27519839 Period Doubling Bifurcation and Chaos in Duffing Oscillator System by Subash B

Contents1 Introdution 31.1 Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Organisation of the thesis . . . . . . . . . . . . . . . . . . . . 92 Population Growth and the Verhulst Model 102.1 The Logisti Map . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Period Doubling Bifuration: . . . . . . . . . . . . . . . . . . . 162.3 Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 The Dung Osillator Numerial Solutions 213.1 Dynamis of Dung Osillator: . . . . . . . . . . . . . . . . . 234 Dung Osillator: Bifuration Diagrams, Lyapunov Expo-nents and Chaos 274.1 Amplitude of the Fore(ampf ) as ontrol parameter: . . . . . . 304.2 Beta() as ontrol parameter . . . . . . . . . . . . . . . . . . 324.3 Lyapunov Spetrum . . . . . . . . . . . . . . . . . . . . . . . 371

CONTENTS 25 Poinar's Setion 436 Summary and Conlusion 48

Chapter 1IntrodutionClassial mehanis deals with how deterministi systems, suh as swingingpendulums and orbiting planets, hange with time. The dynamis of suh asystem are desribed by its state, whih aptures the values of all the variablesthat is needed to predit the future of the system, and a set of rules, oftenin the form of dierential equations, whih say how the state hanges withtime. In a dynamial system we observe the world as a funtion of time. Weexpress our observations as numbers and reord how they hange with time;given suiently detailed information and understanding of the underlyingnatural laws, we see the future in the present as in a mirror. DynamialSystems may be lassied into two main ategories:1. Linear2. Nonlinear 3

CHAPTER 1. INTRODUCTION 4In general, all real systems are nonlinear, However, very often it is the asethat, as a rst approximation to the dynamis of a partiular system, a lin-ear model may be used. Linear models are preferable from a sientist's pointof view as typially they are muh more amenable to mathematial analy-sis. (Hene the disproportionate number of linear systems studied in siene.)Nonlinear systems, in ontrast, are muh more diult to analyse mathemat-ially, and, apart from a few exeptions, analytial solutions are not possiblefor the nonlinear dierential equations used to desribe their temporal evo-lution. In addition, only nonlinear systems are apable of a most fasinatingbehaviour known as haoti motion, or simply haos, whereby even simplenonlinear systems an, under ertain operating onditions, behave in a seem-ingly unpreditable manner. Two things might seem to be suient for thestudy of a deterministi system: (1). Disovering the rules governing it; and(2). Finding suiently a

urate analyti or numerial tehniques to usethese rules to predit its evolution.Indeed, this is true for many pratial purposes. However, understandinga dynamial system involves more than this, and we often want to hara-terise and lassify systems rather than to treat eah in isolation, in order togain some insight into their struture. There are many ways to approah this,but here we onentrate on one or two of the more graphial ones, whih haveome to inreased prominene reently, although their roots were establishedmany years ago.

CHAPTER 1. INTRODUCTION 5Nonlinear dynamis is one of the glorious su

esses of omputational si-ene. It has been explored by mathematiians, sientists and engineers, andusually with omputers as an essential tool. (Even theologists have foundmotivation from the mathematial fat that simple systems an have veryompliated behaviors.) The omputed solutions have led to the disoveryof new phenomena suh as solitons, haos, and fratals. In addition, be-ause biologial systems often have omplex interations, and may not be inthermodynami equilibrium states, models of them are often nonlinear, withproperties similar to other omplex systems. The intriguing properties andtantalising possibilities of haos have thus reated onsiderable interest in themathematis world, thus leading to a mass of new denitions and results inthe general eld of nonlinear dynamis whih enompasses haos and systemstheory.In 1963 Edward Lorenz published his work entitled 'Deterministi nonpe-riodi ow' whih detailed the behaviour of a simplied mathematial modelrepresenting the workings of the atmosphere. Lorenz showed how a relativelysimple, deterministi mathematial model (that is, one with no randomnessassoiated with it) ould produe apparently unpreditable behaviour, laternamed haos. There are several tools available to study the behaviour ofhaoti system. They are1. Bifuration Diagram2. Lyapunov Spetrum

CHAPTER 1. INTRODUCTION 63. Poinare Setion4. Power Spetrum et.,Although there are fany pakages available to get these, we tried to un-derstand the basi onept of these tools by writing universal,user-friendly fortran77 algorithms, that will generate bifurationdiagrams and poinare setions for any system, ourselves.1.1 ChaosThe study of dynamial systems atually dates bak many years but the lastthree deades have seen intensive studies whih have been prompted by thedisovery of haos. Initially, haos was seen purely as a mathematial u-riosity. Although irregular or unpreditable behaviour may have been noted,this was often attributed to random external inuenes. Correspondingly inengineering, and in partiular in eletrial systems, the appearane of haoswas usually regarded as a nuisane and thus designed out where possible.Changes ame about with publiation of seminal works by Lorenz, Feigen-baum, Smale, and May, oupled with numerial simulations by a host ofresearhers, notably early work by Ueda in eletrial engineering, so thatmodern studies have now onrmed that haoti phenomena are ompletelydeterministi, o

urring in a variety of nonlinear problems in physial andnatural systems. Mathematially, the study of haoti systems has proved ex-tremely useful sine the latter form arhetypal dynamial systems exhibiting

