Chaotic Analysis of a Mechanical Oscillator With Bistable Potential by PoincareMapping and Phase Space Analysis

Calvin MilliganPhysics Department, The College of Wooster, Wooster, Ohio 44691, USA

(Dated: May 7, 2015)

The purpose of this lab was to determine the effect that various damping and periodic drivingparameters have on a Duffing mechanical oscillator with two different potential wells. This was doneby experimenting with variations in the system and deriving the equation of motion for the system,x + γx = αx − βx3 + a sin (ωt), which describes the general motion of the system. The equationof motion includes definition of the initial parameters of the system, making it possible to makereasonable predictions about the motion of the system over time. Conclusively, two stable fixedpoints were identified and one unstable fixed point was identified. At these points the system isable to stay at rest when under particular initial conditions and without the influence of an exteriorforce actively driving the system. I discovered that when the amplitude of this specific system isincreased from 4.84 V to 5.27 V the range of velocities and positions also increased for the system.Also, increasing the damping parameter of the system from 2.35 cm to 1.30 cm caused the systemto reach a non chaotic limit. The initial parameters of the system caused large variations in motionthat included changing in velocity around the equilibrium points and changing in the position limits.

I. INTRODUCTION

Chaos occurs in a mechanical system when the motion,energy, and force that a system is subject to are con-stantly fluctuating, causing non-repetitive behavior overa long period of time. Even in chaotic systems it is verypossible to predict the motion of the system for finite pe-riods of time given the initial parameters and conditionsof the system. However reasonable prediction of motion,energy, and force within the system becomes exponen-tially harder to predict as the limits of time increase toinfinity. Henri Poincare was one of the first scientiststo explain chaos around 1900. He postulated that thenatural curvature of orbits could be described geomet-rically using the idea of constantly evolving state vari-ables; state variables being periodically changing vari-ables. Poincare was able to give further intuition intothe matter by showing that reasonable approximationsof these variables could be investigated at a reoccurringposition [2]. At a successive point, a reasonable predic-tion of condition values can be made using the equationof motion to calculate from a previously known value.Other than its application to orbitals, chaotic motion candescribe other nonlinear phenomenon applicable to nat-ural systems. German electrical engineer Georg Duffingconstructed one of the most widely used nonlinear motionequation in 1918, the Duffing Equation [2]. His equationtakes into account the effects upon a simple harmonic os-cillator when manipulated by an periodic driving force, acontinuous damping or drag force, and a nonlinear term,as well as initial parameters associated to these terms.Depending on the given conditions the system may actchaotic over discrete intervals and periodic over other in-tervals. In 1963, Edward Lorenz noticed that when hesimulated a weather pattern on separate occasions usingthe same model he obtained vastly different results [4].He concluded that his initial conditions in each instancewere slightly different. This idea, known as the butterfly

effect, says that the behavior of a chaotic system, wherevalues are obtained with a certain systematic random-ness, changes substantially even with slight changes ininitial conditions. It is important to analyze chaos in or-der to find patterns within a systematically random setof conditional variables that allow us to weed out andpredict particular variable values as time processes. Thepredictions that we are able to make are useful in predict-ing expectations for economic patterns, weather patterns,and mechanical systems such as the oscillator studied inthis experiment.

II. THEORY

It is necessary to derive the equation of motion of thesystem from the kinetic and potential energy equations inorder to consider the the motion of the bistable mechani-cal oscillator and its dependence on the initial conditionsof the system. Just like in any other mechanical system,finding the Lagrangian by taking the difference betweenthe kinetic and potential energies allows us to solve forthe second-order Euler-Lagrange equation to obtain Duff-ing’s historic equation for the damped, driven nonlinearoscillator [3]. Following this logic we get

L = T − V ;d

dt

∂L

∂x=∂L

∂x, (1)

which, for this particular set of variables and initial con-ditions, implies

x+ γx = αx− βx3 + a sin(ωt), (2)

where x is angular displacement, α is a linear elastic con-stant related to the spring constants of the oscillator, βis a non-linear elastic constant related to the force thesprings exert as they stretch, a is the amplitude of theforce dependent on the length of the mechanical arm (see

