Exactly solvable chaos in an electromechanical oscillator

Benjamin A. M. Owens,1,a) Mark T. Stahl,2 Ned J. Corron,3 Jonathan N. Blakely,3

and Lucas Illing41Department of Mechanical Engineering and Materials Science, Duke University, Durham,North Carolina 27708, USA2NASA, Marshall Space Flight Center, Alabama 35812, USA3U.S. Army Research, Development and Engineering Command, Redstone Arsenal, Alabama 35898, USA4Department of Physics, Reed College, Portland, Oregon 97202, USA

(Received 3 May 2013; accepted 17 June 2013; published online 15 July 2013)

A novel electromechanical chaotic oscillator is described that admits an exact analytic solution.The oscillator is a hybrid dynamical system with governing equations that include a linear secondorder ordinary differential equation with negative damping and a discrete switching condition thatcontrols the oscillatory fixed point. The system produces provably chaotic oscillations with atopological structure similar to either the Lorenz butterfly or R€ossler’s folded-band oscillatordepending on the configuration. Exact solutions are written as a linear convolution of a fixed basispulse and a sequence of discrete symbols. We find close agreement between the exact analyticalsolutions and the physical oscillations. Waveform return maps for both configurationsshow equivalence to either a shift map or tent map, proving the chaotic nature of the oscillations.VC 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4812723]

A common perception is that chaos is an unwieldy dy-namical behavior that is best studied using numericalmethods. In general, this view is correct. The vast major-ity of chaotic systems encountered in the fields of physics,biology, fluid dynamics, mechanics, and oceanography—to name a few—have no solutions that can be written outon paper in terms of known simple functions. Therefore,chaotic systems are typically viewed through the lens ofnumerical integration and characterized by estimates oftheir topological features and statistical properties. Whilevaluable, these characterizations are complicated by thefact that the necessary numeric approximations lead tosolutions that are at best shadowed by a true mathemati-cal solution. This lends some uncertainty to results ofnumeric approaches. There is a class of chaotic systems,however, that penetrates this shroud of uncertainty,where exact solutions are known. Far from mathematicalcuriosities, such systems can be implemented in practice.As an example, we show in the current paper how anelectromechanical experimental system, consisting of amagnetic mass on a spring that oscillates inside a coil thatis used to sense and influence the magnet’s motion, resultsin a hybrid system, whose chaotic oscillatory behaviorcan be described by an exact analytical solution.

I. INTRODUCTION

The exploration and characterization of chaos are welldocumented for low-dimensional systems. The complex dy-namics of chaotic deterministic systems is generally charac-terized through numerical integration of the systemequations, yielding estimates of the Lyapunov exponents,

attractor topology, and fractal dimension.1–3 Explorations ofchaos exist in a vast array of fields including fluid dynam-ics,1,4 cardiology,5 neurology,6,7 mechanics,8,9 oceanogra-phy,10 and communications.11,12 Because of the nature ofthese systems, analytical solutions are commonly thought tobe unobtainable.

However, exceptions exist. Exact analytical solutionshave been developed for several dynamical systems showingchaotic behavior. Katsura and Fukuda laid out several exam-ples of one-dimensional chaotic maps with exact solutions.13

Umeno applied chaotic solvability to Belousov-Zhabotinskichemical reactions, matching analytic functions to experi-mental return maps.14 Rollins and Hunt created a simplenonlinear circuit producing a one-dimensional, noninvertiblemap with a period doubling route to chaos that admits exactcalculation.15 Drake and Williams showed equivalentdescriptions of common chaotic systems in the form of lin-ear, time-invariant systems with random inputs, which pro-duced outputs indistinguishable form those of the chaoticsystems.16 Corron explored chaotic ordinary differentialequations possessing an exact symbolic dynamics.17 Morerecently, Corron and Blakely showed a theoretical hybridsystem with chaotic folded band topology and exactsolvability.18

In the present paper, we report the construction andbehavior of an electromechanical chaotic oscillator, whichcan be described by a model that admits exact analytic solu-tions. Following the example from theoretical and purelyelectrical systems,14–19 this oscillator is a hybrid constructionmade up of analog and digital systems. Instead of voltageand current oscillations, however, chaotic oscillations ofproof masses are considered.