CHAPTER 1. INTRODUCTION 7various types of interesting behaviour, some of whih remains unexplainedeven today.A deterministi system is a system whose present state is in priniplefully determined by its initial onditions, in ontrast to a stohasti system,for whih the initial onditions determine the present state only partially,due to noise, or other external irumstanes beyond our ontrol. For astohasti system, the present states reets the past initial onditions plusthe partiular realization of the noise enountered along the way. It wouldbe natural to think that if a system is deterministi, its behaviour shouldbe easily predited. But there are systems where their behaviour turns outto be non-preditable: not beause of lak of determinism, but beause theomplexity of the dynamis require a preision that is unable to be omputed.This an be seen in systems where very similar initial onditions yield verydierent behaviours. Thus, let's say if we have the initial states 2.1234567890and 2.1234567891, after some time the system will be for the rst ase in3.5 and in the other in - 1.7. So, no matter how muh preision we have,the most minimal dierenes will tend in the long time to very dierentresults. This is beause there is an exponential divergene of the trajetoriesof the system (This an formally be measured with Lyapunov exponents).Another interesting property of haoti systems are strange attrators1 If one1An attrator is a part of the state-spae (whih is the set of all possible states of asystem) whih `pulls' the dynamis into it. For example, a simple pendulum with fritionhas a stable attrator in the bottom of the vertial axis, beause wherever the pendulumis, it will end at some time in that point. But there is an unstable attrator in the topof the vertial axis, beause if that is the initial ondition, in theory the pendulum would

CHAPTER 1. INTRODUCTION 8thinks of haoti dynamis, it would be easy to assume that the dynamisfollow no pattern. But if we look arefully there is an amazing pattern,in suh a way that the states will not repeat themselves, but will be in adetermined area of the state-spae. In this thesis, we are onerned onlywith haos in deterministi systems.The notion of omplexity and haos an be realized even in low-dimensionalnonlinear systems. Partiularly, damped and driven osillators are physi-ally realizable examples exhibiting immense varieties of bifurations andhaos. They are often modelled by a single, seond-order, nonlinear dieren-tial equation (or equivalent rst order system) with periodi inhomogeneities.These inlude the Dung osillator, the Bonhoeer-van der Pol osillator,the Dung-van der Pol osillator, the damped driven pendulum, the drivenMorse osillator and so on. Chaos has now been found in all manner ofdynamial systems; both mathematial models and, perhaps more impor-tantly, natural systems. Chaoti motion has been observed in all of the 'real'osillatory systems. In addition, many ommon qualitative and quantita-tive features an be diserned in the haoti motion of these systems. Thisubiquitous nature of haos is often referred to as the universality of haos.stay up there, but the most minimal perturbation would move the pendulum out of theunstable attrator. The unstable attrator repels the dynamis of the system.

CHAPTER 1. INTRODUCTION 91.2 Organisation of the thesisIn the seond hapter, we disuss a simple model for population growthknown as the Verhulst model, whih is a nonlinear model. The famous logistimap is losely related to this. We disuss the periodi doubling bifurationroute to haos in this model. The universal features of this route to haos isnoted.In the third hapeter, we disuss the Dung Osillator. We desribe themethod to numerially solve this equation. The Time Series Plots and PhasePlots are presented.The fourth hapter is devoted to the bifuration sequene and the Lya-punov exponents for this problem when the ontrol parameter is varied. Wehave not used any standard pakage to obtain the bifuration diagram. Wehave written a simple ode to obtain it. We have studied the periodi win-dow in the haoti domain, as well as the transition from hoas to periodibehaviour, when the parameter is inreased. We have used the standardmethod due to Wolf, etal for obtaining the Lyapunov Spetrum. We omparethe bifuration pattern and the maximal lyapunov exponent. The results areonsistent. This hapter is the ore of the thesis.Chapter ve is on Poiar setions for the Dung Osillator. We notiea lear distintion between Poinar setions in the periodi and haotidomains.

Chapter 2Population Growth and theVerhulst ModelThe simplest model of population growth assumes a onstant growth ratewhereby the population inreases in size by a xed proportion in eah timeinterval. This model an be used to explain the important terms used inNonlinear dynamis. This model an be written as,

Pn+1 = Pn + rPn = (1 + r

)Pn = rPnwhere Pn+1 is the population size at time n+ 1, Pn is the population size attime n, r is the growth rate (the same as interest rate in ompound interestalulations), and r is the growth fator = 1+ r. It follows from equation(2)10

CHAPTER 2. POPULATION GROWTHAND THE VERHULSTMODEL11that r = (Pn+1 Pn)Pn = P/Pn usually expressed as a perentage. ThenPn = (1 + r