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Figure 1, label 10), ω is the frequency of the force givenby 2π of the period T that it take the arm to make onerotation, and finally γ is the damping coefficient propor-tional to the velocity of the system. In this particularexperiment I manipulated a, ω, and γ to change the be-havior of the system [1]. Here we can also say that ifthere was no damping parameter γ, our equation of mo-tion would look like

x = αx− βx3 + a sin(ωt), (3)

and the natural frequency of the system would we of theform

ωo =

√βx2 + α

m(4)

where the frequency of the system is going to be chaoticeven in the absence of damping since β is affected by theposition of the non inform mass of the disc and and ran-domized position of x. Under particular conditions thesystem can behave predictably or chaotically. Chaoticbehavior displays that even slight manipulation of theinitial conditions can lead to drastically different results.When the system is underdamped the system will satisfyωo¿γ, where we will see the system act chaotically, as inmost cases explored in this experiment suggest. Whenthe system is overdamped the system will satisfy ωo=γ,where we will see the system reach a limit of motion astime progresses. We can see this sort of motion in Fig-ure 2 in the middle graph. When the system is criticallydamped the system will satisfy ωo < γ, where we willsee the damping parameter outweigh the elasticity of thesystem and the driving force will be too strong [2]. Thisfinal case was not explored to its fullest extent, when themotion of the system flatlines to fixed point over a smallperiod of time, even at large initial conditions of poten-tial and velocity. We do see semblance of this in Figure2 in the bottom graph since the motion is restricted overtime.

Initially looking at the position versus time of the non-linear oscillator helps to understand the motion of thesystem. Looking at Figure 2 we can see that under cer-tain parameters and time constraints the system can actperiodically, chaotically, or some combination of both. Incertain instances we see period doubling, where a motionis repeated upon consecutive intervals, where the oscilla-tor only moves close to one fixed point.

Next, to understand the energy related to the systemwe can look at the potential energy curve as displayed inFigure 3. The general form of the potential equation is

V = −αx2

2+ β

x4

4, (5)

where V is a function of position x and α and β areparameters.

We can study the behavior of chaotic motion under twoseparate conditions, over a discrete intervals of time or

FIG. 1: Displayed is a comprehensive diagram of the bistablemechanical oscillator used in this experiment. Each key com-ponent of the system is annotated by a number. The massattached to the metal wheel (1) is free to rotate about a PascoSmartPulley (2). The springs (5) that manipulate the motionof the system are attached to a string that can rotate thepulley in two directions. Next, there is a power supply (3)with a variable voltage (4) that can be measured by the volt-meter (7). The Pasco Mechanical Oscillator (6) is driven bythe power supply and has a variable amplitude (10). The me-chanical arm (10) of the Oscillator passes through the PascoPhotogate (8), in order to measure the frequency. Finally,there is a damping magnet (9) behind the metal disc that canmove closer to the disc to increase the damping effect.

over a continuous period of time. Over a continuous in-terval we talk about the phase space of the system wherewe relate velocity and position over a related time inter-val. As shown in Figure 3 we observe an eight-shapedcurve, called a separatrix, representing the phase spacerelationship between velocity and position. Each phasebegins and ends at a similar position on the graph andcontinually revolves. The phases create a concentric pat-tern and no successive phase is identical. If we changedthe sampling of the phases in Figure 3 we would see dif-ferent spacing and behaviors between phases. Each phasegives us the range of velocities and positions over contin-uous, but periodic intervals of time. We can easily seewhere the fixed points of the system are located. At the

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FIG. 2: The graphs depicted are position vs. time plotsfor different initial parameters of four separate data runs.The parameters being manipulated correspond to the driv-ing frequency, driving amplitude, and damping position. Thetop plot (blue) has initial parameter of 3.40 cm amplitude,5.27V frequency, and 1.30 cm damping position. The mid-dle plot (red) has parameters of 4.15 cm amplitude, 5.27Vfrequency, and 2.35 cm damping position. Finally, the bot-tom plot (green) has parameters of 4.15 cm Amplitude, 4.84Vfrequency, and 1.30 cm damping position.

stable fixed points the velocity is converging to zero andthe phase lines are moving towards a particular position.These are shown by the holes of the eights. The unstablefixed point is shown where the phase lines are deflectingaway from a particular position near the center of theeight. At the unstable point the potential energy reachesa local maximum. Under particular initial conditions itis possible for the oscillator to rest at the unstable fixedpoint. For certain initial conditions the oscillator canalso move about a single stable fixed point with a limitedangular position.