The experimental system consists of a linear mechanicaloscillator and a hybrid electrical system with analog and dig-ital components. The mechanical oscillator is formed by twoa)benjamin.owens@duke.edu

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connected magnetic proof masses attached to springs. Eachproof mass oscillates within a wire coil allowing electromag-netic coupling between the mechanical and electrical part ofthe system. The primary coil is electrically driven to providea negative damping effect, while the secondary coil exerts aconstant force. The constant electromagnetic forcing isswitched between two values based on the position and ve-locity of the system. The electrical system can be tuned toproduce two types of chaotic oscillations; one with a topo-logical structure similar to the Lorenz butterfly attractor1 theother similar to a R€ossler’s folded-band.2

The mathematical model equations used to describe thesystem provide solutions that are both solvable and provablychaotic. These solutions can be written down in the form of alinear convolution of a symbol sequence and a fixed basisfunction. Given the discrete, measured symbols of the elec-tromechanical system, an analytical solution for the systemcan be constructed that matches well with the observed sys-tem behavior. This match between theory and experimentproves the realizability of mechanical configurations of thesetypes of systems and opens the door for further explorationinto solvable chaotic systems.

The paper is organized as follows. Section II introducesthe experimental system. Section III discusses the governingequations and the analytical solutions. Next, Sec. IV lays outresults from experimental trials and compares the collecteddata to analytic solutions reproduced from the symbolsequences of the experimental results. Conclusions then fol-low in Sec. V.

II. ELECTROMECHANICAL OSCILLATOR

The electromechanical oscillator is a mechanical spring-mass system that is coupled to active electronic circuits. Inthis section, we first describe the mechanical configurationof the spring-mass system and derive a mathematical modelfor the mechanical oscillator. We then describe the design oftwo electromagnetic coils that couple the active electroniccircuits to the mechanical system.

A. Mechanical oscillator

The full mechanical system and schematic are shown inFig. 1. The frame for the system is constructed using steelstrut channels and has a height of 1.75m and a width of 1m.This rigid frame holds a spring-mass system, which isassembled using two opposing spring sets and two separatemasses connected by braided fiber fishing line. Each springset comprises four parallel springs that are suspended frombrackets at the top of the frame. Hanging from each springconfiguration are three joined cylindrical neodymium mag-nets, which provide the mass for the mechanical oscillator. Apair of rod and bolt connectors are attached to both ends ofthe magnets and provide connection points for the fishingline as well as rigid support for the vertical orientation of themagnets. The hanging masses are connected to each otherusing a long line passing through low-friction pulleys at thebottom of the frame. This symmetric configuration was cho-sen to minimize the effects of gravity on the mechanicalsystem.

The complete line of the spring system down throughthe magnets and pulleys on one side and up through magnetsand the springs on the other constitutes the primary axis ofmotion for the system, which we denote as y. To mitigate ra-dial motion away from this axis, all the attaching linesbetween the various mechanical components are held underconsiderable tension. As seen in Fig. 1, each of the left andright segments of the axis of motion line pass through coilsof magnetic wire wound about a section of PVC pipe. Eachcoil is controlled by electrical circuitry and coupled to themagnets by induced magnetic fields in the coils. In Fig. 1,the coil on the left is designated the primary coil, and itsfunction is to induce a negative friction. The coil on the rightis called the secondary coil and provides a switchable fixedpoint. It is the effect of the coils and the active electronicsthat yield the solvable chaotic dynamics.

The eight springs used in the system come fromMcMaster-Carr and have estimated individual constants of

FIG. 1. (a) Labeled image and (b) schematic of a linear oscillator systemwith electrical and mechanical components and measuring equipment.

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19.2N/m and an unstretched length of 6.35 cm. Under experi-mental line tensioning and no electromagnetic coil forcing,the static spring stretch is set to 17.1 cm, which is well belowthe maximum elastic deflection of 26.0 cm and within the lin-ear range of motion for the springs. All six magnets are rare-earth permanent neodymium of type N52 from K&JMagnetics with a maximum residual flux density Br¼ 1.48T.Each magnet measures 2.54 cm in diameter, 1.27 cm in length,and 48.0 g in mass. With the rigid attaching pieces, the magnetsystems on either side have a mass of "460 g for a combinedsystem mass of" 920 g.

