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Date Created: 09/23/15
Chapter 1 Experimental Background The subject of this series of lectures is the analysis of data generated by a dynamical system operating in a chaotic regime More speci cally we describe how to extract from chaotic data topological signatures that determine the stretching and squeezing mechanisms which act on flows in phase space and which are responsible for generating chaotic data In the rst section of this introductory chapter we describe for purposes of motivation a laser that has been operated under conditions in which it behaved chaoticallyi The topological methods of analysis that we describe in this lecture series were developed in response to the challenge of analyzing chaotic data sets generated by this laser In the second section we list a number of questions which we would like to be able to answer when analyzing a chaotic signali None of these questions can be addressed by the older tools for analyzing chaotic data estimates of the spectrum of Lyapunov exponents and estimates of the spectrum of fractal dimensions The question that we would particularly like to be able to answer is this How does one model the dynamics To answer this question we must determine the stretching and squeezing mechanisms that operate togetheri repeatedlyito generate chaotic data The stretching mechanism is responsible for sensitivity to initial conditions while the squeezing mechanism is responsible for recurrent nonperiodic behavior These two mechanisms operate repeatedly to generate a strange attractor with a selfsimilar structure A new analysis method topological analysis has been developed to respond to the fundamental question just statedi At the present time this method is suitable only for strange attractors that can be embedded in threedimensional spacesi However for such strange attractors it offers a complete and satisfying resolution to this question The results are previewed in the third section of this Chapter It is astonishing that the topological analysis tools that we describe have provided answers to more questions than we asked originallyi This analysis procedure has also raised more questions than we have answered e ope that the interaction between experiment and theory and between old questions 2 CHAPTER 1 EXPERIMENTAL BACKGROUND answered and new questions raised will hasten evolution of the eld of nonlinear dynamics 11 Laser with Modulated Losses The possibility of observing chaos in lasers was originally demonstrated by and by Gioggia an Abra ami The use of lasers as a testbe for generating de terministic chaotic signals has two major advantages over uid and chemical systems which until that time had been the principal sources for chaotic data 1 The time scales intrinsic to a laser 10 7 to 10 3 s are much shorter than the time scales in uid experiments and oscillating chemical reactions This is important for experimentalists since it is possible to explore a very large parameter range during a relatively short time E0 Reliable laser models exist in terms of a small number of ordinary differ ential equations whose solutions show close qualitative similarity to the behavior of the lasers that are modeled The topological methods described in the remainder of this work were orig inally developed to understand the data generated by a laser with modulated lossesi A schematic of this laser is shown in Fig lili A C02 gas tube is placed between two infrared mirrors The ends of the tube are terminated by Brewster angle windows which polarize the eld amplitude in the vertical direction Under normal operating conditions the laser is very stable A Kerr cell is placed inside the laser cavity The Kerr cell modi es the polarization state of the electromagnetic eld This modi cation coupled with the polariza tion introduced by the Brewster windows allows one to change the intracavity losses The Kerr cell is modulated at a frequency determined by the operating conditions of the laser When the modulation is small the losses within the cavity are small and the laser output tracks the input from the signal gener atori The input signal from the signal generator and the output signal the measured laser intensity are both recorded in a computer When the modu lation crosses a threshold the laser output can no longer track the signal input At rst every other output peak has the same height then every fourth peak then every eighth peak and so on In Fig 12 we present some of the recorded and processed signals from this part of the perioddoubling cascade and beyond The signals were recorded under different operating conditions and are displayed in ve lines as follows a period 1 b period 2 c period 4 d period 8 e chaosi Each of the four columns presents a different representation of the data In the rst column the intensity output is displayed as a function of time In this presentation the periodl and period2 behaviors are clear but the higherperiod behavior is not The second column displays a projection ofthe dynamics into a twodimension al space the dIdt vsi 1t plane In this projection periodic orbits appear as 11 LASER WITH MODULATED LOSSES 3 i g K 002 PS A 39 z I 5 M D HHllHllIHHHI C Figure 11 This schematic representation of a laser with modulated losses shows the carbon dioxide tube C02 power source PS mirrors Kerr cell signal generator S detector D and computer oscilloscope and recorder A variable electric eld across the Kerr cell rotates its polarization direction and modulates the electric eld amplitude Within the cavity closed loops deformed circles Which go around once twice four times be fore closing In this presentation the behavior of periods 1 2 and 4 is clear Period 8 and chaotic behavior is less clear The third column displays the power spectrum Not only is the periodic behavior clear from this display but the rel ative intensity of the various harmonics is also evident Chaotic behavior is manifest in the broadband power spectrum Finally the last column displays a stroboscopic sampling of the output In this sampling technique the output in tensity is recorded each time the input signal reaches a maximum or some fixed phase with respect to