The other useful type of analysis for a phase space plotis a Poincare map, over a discrete and periodic interval.The idea of a Poincare plot is that particular trends forvelocity and position tend to occur over a periodic in-terval. If we think about the phase space as constructedconcentrically outward about a center phase, you can cre-ate a cross sectional sample by cutting a line perpendicu-larly through all the phases at a particular junction. We

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FIG. 3: The graphs depicted show phase space plots of thedifferent initial values for four separate runs, corresponding tothe driving frequency, driving amplitude, and damping posi-tion. The top left (red) has parameters of 4.15 cm amplitude,5.27 V frequency, and 2.35 cm damping position. The topright (black) has parameters of 4.15 cm amplitude, 5.27 Vfrequency, and 1.30 cm damping position. The bottom rightplot(green) has parameters of 4.15 cm amplitude, 4.84 V fre-quency, and 2.35 cm damping position. Lastly, the bottomright (blue) has initial parameter of 4.15 cm amplitude, 4.84V frequency, and 1.30 cm damping position. The phase wasplotted for each run for identical time intervals of 40 seconds.Each behavior has slightly different patterns due to the inter-vals of time chosen and the starting position of the oscillator.The most different plot was the blue plot which moved ina non chaotic fashion around a single fixed point within thesystem.

can see this concept by observing the colored dotted linesintersecting through the phase spaces in Figure 3. How-ever, it would be possible to receive a slightly differentPoincare map pattern by choosing a different position ofthe driving force at which to record the velocity versesthe position, as shown by the grey intersection in Figure3. In this instance their would be different fractal pat-terns, within the Poincare map that would limit particu-lar velocity and position combinations. Fractal patternsare continuously repeating patterns that rotate about thefixed points. Shifting the position at which the velocitiesand positions are recorded will rotate the fractal patternstoward the fixed points but will keep similar shape un-der the same initial conditions. By changing the initialconditions the fractal patterns can rotate, distort and ex-pand, however the fixed points will still remain the mostdense regions of the map. In these maps each successivepoint depends on the previous point, creating a type ofchain reaction that can distinctively change the behaviorof these maps.

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III. PROCEDURE

This lab was conducted using DataStudio for data col-lection. The equipment used is displayed in Figure 1with detail. In the bistable oscillator, there is a disc ofnonuniform weight suspended above a flat surface andattached to a SmartPulley. The pulley and the disc areable to radially move by the way of a configuration of twostrings. Both strings are attached to two springs; one isstationary and the other is attached to a mechanical os-cillator. The mechanical oscillator rotates, changing thedriving force upon the pulley system.The Pasco Smart-Pulley and the Pasco Photogate are connected to thePasco Interface box that transmits data to the DataStu-dio software for Macintosh. The software records severaldifferent measurements to create the Poincare, the phasespace, the position vs. time, and the potential energy di-agrams. Variations of initial parameters were establishedfor five separate runs. The changing parameters includedthe amplitude of the driving force (3.40 cm and 4.15 cm),period of the driving force (5.27 V and 4.84 V), and posi-tion of the damping magnet (2.35 cm and .30 cm). Theseparameters are summarized in Table 1.

TABLE I: This is a summary of the data runs performed inthis experiment. Five data runs were collected using variousparameters, as specified.