B. System model

To derive the equations of motion, a couple of considera-tions are made to simplify the system. First, the pulleys areassumed to be massless and frictionless, allowing any noncon-servative friction losses or inertial effects resulting from theirmotion to be ignored. Next, because the two sides are attachedby a single line, the tension is assumed to be equal and con-stant throughout the line. This also means that the oscillationsof the magnets on each side can be assumed to be equal alongthe primary axis and centered about a point set by the line ten-sion and the weights of the respective sides of the system.

Each set of magnets, considered now as a single, longmagnet, is cylindrical, uniformly magnetized in the verticaldirection, and moves along the central axis of a coil, i.e.,along the vertical axis of motion y. When a current I isapplied to the coil, it generates a magnetic field Bcoil thatinteracts with the magnet, resulting in a force Fmag.Throughout, we assume that the current I varies slowly intime such that Bcoil can be treated as a steady state field.Cylindrical symmetry of the coil’s magnetic field togetherwith the vertical alignment of the magnet implies that theresulting magnetic force is in the vertical direction,

Fmag ¼ Fmagy: (1)

Taking into account previous considerations, we canapply Newtonian mechanics to get the equation of motion

m€y þ c _y þ ky ¼ Fmag; (2)

where Fmag is the effective magnetic force, k the effectivespring constant of the system, and c the lumped viscousdamping coefficient.

In this system, the magnetic force due to the primarycoil Fpc is directly proportional to the velocity of the oscillat-ing magnet and represents a negative viscous friction term,whereas the force due to the secondary coil Fsc represents anoffset term that is switched between two constant values as afunction of the magnet’s position and velocity. The totalmagnetic force is therefore

Fmag ¼ Fpc þ Fsc ¼!2

RN_y þ Fscðy; _yÞ; (3)

where ! is a geometry dependent coefficient and RN is themagnitude of the negative resistor, both detailed in Sec. II.Using Eq. (3), Eq. (2) can be written as

€y & 2fxn _y þ x2nðy& ydðtÞÞ ¼ 0; (4)

where the resonance frequency is x2n ¼ k=m; the effective

damping ratio is f ¼ ð!2=RN & cÞ=ð2mxnÞ and yd(t) is atime-varying oscillation offset caused by Fsc. The time-varying offset yd is modeled by a guard condition and is afunction of the oscillator amplitude. It is

_y ¼ 0 ) yd ¼ YB þ ðYT & YBÞ ' Hðy& Y0Þ; (5)

where YB and YT are the bottom and top set points, Y0 is athreshold, and the Heaviside function H is defined as

Hðy& Y0Þ ¼1 if y ( Y00 if y < Y0:

!(6)

The choice of the threshold Y0 enables selection of the but-terfly or folded-band oscillator. To obtain the butterfly topol-ogy, the threshold is the midpoint Y0¼ (YT þ YB)/2. For thefolded band, the threshold can be either set point Y0¼YT orY0¼ YB. The guard condition is realized using a position sen-sor and control electronics that switch the fixed points bycontrolling the force FSC due to the secondary coil.Additionally, the electronics controlling the primary coil canbe tuned for different levels of negative damping on themoving masses. The design and implementation of the mag-netic coils are described in the remainder of this section.

C. Coil design

The coils are made by winding a single continuous cop-per wire in many layers onto a PVC pipe, one-layer on top ofthe other and each layer shortened with respect to the previ-ous one by starting it 2.54 cm further away from the coil’stop end. With this winding, we approximate a coil for whichthe number of turns per unit length, n(y), increases linearlywith distance,

nðyÞ ¼ cny; y 2 ½0; Lc*: (7)

Here, the origin of the coordinate system is at the top of thecoil, the y-axis is pointing downward, Lc is the length of thecoil, and cn ¼ 2Nturns=L2c " 16 cm&2 is a constant.