)nP0 = rnP0 (2.1)where P0 is the initial population at time t = 0. However, in this simplemodel (with r onstant) the population grows without bounds, i.e. as ntends to innity so does the population, P . Clearly this does not happenfor real populations, whih are limited in their growth by external fatorssuh as food, spae, disease, et. Therefore, there must exist a maximumpossible population size Pmax. To take this maximum population size intoa

ount in a population model, a modied growth rate, r, is used, whih isproportional to the dierene between the population size at time n (i.e. Pn)and the maximum possible population size Pmax, expressed as follows:

r = a(Pmax Pn) (2.2)where a is a onstant. Now, when Pn = Pmax, it follows that r = 0 andno further growth takes plae. Substituting equation (2.2) into equation (2)leads toPn+1 = Pn + a(Pmax Pn)Pn (2.3)

CHAPTER 2. POPULATION GROWTHAND THE VERHULSTMODEL12and this is known as the Verhulst model of population growth. If we expandequation (2.3) we obtainPn+1 = Pn + aPmaxPn P

2

n (2.4)and we see that the last term provides a negative nonlinear feedbak to theequation. It is this term whih ause the Verhulst model to posses a rihvariety of behaviour inluding haoti motion. Often, it is more usual to userelative population sizes (denoted by lower ase p) where the populations areexpressed as a fration of the maximum possible population size. Equation(2.3) then beomespn+1 = pn + apn(1 pn) (2.5)where

pn+1 =Pn+1Pmax

pn =PnPmax

pmax =PmaxPmax

= 12.1 The Logisti MapEquation (2.5) is the Verhulst model normalized by the maximum populationsize. A slightly simpler ousin of the Verhulst equation known as the logistimap. To obtain this map, we simply remove the rst term from equation

CHAPTER 2. POPULATION GROWTHAND THE VERHULSTMODEL13(2.5). In its more usual form, in terms of x instead of p, the logisti map iswritten asxn+1 = axn(1 xn) (2.6)where the urrent value of the variable x, i.e. xn, is mapped onto the nextvalue, i.e. xn+l.The logisti map is a non-invertible map. Non-invertiblemeans that although we may iterate the map forward in time with eah

xnleading to a unique subsequent value, xn+1, the reverse is not true, i.e.iterating bakwards in time leads to two solutions for xn for eah value ofxn+1. By repeatedly iterating the logisti map forward through time, wemay observe various behaviours of the iterated solutions. The sequene ofiterated solutions to the map is alled an orbit. The behaviour of su

essiveiterates of the logisti map depends both on the ontrol parameter a, andthe initial ondition (or starting point) used for the iterations. The funtionorresponding to the logisti map (equation (2.7)), known as the logistifuntion or the logisti urve, is the paraboli urve.

f(x) = ax(1 x) (2.7)When iterating the logisti map, the iterated solutions eventually settle downto a nal behaviour type. The sequene of iterated solutions produed bythe initial iterations is known as the transient orbit of the system. The -nal sequene of iterated values that the iterations tend to is known as thepost-transient orbit. Transient behaviour is obvious in gure 2.1(a) where

CHAPTER 2. POPULATION GROWTHAND THE VERHULSTMODEL14

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0 5 10 15 20 25 30 35 40

x_n

n

a=0.9 decay to zero

(a) a=0.9 (deay to zero) 0.4 0.45 0.5

0.55

0.6

0.65

0 5 10 15 20 25 30 35 40

x_n

n

a=2.6 Steady State Period 1

(b) a=2.60 (steady state period 1) 0.5

0.55

0.6

0.65

0.7

0.75

0.8

0 5 10 15 20 25 30 35 40

x_n

n

a=3.20 Period 2

() a=3.20 (period 2) 0.3 0.4 0.5 0.6

0.7

0.8

0.9

0 5 10 15 20 25 30 35 40

x_n

n

a=3.52 Period 4

(d) a=3.52 (period 4) 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30 35 40

x_n

n

a=4.00 Choatic

(e) a=4.00 (hoati)Figure 2.1: Interated solutions of the Logisti Equation plotted for seletedvalues of the ontrol parameter

CHAPTER 2. POPULATION GROWTHAND THE VERHULSTMODEL15the orbit tends asymptotially to zero. The transient orbit is also notieablein gure 2.1(b), espeially over the rst ten or so iterations. However, by thetwentieth iteration there is no notieable hange in the iterated solutions. Infat the orbit tends asymptotially to the xed point of 0.6153.... In gure2.1(c), the transient behaviour seems to disappear rather fast with only theinitial ondition appearing to be notieably 'o the post-transient period 2orbit. In gure 2.1(d), a post-transient period 4 orbit seems to be well es-tablished by the tenth iteration. Transient behaviour is not at all notieablein gure 2.1(e) due to the errati nature of the post-transient, haoti orbit.This is quite often the ase for haoti motion where the omplex strutureof the post-transient behaviour may make it diult to observe the initialtransient behaviour. In general, the post-transient orbits are approahedasymptotially by the iterates of the logisti map. In pratie, a nite res-olution (i.e. a

uray to within a given number of deimal plaes) of thepost-transient orbit is usually all that is required. The resolution obtainedis diretly related to the number of iterations undertaken. This is illustratedin gure 2.1(b) for the a = 2.6 ase. Note that if one starts the iterationproedure on a post-transient orbit then no transient orbit will o

ur. Wesee from the gure that the logisti map settles down, through a deayingosillation, to the single value of 0.6153... . This nal solution is known asa period 1 orbit, as the iterates tend to a xed value where xn+1 = xn forlarge n. If the solutions to the logisti map repeat every seond value, i.e.xn+2 = xn. This is known as a period 2 orbit. The period 2 orbit is shown