Run Drv. Amp. (cm) Freq. (V) Dmp. Pos. (cm)

1 4.15 4.84 2.35

2 4.15 4.84 1.30

3 4.15 5.27 2.35

4 4.15 5.27 1.30

5 3.40 5.27 1.30

Data was collected in 1:30hr increments. In order tocreate the position verses time diagram the interface sim-ply tracked the angular position of the smart pulley,at 0.01s per measurement. The phase space plot gavethe angular velocity verses the angular position and wasmeasured over a continuous interval. In contrast, thePoincare map plotted the angular velocity verses the an-gular position at the discrete interval of time when themechanical oscillator crossed the path of the Photogate.Finally, the potential energy curve plotted the potentialenergy of the system as the angular position of the os-cillator changed. All the data collected was exported toIgor Pro for analysis and manipulation. All figures wereconstructed using Igor Pro, excluding Figure 1 and Table1.

IV. DATA AND ANALYSIS

Figure 2 gives the angular positions verses time graphs.The last graph shows us an example of, not chaotic, butperiodic motion when the oscillator is constrained to a

small interval with a repeating pattern of motion. This isthe simplest case where position over time is predictable.The first graph displayed has a larger frequency and asmaller amplitude with a damping magnet. We see thatthe system oscillates around the two fixed points withoccasional oscillations between the fixed points and fre-quent consecutive oscillations around a single fixed point.The second graph shows a transition between chaotic andperiodic motion. This occurs because the damping driv-ing force drives the system when it is at various positionsand directions for the first 60s, but it eventually reacheda predictable pattern when the motion of the oscillatorwas in a small interval.

The next set of plots, Figure 3, shows the phase spacesof four different parameter sets. The red and black phasespaces have different damping parameters as well as thegreen and blue phase spaces. The black and blue have dif-ferent driving frequencies as well as the red and the green.First, we see as the damping parameter increases there ismore amplitude of motion around the fixed points shownfrom red to black. In other words the oscillations becomelarger since the damping parameter forces the system totake on a smaller range of velocities and positions. Be-tween the green and the blue we see that as the dampingincreases the motion of the system tends to center arounda single fixed point with a smaller range of velocities andpositions. This same occurrence happens between theblack and the blue when the frequency is decreased. Itis observed that a particular fixed point, represented bythe larger circular pattern on the left, created a largerrange of position and velocity, with the exception of thebottom right phase plot, which had a discrete range ofmotion, considering it was non chaotic.

In Figure 4 we see the Poincare maps for 4 separateparameter sets. The black and the blue plots now differin the amplitude of the driving force. We see that therange of velocities and positions expands slightly as theamplitude increased. There also seems to be a shift inthe fractal patterns as the amplitude increased. The dif-ference shows more empty spaces where the velocity andthe position are not expected to be, within the range ofvalues for the blue map. In sum the green map coversa wider range of values then the blue, which has lesseramplitude. The red and the black differ in damping pa-rameters. As the damping increases we see that there ismore empty space near the fixed points even though thepatterns seem to be very similar in shape. This was asimilar observation in the phase plot for a similar changein damping.

To give a sense for how the Poincare plots are devel-oped over a period of time refer to Figure 5, showing theprogression of the point values for a particular parame-ter set. Figure 6 shows the Poincare plots for a systemwith damping and without damping. This shows that thebottom right plot, that includes damping, reaches a morediscrete range of values for position and velocity, basedon the complexity of the fractal patterns and the smallerdispersion of points. The green plot covers a wide range

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FIG. 4: The graphs depicted show Poincare plots for the dif-ferent initial values of four separate data runs, correspondingto the driving frequency, driving amplitude, and damping po-sition. The top left (red) has parameters of 4.15 cm ampli-tude, 5.27 V frequency, and 2.35 cm damping position. Thetop right (black) has parameters of 4.15 cm amplitude, 5.27 Vfrequency, and 1.30 cm damping position. The bottom rightplot(green) has parameters of 4.15 cm Amplitude, 4.84 V fre-quency, and 2.35 cm damping position. Lastly, the bottomright (blue) has initial parameter of 3.40 cm amplitude, 5.27V frequency, and 1.30 cm damping position. The fractal pat-terns tend to move around the known stable fixed points andhave slight distortion under changing parameters. Each plotcontains 7 500 data points which varies in time scale for thetwo different frequency values.