To obtain an expression for the magnetic force, wemodel the coil, with its dense helical windings, as a contin-uum of current-carrying circular loops. By Newton’s thirdlaw, the force of a single such loop on a magnet is identicalin magnitude and opposite in direction to the force of themagnet on the loop. The force of the entire coil on the mag-net, Fmag, can therefore be obtained by summing the contri-butions from the force of the magnet on all loops,

Fmag ¼ &ðJcoil + BmagdV

¼#2prcI

ðLc

0

nð"yÞBq;magðrc; "y & yÞd"y$y; (8)

where Jcoil is the coil current density, I is the current in thecoil’s wire, rc is the coil radius, and Lc is the coil length.Bq,mag(rc,"y & y) is the radial component of the magnet’s

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magnetic field expressed in cylindrical coordinates, whilethe magnet’s geometric center is at position y. The forcedepends on the geometry of the coil through nð"yÞ and on thegeometry of the magnet through Bq,mag. These dependenciesare lumped into a single function !, which allows us toexpress the magnetic-force as

Fmag ¼ !ðyÞI: (9)

In the limit where the permanent magnet can be approxi-mated as a dipole magnet with dipole moment l ¼ ly andthe coil geometry is assumed to be described by Eq. (7), thisfunction becomes

!ðyÞ ¼ ll0cn2

ðLc & yÞ3 & yr2c#r2c þ ðLc & yÞ2

$3=2þ yffiffiffiffiffiffiffiffiffiffiffiffiffiffi

y2 þ r2cp

0

B@

1

CA: (10)

A simplified expression is obtained for the case of a longcoil (Lc , rc) and magnet positions away from the coil edges(Lc> y, rc). In this case,

!ðyÞ " ll0cn: (11)

Figure 2(a) depicts the dimensionless quantity !(y)/(ll0cn) for three cases. The (red) dashed line corresponds toEq. (10). The (black) dashed-dotted line is the result of a nu-merical integration that uses Eq. (7) to model the coil andthe exact expression for the magnetic field of a 3.81 cm longcylindrical magnet with a 2.54 cm diameter. The differenceis due to the dipole approximation for the finite-size cylindri-cal magnet. In the limit where the size of the cylindricalmagnet is decreased to zero, the prediction in Eq. (10) isrecovered. The (blue) solid line is the numerical result for

the case where the cylindrical magnet is considered and thephysical coil is mimicked by summing over contributionsfrom the discrete coil layers.

To characterize the coils, each was supplied with a con-stant current I, the resulting on-axis magnetic field wasmeasured, and the normalized field gradient determined, asshown for the primary coil in Fig. 2(b) by (black) circles.Since the force on a vertically aligned dipole magnet isFmag¼ l@Bcoil/@y y, one finds that

!ðyÞ ¼ lI

@Bcoil

@y; (12)

the normalized field gradient should correspond to !(y), inthe dipole approximation. As seen in Fig. 2(b), the measuredresult is indeed close to the theoretical prediction given bythe dipole approximation in Eq. (10). A better matchbetween data and theory is obtained when the layered struc-ture of the coil is taken into account. The result of such a cal-culation is shown in Fig. 2(b) as a solid (blue) line.

As is evident from Fig. 2, !(y) is constant, with!¼ ll0cn, in a region extending from 3 to 20 cm, nearly twothirds the coil length. The chaotic oscillations of our magnetsare entirely contained within this region. Therefore, byapplying a constant direct current to the secondary coil, theconstant offsetting force Fsc in Eq. (3) can be obtained.

D. Primary coil

The goal of the primary coil is to provide a physicalrealization of negative damping in the spring-mass system.To this end, the coil is connected to an active electronic cir-cuit that provides a negative resistance to the electric poten-tial induced by a magnet moving along the coil’s axis. Thenegative resistance is the source of power that is required tosustain the mechanical oscillations in the system.

To implement the velocity dependent primary coil forceFpc, we utilize the emf e that a moving magnet inducesacross the coil. Theoretically, the induced emf is obtainedfrom Faraday’s law by considering the change in time of themagnetic flux through a horizontal circular loop of radius rclying within the coil walls at position "y

e ¼ & dUð"y & yÞdt

¼ @Uð"y & yÞ@"y

_y; (13)

where y denotes the position of the magnet. Since there areno magnetic monopoles, there is a zero total flux through theclosed surface of a Gaussian pillbox formed by two suchloops separated by an infinitesimal distance d"y. The differ-ence dU between the magnetic flux through the top loop andthe flux through the bottom loop is therefore exactly bal-anced by the flux through the sidewall of the pillbox. Thisimplies that @U/@"y¼&2prcBq,mag(rc, "y&y). As there aren("y) loops at position "y, the total emf across the coil is

e ¼ &#2prc

ðLc

0

nð"yÞBq;magðrc; "y & yÞd"y$

_y

¼ &!ðyÞ _y: (14)

FIG. 2. (a) Normalized ! as a function of position: dashed line (red), dipolemagnet, Eq. (10) assuming the dipole approximation; dashed-dotted line(black), numerical calculation using a cylindrical magnet; solid line (blue),cylindrical magnet and layered coil. The shaded region indicates the extentof the coil. (b) Normalized gradient of the coil’s on-axis magnetic field:Circles (black), estimates from measured magnetic field data for the primarycoil; dashed line (red), Eq. (10); solid line (blue), layered coil.