CHAPTER 2. POPULATION GROWTHAND THE VERHULSTMODEL16graphially in gure 2.1(c) where the rst twenty iterated solutions to thelogisti map are plotted against n.The attrator is the set of points approahed by the orbit as the numberof iterations inreases to innity. The posttransient sequenes of iteratedsolutions to the map lie on the attrator. If the system settles down to aperiodi orbit, then the system is said to have a periodi attrator, e.g. aperiod 1 attrator, period 2 attrator or period 4attrator as seen above. If,on the other hand, the system behaves haotially with an aperiodi orbit,then the system is said to have a haoti attrator, more ommonly referredto as a strange attrator. The logisti map with a = 4.0 has a strangeattrator.2.2 Period Doubling Bifuration:So far we have observed periodi attrators, of periods 1, 2 and 4, and ahaoti attrator for the logisti map. These were obtained simply by hang-ing the ontrol parameter a. To examine the global eet of the ontrolparameter a on the behaviour of the logisti map it is useful to plot the post-transient solutions of the logisti map against a. Suh a plot is given in Fig.2.2. As we already know, the orbits deay to zero for values of a between 0.0and 1.0. Values of a between 1.0 and 3.0 produe post-transient solutions tothe logisti map whih are period 1 xed points. These period 1 xed pointsinrease in value as a is inreased and form the ontinuous line in Fig. 2.2,

CHAPTER 2. POPULATION GROWTHAND THE VERHULSTMODEL17

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0.1

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1

2.8 3 3.2 3.4 3.6 3.8 4

x m

ax

a

Bifurcation Diagram for Logistic Map

Figure 2.2: Bifuration Diagram for Logisti Maprising from the a axis at a = 1.0 to the rst period doubling bifuration at a= 3.0. For values of the ontrol parameter between 3.0 and 3.449490 . . . aperiod 2 attrator exists (see Fig. 2.3). Between 3.449490 . . . and 3.544 090. . . a period 4 attrator exists, followed in sequene by attrators of period8 (between 3.544 090 . . . and 3.564407 . . . ), period 16, period 32 and soon. This sequene arries on, doubling in period eah time, until an inniteperiod is reahed at a nite value of the ontrol parameter of a = 3.569 945. . ., at whih point the behaviour is haoti and a strange attrator exists.This sequene is known as the period doubling route to haos. The rstfew period doubling bifurations may be seen in Fig. 2.3, whih ontains ablow-up of the diagram of Fig. 2.2 in the region of the ontrol parameter

CHAPTER 2. POPULATION GROWTHAND THE VERHULSTMODEL18period 1

Period Doubling Bifurcation

period 2

period 4

Figure 2.3: xxxxxxxxbetween 3.0 and 4.0. The period doubling sequene outlined above o

ursthrough the bifuration (splitting into two parts) of the previous xed pointswhen they beome unstable. This splitting is sometimes known as a pithforkbifuration due to its shape (although this term is not used here in its stan-dard mathematial sense). This is shown shematially in Fig. 2.3. At eahperiod doubling bifuration point, the previously stable attrating periodixed point beomes unstable, that is, it begins to repel iterated solutions inits viinity, and two new stable xed points emerge. Unstable xed pointsare shown dashed in Fig. 2.3. These period doubling bifurations give theirname to the bifuration diagram of Fig. 2.2. Fixed points whih attrat thesolutions to a map are known as stable; those whih repel nearby solutionsare unstable; and those whih neither attrat nor repel nearby solutions aresaid to be neutrally stable, or indierent. Loal stability depends on the

CHAPTER 2. POPULATION GROWTHAND THE VERHULSTMODEL19absolute value of the slope of the map funtion at the xed point, i.e.stable : |f (xn)| < 1neutrally stable : |f (xn)| = 1unstable : |f (xn)| > 12.3 UniversalityAround Otober, 1975, Mith Feigenbaum was studying the properties ofthe logisti map with a programmable poket alulator. As he saw that ittook lots of strength to try to nd the bifuration points alulating everypossible value of a (there were not so fast and not so many omputers...),he studied the geometrial onvergene of the doubling of periods, sine itseemed to be some regularity. So, we an all the rst bifuration point a0,whih for the logisti map is 1. The seond bifuration point a1 = 3. Andso, a2 = 3.4495..., a3 = 3.5441..., a4 = 3.5644... So, the ratio of hange is

an an1an+1 anSo, Feigenbaum (1983) alulated the limit

limn

an an1an+1 an

= = 4.6692016091029..