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FIG. 5: This shows the succession of the Poincare map at10 points, 100 points, 1000 points, and 10 000 points. Theparameters for this data set are; 4.15 cm Amplitude, 4.84 Vfrequency, 2.35 cm damping position

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FIG. 6: The graphs depicted show Poincare plots of two dif-ferent initial values for damping position.The bottom plot(green) has parameters of 4.15 cm amplitude, 4.84 V fre-quency, and 2.35 cm damping position. The top plot (blue)has initial parameter of 4.15 cm amplitude, 4.84 V frequency,and 1.30 cm damping position. The bottom was plotted over7 500 data points while the top was plotted over only 1000data points because it had a very small range of velocity andposition values as time passed.

of values but we can see that the most dense populationsoccur at stable fixed points with a probable velocity atthe fixed points.

Figure 8 displays a potential energy plot and best fit fora large amplitude low frequency system without damp-ing. We see that the expected potential energy equationdoes not hold true because the double potential wellswere not exactly equal in position and the spring con-stants of the oscillator were not exactly equal. We dohowever get a sense for where the two stable and the un-stable points should occur at. Setting the derivative ofthe potential function equal to zero and solving for theminimum and local maximum values gives us a roughidea of where the critical points of the system occur. Thepotential function for every parameter set was approxi-mately equal since the fixed points did not change.

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FIG. 7: This graph displays the potential plot for the pa-rameters; 4.15 cm amplitude, 4.84 V frequency, and 2.35 cmdamping position. The figure also includes an approxima-tion fit that shows the stable and unstable fixed points whenthe slope of the fit line approaches zero. The green potentialcurve-fit has the form of V (x) = −k1(x−xo)2+k2(x−xo)4+ko,where ko=-223± 6, k1= (6.6± 0.8)× 10−3, k2= (1.9± 0.2)×10−7, and xo= -220. The value ko corresponds to a shift inthe y-axis and the value xo corresponds to a shift in the x-axisrelated to the position of the unstable fixed point.

V. CONCLUSIONS

Albert Einstein once said, ”As far as the laws of math-ematics refer to reality, they are not certain, and as far

as they are certain, they do not refer to reality [3].” Thisstatement give us a sense of what chaotic motion tells us,which is that we can explore different forms of the data ina system such as the one we explored in this experiment,but we cannot rigorously define the motion in a mathe-matical sense. By using data analysis, such as the phaseplot (continuous velocity vs. position), the Poincare plot(periodic velocity vs. position), and the potential energyplot (energy vs. position), we can make reasonable pre-dictions about how a system will act by observing thepatterns of motion around the fixed points. The motionof the bistable oscillator that was analyzed in this exper-iment gave us some sense of the relative patterns thatoccur when the parameters of the system are changed.We know that the non-linear changing of the elastic con-stant β and the linear elastic constant α directly affectthe magnitude of the potential energy for the system atvarious angular positions of x. Also we found that thenatural frequency ωo could be compared to the dampingparameter γ in order to explain the conditions of the sys-tem. This comparison gave a reasonable explanation forthe non chaotic behavior of the oscillator of amplitude4.15 cm, frequency of 4.84 V, and 1.30 damping position.The motion of the system was not chaotic over all pa-rameter sets. Subtle changes in the parameters gave usnoticeably different behavior. There were distinctive dis-tortions of the Poincare plots as the parameters changedthat allowed us to make inferences on how the parameterschanged the motion of the system.

[1] The College of Wooster, Department of Physics, JuniorIndependent Study Lab Manual, 57-60, (Spring 2015).

[2] N. Mann, Classical Mechanics Lecture 28 Notes: TheDamped Driven Simple Harmonic Oscillator, 104-9 (Fall2014).

[3] B. Davies, Exploring Chaos: Theory and Experiment, 57-60, (Purseus Books 1999).

[4] G. Williams, Chaos Theory Tamed, 10-520, (JosephHenry Press 1997).