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This emf e is used to induce an opposing current in the coilvia a negative resistance, implemented using an active elec-tronic circuit.

A schematic of the negative resistance circuit used todrive the primary coil is shown in Figure 3. The dotted boxrepresents the primary coil, and the elements within the boxdepict an equivalent electrical model for the actual physicalcoil. For the model, we directly measure the series resistanceR¼ 65 X and inductance L¼ 220 mH. The induced emf evaries in proportion to the axial velocity of the magnet in thecoil per Eq. (14). A LM675T power operational amplifieris used to drive the coil via the 33 X power feedback resistor.Two capacitors are included to suppress high-frequencyoscillations. The remaining discrete components aredesigned to compensate for the coils internal impedance andprovide a negative resistance. The 22H inductor is realizedby using active components in a general impedance con-verter (GIC) circuit.20

At low frequency, the circuit exhibits a negative resist-ance as desired. Neglecting the effects of the capacitors,analysis of the circuit reveals that the current in the coil isgiven by

i ¼ & eRN

; (15)

where the magnitude of the negative resistance is

RN ¼ RG

100& R; (16)

and RG is an adjustable resistance implemented with trimmerpotentiometers. In practice, decreasing RG corresponds toincreasing oscillator gain; however, stability of the feedbackcircuit requires RG> 100R, and the negative resistor fails forRG less than this threshold. Substitution of this current intoEq. (3) gives a velocity dependent force

Fpc ¼!2

RN_y " ðl0lcnÞ

2

RN_y; (17)

as desired.

E. Secondary coil

The function of the secondary coil is to shift the oscilla-tor’s fixed point between two levels depending on the guardconditions from Eqs. (5) and (6) and the threshold defini-tions. The shift is provided by inducing a constant force onthe secondary magnet in the spring-mass system using a con-stant current through the secondary coil.

Figure 4 shows a schematic of the electronic circuitused to implement the guard condition and drive the cur-rent in the secondary coil. To detect the position and ve-locity of the mechanical system, a commercially availablelinear variable differential transformer (LVDT) positionsensor (Measurement Specialties DC-EC 2000) is used.The output voltage !(t) from the position sensor drivestwo comparators that detect the logical conditions ! > V0

and dvdt < 0; where V0 is an adjustable threshold that ena-

bles tuning between shift and folded band dynamics. Alow-pass filter (LPF) is used to suppress any fast electricaloscillations due to switch bounce or noise in the mechani-cal sensor. A digital latch comprising discrete AND andOR gates realizes the logical output of the guard condition,which determines the polarity of a signal applied to thesecondary coil.

The secondary coil is driven by a power operational am-plifier (LM675T) configured as a comparator to distinguishlogic levels. The amplifier is powered with 615V, and itsoutput supplies the coil with &13.6V and 13.2V for s(t)¼ 0and 1, respectively. The series resistance of the secondarycoil is 40 X, implying a 670mA change in the direct currentat a transition in the logic state.

To change the particular configuration of the system,the only parameter varied is V0. When V0 is set to a volt-age in the middle of voltages for the upper and lower os-cillation centers, the system will show shift dynamics.This center threshold V0 represents the physical Y0 valueof Eq. (5).

III. SIMPLIFIED SYSTEM EQUATIONS

In order to analyze the electromechanical system,including guard conditions, governed by Eqs. (4)–(6), it isconvenient to use a dimensionless model obtained by usingthe following substitutions:

FIG. 3. Negative resistance circuit used for driving the primary coil.Elements in the dashed box model the circuit characteristics of the primarycoil with R¼ 65 X and L¼ 220 mH.

FIG. 4. Position sensor and logic circuitry used to implement guard condi-tion and discrete switching with the secondary coil. The LVDT positionsensor outputs a voltage that varies linearly with the displacement of thespring-mass system, and V0 is an adjustable threshold used to select the but-terfly or folded-band oscillations.