CHAPTER 2. POPULATION GROWTHAND THE VERHULSTMODEL20Basing himself on work on universal series by Metropolis, Stein, and Stein(1973), Feigenbaum found that the limit of ratios was the same for thesine map(a sin(x)), and atually, for any other funtion with doublingof periods. Thus, Feigenbaum's , turned out to be an universal onstant.Independently of the funtion, will always be the ratio of the bifurationsleading to haos. Feigenbaum's is also present in the famous Mandelbrotset. But Feigenbaum also found another universal onstant. If you alulatethe limit of the ratios of the distane from x = 0.5 (whih is the ritialpoint in the logisti map), to the nearest point of the attrator yle, it willalso be an universal onstant, alled Feigenbaum's . When a point of theattrator yle equals 0.5, it is alled a superstable orbit, beause it is whenit onverges most quikly to the attrator. So we havelimn

anan+1

= = 2.502907876...The ratio of distane between the values of a where there are superstableorbits is also .

Chapter 3The Dung Osillator NumerialSolutionsMany systems in nature have several stable states separated by energy bar-riers. When the system an move among the stable states, the dynamis anbeome quite omplex. A simple model that illustrates some of these fea-tures is the Dung double-well osillator. This model was rst introduedto understand fored vibrations of industrial mahinery [Dung, 1918. Inthis model, a partile is onstrained to move in one spatial dimension. Anexternal fore ats on the partile. The fore is desribed by

F = kx x3 (3.1)21

CHAPTER 3. THE DUFFINGOSCILLATOR NUMERICAL SOLUTIONS22

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-1.5 -1 -0.5 0 0.5 1 1.5

V(x

)

x

Potential Energy Diagram for Duffing Oscillator k=1 ,b=1"

((1*(x**4)/4) - (1*(x**2)/4))

Figure 3.1: plot of the potential energy funtion for dung osillatorThe name double-well enters beause the orresponding potential energyfuntion as a double well struture. Formally, the potential energy funtionis written asU(x) =

1

2kx2 +

1

4x4 (3.2)Figure above shows a plot of the potential energy funtion. We see thatthere are two stable equilibrium states at x = k

.

CHAPTER 3. THE DUFFINGOSCILLATOR NUMERICAL SOLUTIONS233.1 Dynamis of Dung Osillator:The General Equation of the Dung Osillators isx+ x kx+ x3 = F cos(wt) (3.3)we an rewrite equation(3.3) as

x+ damp x freqos x+ x3 = ampf cos(wt) (3.4)The equations desribing the dynamis in state spae are usually written asx = hf(y) (3.5)y = hf(kx x3 y + F cos(wt))where the term represents damping proportional to the veloity of thepartile. The motion of the partile in this situation is relatively simple. Ifstarted o with a ertain amount of kineti energy, the partile osillatesbak and forth, gradually losing energy via damping and nally omes torest at the bottom of one of the wells.

4thorder Runge-Kutta algorithm for the solutionsIn order to study the dynamis of Dung Osillator represented by the se-ond order dierential equation(3.3), we have to integrate out the equation(3.3).

CHAPTER 3. THE DUFFINGOSCILLATOR NUMERICAL SOLUTIONS24We use the fourth order Runge-Kutta method for intergration, where the so-lution of a dieretial equation dxdt

= f(t, x) is given byxn+1 = xn +

k16

+k23

+k33

+k46

+ (h5) (3.6)where,k1 = hf(tn, xn)

k2 = hf(tn +h

2, xn +

k12) (3.7)

k3 = hf(tn +h

2, xn +

k22)

k4 = hf(tn + h, xn + k3)we wrote an algorithm whih an solve seond order dierential equationwhih is shown in appendix( ). For implementing the Runge-Kutta method,the Dung equation is written in terms of two rst-order equations for xand y whih are of the form Eqn. 3.5The Fortran 77 program will produea .dat le whih has solutions of the equation as t x y. Using that le wesimulate the system using Gnuplot - a plotting tool for sienti work. wex out parameters at k = 1, = 1, = 0.5 and vary the amplitude of thefore F, we plot the time-series x(t) against t, and the phase plots x versusx for various values of F in Fig. 3.2 - 3.6 [ht we learly observe period 1,period 2, period 4, and haoti osillatios as the parameter f is varied. So,learly there is period-doubling bifuration route to haos in the system.

CHAPTER 3. THE DUFFINGOSCILLATOR NUMERICAL SOLUTIONS25 0

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100 105 110 115 120 125 130 135

