for the magnetic pendulum as dampingis increased. The second and third columnsshow successive magnifications. Other pa-rameters and domains of the (x0, y0) planeare the same as those in fig. 1.The basins of araction have three-fold rotationsymmetry about the origin, which is a conse-quence of the equilateral-triangle arrangement ofmagnets and the initial condition u(0) = 0. Theirdetails also depend crucially on system parame-ters. One might consider what happens, for in-stance, when the level of damping is increased (seefigure to the right) through b = 0.125 (first row),b = 0.150 (second row), b = 0.175 (third row), andb = 0.200 (boom row). The paern becomes lesscomplex and, accordingly, the pendulum less sen-sitive to small fluctuations in x0. However, theirkey features remain intact: the persistence of self-similar structure (fractality) and complex bound-aries that tend to involve all three colours. TheWada-type property is still present in the right-hand column of the last two rows, but it is not ob-vious from these figures.A helpful way to quantify just how strongly thelong-term state of a system depends upon smallfluctuations at its input is to estimate the fractaldimension 1 < D ⩽ 2 of the basin boundaries.One selects a set of NΓ points in a domain Γ of the(x0, y0) plane and tests each of them in turn forthe property of final-state sensitivity (FSS) by considering a triplet of initial conditions: say (x0, y0),(x0+ ε, y0), and (x0− ε, y0), where 0 < ε ≪ O(1) can be interpreted as an error or as a limit to ourexperimental precision. When the winning magnet is the same for all three trajectories, then thefinal state is independent of ε and (x0, y0) is, accordingly, free from FSS. Alternatively, think of ε asthe radius of a small disc centred on (x0, y0), somewhere within which the ‘true’ initial conditionlies. FSS, as demonstrated in fig. 2, appears whenever that disc impinges on a basin boundary andthus overlaps more than one colour.If the total number of points possessing FSS for a given ε is denoted by Nε , one finds that thatNε/NΓ ∼ εα . The parameter α ≡ 2 − D is known as the uncertainty exponent and it satisfies the13 autumn 2018chalkdustinequality 0 ⩽ α < 1. Finally, we obtain the uncertainty dimension D, whereD = 2− d log10(Nε/NΓ)d(log10 ε)(3)and a larger D is indicative of increased susceptibility to initial fluctuations. The paerns shownin the middle panes of figs 1 and 3 turn out to have dimensions in the range D ≈ 1.32 (for b = 0.1)to D ≈ 1.16 (for b = 0.2). It follows that the basin boundaries for lightly-damped pendula tendto be associated with higher values of D. More generally, we now see a connection between thedimension of an abstract fractal paern (which, crucially, need not be an integer such as 1 or 2)and the physical property of FSS.Concluding remarksIn this article, we have started to unpick some of the intriguing behaviour exhibited by what is, inreality, a simple toy—one that never fails to capture the imagination of university students in lec-tures and Ucas applicants (and their parents alike!) at open days. The apparently erratic swingingand unknowable terminus of a bob are not quite so ‘random’ as one might first suppose from afew naïve observations. The essential ingredient giving rise to all this rich and diverse behaviour isthe interplay between the three constituent feedback loops (here, due to gravity, dissipation, andmagnetism).Although we have considered the barest of bare-bones models (from pretending gravity providesa restoring force proportional to −x, to suppressing the fully-vectorial character of magnetic in-teractions), the beautiful complexity of nature survives and we simply cannot get rid of it. That, itseems to us, is a mind-blowing conclusion!James Christian and Holly Middleton-SpencerJames is a lecturer in physics at the University of Salford with research interests covering various theoreticalaspects of electromagnetics and fluids. When not writing papers or teaching, he spends most of his timeplaying with fractals and wishing he were a mathematician instead.Holly is James’s long-suffering graduate student and has just completed an MSc on waves scaering fromfractal screens. She will shortly be starting a PhD in applied maths at the University of Newcastle, studyingvortices in superfluids. In the end, she’ll be a real mathematician.Did you know...…that if you walk randomly on a 2D laice, you will eventually end up back where youstarted, but if you do the same on a 3D laice you might not. Or, as Shizuo Kakutani putsit: “A drunk man will find his way home, but a drunk bird may get lost forever.”Did you know...…that primes are so scary that there is a prime named aer one of the seven princes of hell:Belphegor’s prime (1000000000000066600000000000001).chalkdustmagazine.com 14

The magnetic pendulum : a tabletop demonstration of chaos

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The magnetic pendulum : a tabletop demonstration of chaos

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