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s ¼ t

T; u ¼ y& C2

C1; and s ¼ yd & C2

C1; with period

T ¼ 2p

xn

ffiffiffiffiffiffiffiffiffiffiffiffiffi1& f2

p : (18)

With these substitutions, Eq. (4) simplifies to the dimension-less form

€u & 2b _u þ ðx2 þ b2Þðu& sÞ ¼ 0; (19)

where x¼ 2p is the oscillation frequency, b ¼ 2pf=ffiffiffiffiffiffiffiffiffiffiffiffiffi1& f2

pis a positive coefficient that quantifies the degree of

negative damping, and s is the discrete switching point state.Parameters C1 and C2 are dependent on the system beingmodeled. For the butterfly oscillator, C1¼ (YT &YB)/2 andC2¼ (YT þ YB)/2. For the folded band, C1¼YT &YB andC2¼YB. With these substitutions, s is a system dependentvariable governed by a guard condition and state characteri-zation of either

_u ¼ 0 ) s ¼ sgnðuÞ (20)

or

_u ¼ 0 ) s ¼ Hðu& 1Þ (21)

for the Lorenz-butterfly or R€ossler folded band topologies,respectively. Note, these equations follow the form found inRefs.17 and 19 and can therefore be similarly solved.Significantly, exact analytical solutions to both hybrid sys-tem configurations can be found.

A. Butterfly

It has been shown previously that Eq. (19) with guardcondition (20) can be analytically solved.17,19 Without lossof generality, we consider an initial condition coincidingwith a triggered guard condition, _u¼ 0, such that juj- 1. Assuch, the solution can be written as a linear convolution

uðtÞ ¼X1

n¼0

sn ' Pðt& nÞ; (22)

and sðtÞ ¼X1

n¼0

sn ' /ðt& nÞ; (23)

where sn is a weighting factor. The basis functions are shownin Fig. 5(a) and defined as

PðtÞ ¼ð1& e&bÞebtqðtÞ; t < 0

1& ebðt&1ÞqðtÞ; 0 - t < 1

0; t ( 1;

8>><

>>:(24)

with

qðtÞcosxt& bxsinxt; (25)

and

/ðtÞ ¼1; 0 - t < 1

0; otherwise:

(

(26)

Note here that it is assumed that the solution satisfiesguard conditions at t¼ n and nþ 1/2 such that sgn(u(n))¼ sgn(u(nþ 1/2)) for all integer n. The weighting fac-tors sn are given as a sequence of binary symbols 61 corre-sponding to the switching state at each integer guardcondition or sn¼ sgn (u(n)). This definition means thatswitching transitions occur only at integer dimensionlesstime n.

In addition to the exact solution, it is useful to considera return map. This map is a stroboscopic sampling of u forsuccessive returns at integer times, yielding,

unþ1 ¼ ebðun & snÞ þ sn; (27)

where unþ1¼ u(nþ1) and un¼ u(n). This recursion relationcorresponds to a shift map with slope eb for all b> 0.This map is closed on the interval junj< 1 and for negativedamping 0<b< ln 2. A typical map is shown in Figure 6.Since this iterated map is piecewise linear with constantslope> 1, the map is necessarily chaotic, indicating that thehybrid system is likewise chaotic, having a correspondingLyapunov exponent k¼b. This configuration has topological

FIG. 5. Basis pulses for the solvable butterfly (a) and folded-band (b)oscillators.

FIG. 6. Theoretical return map for the solvable oscillator exhibiting a butter-fly attractor.

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characteristics similar to that of the famous Lorenz attractoras will be shown graphically in the experimental and analyti-cal comparisons in Sec. IV.