Dis

plac

emen

t

Time

T Vs X freqos = -1 damp = 0.5 beta = 1 w = 1 ampf=0.34

p34

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Vel

ocity

Displacement

freqos = -1 damp = 0.5 beta = 1 w = 1 ampf=o.34

p34 u 2:3

Figure 3.2: Periodi Osillations of period 1

0

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ocity

Displacement


p35.dat u 2:3

Figure 3.3: Periodi Osillations of Period 2 0.2

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p357

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ocity

Displacement


p357.dat u 2:3

Figure 3.4: Periodi Osillations of Period 4

CHAPTER 3. THE DUFFINGOSCILLATOR NUMERICAL SOLUTIONS26

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p365.dat

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eloc

ity

Displacement


p365.dat u 2:3

Figure 3.5: Choati Osillations onned to right well

-1

-0.5

0

0.5

1

150 200 250 300 350 400

Dis

plac

emen

t

Time


p42_chaos

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1.5 -1 -0.5 0 0.5 1 1.5

Vel

ocity

Displacement


p42_chaos u 2:3

Figure 3.6: Double band haoti Attrator

Chapter 4Dung Osillator: BifurationDiagrams, Lyapunov Exponentsand ChaosBifuration means a splitting into two parts. The term bifuration is om-monly used in the study of nonlinear dynamis to desribe any sudden hangein the behavior of the system as some parameter is varied. The bifurationthen refers to the splitting of the behavior of the system into two regions: oneabove, the other below the partiular parameter value at whih the hangeo

urs.We wrote an algorithm to plot bifuration diagram for various parametersof the system. In general, bifuration diagram is plotted with the maximain x(t) against ontrol parameter of the system. Sine we are only interested27

CHAPTER 4. DUFFING OSCILLATOR: BIFURCATION DIAGRAMS, LYAPUNOV EXPONENTS AND CHAOS28Algorithm 1 Taking Xmax from Time SeriesK = 0NMAX = 50WHILE K < NMAX DOCALL RK4 = T,X,YX1 = X2Y1 = Y2X2 = X3Y2 = Y3X3 = XY3 = YIF X1 > X2 AND X2 < X3PRINT T-H,X2J = J+1ENDENDin the long term behaviour of the system, one has to allow the system tosettle down and then the solutions of the system has to be taken. From allthe solutions that we got by integrate out the equation using Fourth orderRunge-Kutta method. Inorder to get the x maximas for a partiular ontrolparmeter, we take three onseutive solutions of the system xn1,xn,xn+1 andwe heked whether the middle one 'xn' is bigger than the other twoxn1 < xn < xn+1 (4.1)if so that 'xn' is the maxima. This ondition (4.1) will satisfy only when

xn = xmax. One an hek this easily by plotting both the 'x t' data leand 'xmax t' data le together as shown below. After getting the xmax for

CHAPTER 4. DUFFING OSCILLATOR: BIFURCATION DIAGRAMS, LYAPUNOV EXPONENTS AND CHAOS29

0.2

0.4

0.6

0.8

1

1.2

1.4

1260 1270 1280 1290 1300

Dis

plac

emen

t

Time

freqos = -1 damp = 0.5 beta = 1 w = 1 ampf =0.34

-1.5

-1

-0.5

0

0.5

1

1.5

1250 1300 1350 1400 1450 1500

Dis

plac

emen

t

Time

freqos = -1 damp = 0.5 beta = 1 w = 1 ampf =0.42

Figure 4.1: Xmax and Time series

CHAPTER 4. DUFFING OSCILLATOR: BIFURCATION DIAGRAMS, LYAPUNOV EXPONENTS AND CHAOS30a partiular ontrol parameter say ampf = 0.34, a general do loop will dothe ontrol parameter to vary a

ording to your inrement value. Now wehave to take xmax and ontrol paratmeter to reate a data le, sine these arethe values needed for bifuration diagram. For Dung Osillator we havetwo ontrol paratmeters Amplitude of the Fore(ampf) Coeient of theNonlinear term()4.1 Amplitude of the Fore(ampf ) as ontrolparameter:In Fig. ?? - 4.6, the parameters k,, are xed at -1, 1, 0.5 and F only isvaried. The amplitude range is indiated in eah gure. Fig. ?? gives theplot for F in the range 0.34 - 0.375 we learly have the same period-doublingroute to haos as was observed in the logisti map. Fig. 4.2 amd Fig. 4.3feature large ranges of F where there are large periodi regimes betweenhaoti regions. When a small part is enlarged as in Fig. 4.4, we an notieself-similarity. The transition from haos to period 1 orbit is seen in Fig. 4.6


1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

0.34 0.345 0.35 0.355 0.36 0.365 0.37 0.375

x m

ax

ampf

freqos = -1 damp = 0.5 beta = 1 w = 1

(a) F as Control Parameter (range 0.1 : 1)

1.2

1.3

1.4

1.5

1.6

1.7

11.95 12 12.05 12.1 12.15 12.2

x m

ax

ampf


11.92 to 12.3

Figure 4.6: Chaos to Periodi


-2

-1

0

1

2

3

4

0 1 2 3 4 5 6 7 8 9 10 11

Amplitude of Force as control Parameter

0.1_10

(b) F as Control Parameter (range 0.1 : 1)Figure 4.2: Control Parameter as Amplitude of the Fore (range 0.1 : 10)4.2 Beta() as ontrol parameterIn gures 4.7 - 4.10 We have the bifuration diagram for various ranges of .In Fig. 4.7 it is interesting to observe that there is no haos after 7.75and upto 100. Fig. 4.10 is partiularly noteworthy. It displays the transitionfrom haos to periodiity and self-similarity.