B. Folded band

Solutions can be similarly derived for the switching defi-nition given by Eq. (21). Again we consider an initial condi-tion coinciding with a triggered guard condition. Under thisassumption, the solution states can be written as linear con-volutions of weighted basis functions

uðtÞ ¼X1

m¼0

rm ' Q t& m

2

& '; (28)

sðtÞ ¼X1

m¼0

rm ' w t& m

2

& '; (29)

where rm is a weighting factor. The basis functions areshown in Fig. 5(b) and defined as

QðtÞ ¼

ð1þ e&b=2ÞebtqðtÞ; t < 0

1þ ebðt&1=2ÞqðtÞ; 0 - t <1

2

0; t ( 1

2;

8>>>>><

>>>>>:

(30)

with

wðtÞ ¼1; 0 - t <

1

2

0; t < 0 or t ( 1

2:

8>><

>>:(31)

In this case, the weighting factors rm are set to the discretestate of s(t) for m/2 - t < mþ 1/2, and it follows that theyare restricted to the set{0,1}. This solution is not valid for allpossible oscillations of u but is limited to initial conditionsbounded by

juð0Þj- 1þ e&b=2

1& e&b=2; (32)

which corresponds to small initial displacements of the phys-ical system.

For the folded-band system, the return map used to helpunderstand the system is based on successive maxima,instead of returns at integer times. For some arbitrary initialcondition bounded by Eq. (32), a successive maxima returnmap is given by

unþ1 ¼ebun; 0 < un - 1;

&e3b=2un þ eb þ e3b=2; 1 < un < 1þ eb=2

ebun & eb & eb=2; un > 1þ e&b=2;

8>><

>>:

(33)

where un is the initial local maxima and unþ1 is the nextmaxima. This map is shown as three piecewise linear

segments (a), (b), and(c), as seen in Fig. 7. Note that unlikethe previous case, the return times are dependent on the sys-tem state, yielding

Dt ¼ tnþ1 & tn ¼

1; 0 < un - 1;

3

2; 1 < un < 1þ e&b=2;

1; un > 1þ e&b=2:

8>>><

>>>:(34)

The return time difference for 1< un< 1þ e&b/2 is due to thefolding mechanism as the system oscillates about a separatefixed point for an additional half cycle.

The properties of the return map (33), and hence the dy-namics of the mechanical oscillator, depend on the particularvalue of the negative damping.18 For small negative dampingsuch that b<b1, where

eb - 1þ e&b=2; b1 " 0:8 ' ln 2; (35)

the iterated map is restricted to linear segments (a) and (b) inFig. 7. In this case, the oscillator exhibits a folded-band to-pology similar to R€ossler’s simply folded band.2 For b1< b< b2 where

eb2 ¼ 2þ e&b2=2; b2 " 1:39 ' ln 2; (36)

the iterated map is bounded and visits segments (a), (b), and(c). In this case, the oscillator exhibits a second fold in its dy-namics. In either case, the slope of the iterated map (33) iseverywhere greater than one, so the bounded oscillations arenecessarily chaotic. For b> b2, the map is not closed andoscillations are unbounded.

IV. RESULTS AND DISCUSSION

The hybrid dynamical system was tuned with two con-figurations, producing two distinct experimental trajectoriescoinciding with those in the theoretical analysis above. Eachanalysis examines a large, continuous time data set to pro-duce time-delay embedded phase projections and returnmaps corresponding to Figs. 6 and 7. Additionally, a smallersubset of these data sets were used to find a sequence of sym-bols for the discrete data. The symbols were then used to re-create an analytical waveform in dimensionless time, scaled,and compared to the physical experimental results.

FIG. 7. Theoretical successive-maxima return map for the solvable folded-band oscillator.

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Position measurements were obtained using the LVDTposition sensor, which was sampled using a data acquisitionsystem. Remote placement of the position sensor was facili-tated using a second string under tension and a mechanicallinkage, which preserved a linear scaling of the true positionfor small displacements typical of the observed oscillations(see Fig. 1). The sub-Hertz oscillation data were significantlyoversampled using 100Hz digital sampling rates.

A. Butterfly

In the first configuration, the voltage V0 and the resist-ance RG were tuned to obtain chaotic oscillations exhibitingthe Lorenz butterfly dynamics. A 5-min record of the posi-tion of the spring-mass system as well as the electronicswitching state were captured and stored. Figure 8 shows theattractor obtained from this measurement using time-delayembedding (Dt " 0.15 T) of the position data. A typical seg-ment showing the continuous and switching states is shownin Figure 9. From the data, we estimate the period for theunderlying harmonic oscillator at T" 0.46 s.

Figure 10 shows a return map constructed using positiondata sampled at guard conditions corresponding to possible

switching events. We note that this return map shows a shiftdynamic consistent with the desired butterfly topology.19

From the average slope of each segment of the return map,we estimate the negative damping parameter to be b¼ 0.47.