-3

-2

-1

0

1

2

3

4

5

6

20 25 30 35 40 45 50

x m

ax

ampf


20 to 50

Figure 4.3: Control Parameter as Amplitude of the Fore (Range 20 : 50)

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

36.5 37 37.5 38 38.5 39 39.5 40 40.5

x m

ax

ampf


36_41

Figure 4.4: Control Parameter as Amplitude of the Fore: Magnied, show-ing self similarity


-2

-1

0

1

2

3

4

5 10 15 20 25 30

x m

ax

ampf


5 to 25

Figure 4.5: Control Parameter as Amplitude of the Fore (Range 5 : 25)

-1

-0.5

0

0.5

1

1.5

2

2 3 4 5 6 7 8

Control Parameter is Beta

0.1_10

0.5

0.6

0.7

ampf = .37 freqos = -1 damp = 0.5 w = 1

bif_8_100.dat

Figure 4.7: Beta range :


-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7

x m

ax

beta


beta=2 to 2.7 ampf=0.37

Figure 4.8: Beta () Range 2 : 2.7

-0.48

-0.47

-0.46

-0.45

-0.44

-0.43

-0.42

-0.41

-0.4

2.5 2.505 2.51 2.515 2.52 2.525 2.53 2.535

x m

ax

beta


beta=2.48to2.55_ampf

Figure 4.10: Chaos to period and Self similarity


-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

2.47 2.48 2.49 2.5 2.51 2.52 2.53 2.54 2.55

x m

ax

beta


beta=2.48to2.55_ampf

Figure 4.9: Chaos to Period in Beta

CHAPTER 4. DUFFING OSCILLATOR: BIFURCATION DIAGRAMS, LYAPUNOV EXPONENTS AND CHAOS374.3 Lyapunov SpetrumA slight hange in the initial onditions of the haoti system soon resultsin an entirely dierent time series. This phenomenon is known as sensitivedependene on initial onditions and is a hallmark of haoti motion. Thisproperty is highlighted in gure 4.11 whih ontains two displaement-timeseries for the Dung osillator with very lose initial onditions. One os-illation begins at x0 = 0, y0 = 1 and the other at x0 = 0, y0 = 1.1. Asan be seen from the gure, the two solutions initially appear to follow anidential path (g(4.11)). However, as time passes, the solution paths be-gin to diverge, leading to ompletely dierent long term behaviours (gure4.11). Over small sales, the drifting apart of the two solutions is in fat ex-ponential. This divergene property may be quantied using harateristiexponents known as Lyapunov exponents.


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 10 20 30 40 50

Dis

plac

emen

t

Time

Varying initial Values x=0, y= 1,1.1 freqos = -1 damp = 0.5 beta = 1 w = 1 ampf=0.365

x=0,y=1x=0,y=1.1

Figure 4.11: Sensitive dependene of initial pointsLet 0 be the observed seperation of two very lose points on seperatetrajetories on the attrator. After an elapsed time t, the trajetories havediverged and their seperation is now t. As this seperation is exponential,we may writet = 0e

t (4.2)where is the Lyapunov exponent. Rearranging equation(??), we obtain =

1

tln

[

t0

]

Nearby trajetories on strange attrators diverge, while still remaining in

CHAPTER 4. DUFFING OSCILLATOR: BIFURCATION DIAGRAMS, LYAPUNOV EXPONENTS AND CHAOS39t

0

time delay t

Initial Trajectory

Seperation

Final Trajectory

Seperation

Trajectory 1

Trajectory 2

Figure 4.12: Trajetory modela bounded region of phase spae due to the folding proess. In addition,trajetories near to the attrator (but not on it) are attrated to it, i.e. theyonverge on to it. We have divergene in some diretions and onvergenein others. To fully haraterize the divergene and onvergene propertiesof an attrator we require a set of Lyapunov exponents, one for eah or-thogonal diretion of divergene/onvergene in phase spae. The number ofLyapunov exponents required to dene the attrator is equal to the dimen-sion of its phase spae. Chaoti attrators have at least one nite positiveLyapunov exponent. On the other hand, random (noisy) attrators have aninnite positive Lyapunov exponent, as no orrelation exists between onepoint on the trajetory and the next (no matter how lose they are), i.e. thedivergene is instantaneous. Stable periodi attrators have only zero andnegative values of . Thus, the Lyapunov exponent is a valuable measurewhih may be used to ategorize haoti attrators. If we alulate the Lya-

CHAPTER 4. DUFFING OSCILLATOR: BIFURCATION DIAGRAMS, LYAPUNOV EXPONENTS AND CHAOS40punov exponent for orthogonal diretions of maximum divergene in phasespae, we obtain a set of Lyapunov exponents (1, 2, 3, ..., n), where n isthe dimension of the phase spae. This set of Lyapunov exponents is knownas the Lyapunov spetrum and is usually ordered from the largest positiveLyapunov exponent, 1, down to the largest negative exponent, n i.e. max-imum divergene to maximum onvergene. The Lyapunov spetrum may befound by monitoring the deformation of an innitesimally small hypersphereof radius on the attrator. Through time, the sphere is strethed in thediretions of divergene (positive s) and squeezed in the diretions of on-vergene (negative ) taking up the shape of an ellipsoid. The exponentialrates of divergene or onvergene of the prinipal axes of this ellipsoid givethe spetrum of Lyapunov exponents.


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

lam

bda

ampf

Lyapunov Spectrum control parameter = ampf

-1

-0.5

0

0.5

1

1.5

2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 4.13: Control Parameter as Amplitude of the Fore (range 0.1 : 1)


-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 2 4 6 8 10

lam

bda

ampf


-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

10 15 20 25 30 35 40 45 50

lam

bda

ampf


-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

50 60 70 80 90 100

lam

bda

ampf


Figure 4.14: Control Parameter as Amplitude of the Fore (range 10 : 100)

Chapter 5Poinar's SetionAs the name indiates, Poinar's setion is a setional view of the trajetoriesof the system. It is a tool developed by Henri Poinar (1854 1912) fora visualization of the ow in a phase spae of more than two dimensions.The Poinar setion has one dimension less than the phase spae. ThePoinar map maps the points of the Poinar setion onto itself. It relatestwo onseutive intersetion points. Note, that only those intersetion pointsounts whih ome from the same side of the plane. A Poinar map turnsa ontinuous dynamial system into a disrete one. For a Nonlinear System,depending on the strength of nonlinearity, the system shows periodi or quasiperiodi or haoti behavior. We expet that in the periodi regime the periodof system should be related to the period of external fore. If T is the periodof external fore then system period will be nT , where n is an integer. Inthe haoti regime n tends to innity. The number distint points(over a43

CHAPTER 5. POINCAR'S SECTION 44long period of time) plotted on Poinare map indiates the period of thesystem. Hene in haoti regime the Poinare map tries to ll a subset ofphase spae. After integrating out the equation(3.4), as we know the periodof the system is hightly dependent on the period of the external fore. Nowthe Dung osillator an be onsidered as an autonomous three dimensionalsystems with the time t as the third independent variable,Z = t.

dZ

dt= 1.Hene in order to get the poinare setion, we sampled the solution at every

t =2

(5.1)where is the time period of the fore. We have plotted poiare setion ofvarious regions of the bifuration diagrams and results are quite well satised.

CHAPTER 5. POINCAR'S SECTION 45

-4

-2

0

2

4

-4 -2 0 2 4

freqos = -1 damp = 0.5 beta = 1 w = 1 ampf=0.34 period 1

(a) 1 1.5 2

2.5

3

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

freqos = -1 damp = 0.5 beta = 1 w = 1 ampf=5 period 2

(b) 0.6

0.8

1

1.2

1.4

1.6

1.8

-0.1 -0.05 0 0.05 0.1 0.15 0.2

freqos = -1 damp = 0.5 beta = 1 w = 1 ampf=6p5 period 2

() Figure 5.1: (a) (b) ()xxxxxxxxxxxx


-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-2 -1.5 -1 -0.5 0 0.5

ampf = 8.5

Figure 5.2:


-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

-1.5 -1 -0.5 0 0.5 1

freqos = -1 damp = 0.5 beta = 1 w = 1 ampf=0.42 chaotic

Figure 5.3:In Fig. 5.1 there are 1, 2, and 4 points. Whent =

2

wn, n = 1, 2, 3, ... (5.2)(a) orresponds to phase point returning to the same point after t = 2

w,(b) orresponds to phase point going over to another point after t = 2

wandreturning to itself after t = 4

w, and so on. In Fig. 5.2 and Fig. 5.3 we dontobserve the phase point returning to itself at all, even after large t. Thisindiates that when we keep inreasing t, the number of points would to toinnity, indiating total lak of periodiity, or haos in the system.

Chapter 6Summary and ConlusionThe aim of this dissertation was to learn about haos diretly through nu-merial studies of a nonlinear system exhibiting haos. Towards this end, wehose the Dung Osillator. We solved the system numerially, using a 4thorder Runge-Kutta routine. Time-series, Phase plots, Bifuration diagrams,Lyapunov exponents, and Poinar Setions were studied. Period doublingbifuration route to haos was observed. There was onsisteny among theresults from the bifuration sequenes, Lyapunov exponents and Poinarsetions.The tools we have developed an be used for the study of any similarnonlinear systems. A beginning has been made in understanding haos. Moresophistiated tools are needed for understanding more realisti and therefore,more omplex systems.

48

Bibliography[1 Steven H.Strogatz, Non-linear dynamis and haos, Addison-Wesley Pub-lishing Co. Newyork(1994)[2 Robert C.Hilborn, Chaos and Nonlinear Dynami, An Introdution forSientists and Engineers (Seond Edition), Oxford University Press[3 Paul S Addison, Fratals and Choas, An Illustrated Course, IOP Publish-ing, Bristol and Philadelphia.[4 M. Lakshmanan and S. Rajasekar, Nonlinear Dynamis, Integrability,Chaos, and Patterns, (Springer - Verlag Heidelberg. Indian Edition. 2005)[5 , Fratals and Choas, An Illustrated Course, IOP Publishing, Bristol andPhiladelphia.

49

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