To show the analytically solvable nature of the mechani-cal oscillator, we overlay the corresponding analytic solutionin dimensional form in Figure 9. For this solution, we esti-mate the parameters YT¼ 11.03mm and YB¼&9.31mm.For the segment of measured waveform, the correspondingsymbol sequence is read from the measured switching state.The analytic solution (22) and (23) is calculated from thissymbol sequence and shown in the plots as dashed curves. Astrong correlation between measured and analytic waveformsindicates that the mechanical system is reasonably well mod-eled by the exactly solvable chaotic oscillator exhibiting thebutterfly topology.

B. Folded-band

In the second configuration, system parameters weretuned to obtain chaotic oscillations exhibiting folded-banddynamics. As with the first configuration, an extended timeseries of position and switching state was collected. Figure 11shows the attractor obtained from this data with the same

FIG. 8. Time-delay embedding (Dt " 0.15T) of the position data for the firstsystem configuration to illustrate the butterfly attractor topology.

FIG. 9. A typical segment of experimental data and analytic solution for theoscillator exhibiting a butterfly attractor. Solid line (green) is the measuredwaveform, dashed (blue) is the analytical solution derived from the experi-mental symbol sequence.

FIG. 10. Experimental return map (blue) for the oscillator exhibiting a but-terfly attractor and with least squares fit slope lines (black) for each segment.

FIG. 11. Time-delay embedding (Dt " 0.15T) of the position data for thesecond system configuration to illustrate the folded-band topology.

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time-delay embedding as the first configuration. Figure 12shows a typical segment of the two waveforms. The esti-mated period from the harmonic oscillations is unchangedfrom the first configuration, T " 0.46 s.

Figure 13 shows an iterate map with position datasampled at the successive local maxima of the response. Thismap shows a piecewise tent consistent with folded-band top-ologies.18 The corresponding lines represent a least squaresfit for the subsets of data that comprise each segment.Estimated b values from the three segments are 1.04 ' ln 2,0.93 ' ln 2, and 0.2771 ' ln 2, respectively. The large error inthe furthest right segment is a result of its short length andthe somewhat arbitrary act of where to cut data segments,especially in noisy regions of the data.

As with the first configuration, we overlay the corre-sponding analytic solution in dimensional form in Figure 12for the second configuration. For this solution, tuning param-eters are adjusted such that YT¼ 12.53mm and YB¼ 0mm,with b¼ 0.72 estimated from the first branch of the returnmap. The analytic solution (28) and (29) is calculated from

the corresponding symbol sequence read from the measuredswitching state. This is shown in the plots as dashed curves.Again, a strong correlation is apparent between the measuredand analytic waveforms, indicating that the system is wellmodeled by the exact folded-band solutions.

Discrepancies between the two waveforms are largely afunction of phase slip errors. Oscillations that just cross theguard condition threshold have a necessarily small amplitudefor oscillation about that upper set point. This lends them alarger sensitivity to timing errors inherent in the system.These errors are to some extent a function of the electromag-netic switching event itself which induces a considerable, in-stantaneous torque and a considerable lateral force if themagnet is at all radially displaced from the axis of motion.

V. CONCLUSIONS

In this article, we detail the construction and testing ofan experimental dynamical system that shows exactly solva-ble chaotic oscillations. Detailed theoretically in previousarticles and replicated in electric systems, this system is anequivalent mechanical and electrical manifestation. With theexecution of negative damping through the electromagneticcoupling between the magnet and coil and a simple switch-ing condition, a chaotic oscillator is realized. Additionally,these oscillations can be represented as a linear convolutionof basis functions. Figures show a clear tent map anddynamic shift map that indicate the chaotic nature of theoscillating system. While the approximations are less accu-rate than the electrical equivalents in the literature, symboldynamics can be used to replicate the waveform with consid-erable agreement. This match between theory and experi-ment proves the realizability of mechanical configurations ofthese types of systems and opens the door for further explo-ration into solvable chaotic systems.

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FIG. 12. A typical segment of experimental data and analytic solution forthe folded-band oscillator. Solid line (green) is the measured waveform,dashed (blue) is the analytical solution derived from the experimental sym-bol sequence.

FIG. 13. Experimental successive-maxima return map (blue) for the folded-band oscillator with leastsquares best fit slope lines (black).

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