the maximum There is one sample per cycle In period 1 behavior all samples have the same value In period 2 behavior every other sample has the same value The stroboscopic display clearly distinguishes be tween periods 1 2 4 and 8 It also distinguishes periodic behavior from chaotic behavior The stroboscopic sampling technique is equivalent to the construction of a Poincare section for this periodically driven dynamical system All four of these display modalities are available in real time during the experiment The laser with modulated losses has been studied extensively both experi mentally and theoretically The rate equations governing the laser intensity I 4 CHAPTER 1 EXPERIMENTAL BACKGROUND s quotat VW m n 39u oli s quot3 quotw a v1vounua at I39 a tquot00quot 0 squott gquot 39ii 39n 5 39i 0 i39 39 g 0 39 390 3 i D a Q K I a 0 O I 39 4 Q av t t quot T u v 2 I 3 o t II 39 Min 4 3 quot 39 C I v t 39 u on 1 39 a 39 c 39 p quoto v vi a I u CO 339 vnlhallalt llgl 39 Figure 12 Each column provides a different representation of the experimental data Each row describes different experimental conditions The rst column shows the recorded intensity time signal I The second column presents the phase space projection dItdt vs I The third column shows the power spectrum of the recorded intensity signal The frequencies of the Fourier components in the signal and their relative amplitudes jump out of this plot The last column presents a stroboscopic plot Poincare section This is a record of the intensity output at each successive peak or more generally at some constant phase of the input signal The data sets were recorded under the following experimental conditions a period 1 h period 2 c period 4 d period 8 e chaotic Reprinted with permission from Tredicce et al 11 LASER WITH MODULATED LOSSES 5 and the population inversion N are d E 7k0117 N mcoswt 1 1 MN 7 N0 N0 71gtIN Here m and w are the modulation amplitude and angular frequency respectively of the signal to the Kerr cell N0 is the pump parameter normalized to N0 1 at the threshold for laser activity and kg and 7 are loss rates In dimensionless scaled form this equation is du E 2 7 AcosQTu 12 d2 E 17 612 7 1 egzu The scaled variables are u I 2 k0HN 71 t m A kom 61 1Hk0 and H2 17k0N0 7 1 The bifurcation behavior exhibited by the simple models 11 and 12 is qualitatively if not quantitatively in agreement with the experimentally observed behavior of this laser A bifurcation diagram for the laser model 12 is shown in Fig 13 The bifurcation diagram is constructed by varying the modulation amplitude A and keeping all other parameters xed The overall structures of the bifurcation diagrams are similar to experimentally observed bifurcation diagramsi This gure shows that a periodl solution exists above the laser threshold No gt 1 for A 0 and remains stable as A is increased until A N 08 It becomes unstable above A N 08 with a stable period2 orbit emerging from it in a perioddoubling bifurcationi Contrary to what might be expected this is not the early stage of a perioddoubling cascade for the period2 orbit is annihilated at A N 085 in an inverse saddlenode bifurcation with a period2 regular saddlei This saddlenode bifurcation destroys the basin of attraction of the period2 orbiti Any point in that basin is dumped into the basin of a period 4 2 X 21 orbit even though there are two other coexisting basins of attraction for stable orbits of periods 6 3 X 21 and 4 at this value of Al Subharmonics of period n P7 n 2 2 are created in saddlenode bifurcations at increasing values of A and I P2 at A N 01 P3 at A N 03 P4 at A N 07 P5 and higher shown in the inset All subharmonics in this series up to period n 11 have been seen both experimentally and in simulations of 12 The evolution perestroika of each of these subharmonics follows a standard scenario as T increases 6 CHAPTER 1 EXPERIMENTAL BACKGROUND Ptoquot MC PO P7 39 39 39 39 39 quot PB 39 PSC L l l l I a a a 9 3 PA 3 in 5 P3 l 2 a i P Q C i g b o u Q C a r 92 r 11 P54 39 o g O lidihoonc occcoo 1oo 0a ococaoaccentconcoaucocuo uQQOO P Wdwnwl uIIITIII E 7 I 1 25 Figure 13 The bifurcation diagram for the laser model 12 is computed by varying the modulation amplitude A Stable periodic orbits solid lines7 regu lar saddles dotted lines7 and strange attractors are shown Period n branches Pa 2 2 are created in saddle node bifurcations and evolve through the Feigen baum period doubling cascade as the modulation amplitude increases There are two apparently distinct stable period 2 orbits Hovvever7 these are connected by an unstable period 2 orbit dotted extending from A 2 01 to A 2 08 and thus constitute a single period 2 orbit Which is a snake A period 3 snake is also present Two distinct stable period 4 orbits are present and coexist over a short range of parameter values 07 lt A lt 08 The inset shows a sequence of period n orbits Newhouse orbits for n 2 5 The Smale horseshoe mechanism predicts that as many as three inequivalent pairs of period 5 orbits could exist The locations of the two additional pairs have been shown in this diagram at A 2 065 and A 2 25 Parameter values 61 003 62 00097 S2 15 11 LASER WITH MODULATED LOSSES 7 1 A saddlenode bifurcation creates an unstable saddle and a node that is initially stable to Each node becomes unstable and initiates a perioddoubling cascade as A increases The cascade follows the standard Feigenbaum scenario The ratio of A intervals between successive bifurcations and of geometric sizes of the stable nodes of periods n X 2k have been estimated up to k S 6 for some of these subharmonics both from experimental data and from the simulations These ratios are compatible with the universal scaling ratiosi 9 Beyond accumulation there is a series of noisy orbits of period n X 2k that undergo inverse periodhalving bifurcations This scenario has been predicted by Lorenz Additional systematic behavior has been observed Higher subharmonics are generally created at larger values of Al They are created with smaller basins of attraction The range of A values over which the Feigenbaum scenario is played out becomes smaller as the period n increases In addition the subharmonics show an ordered pattern in phase space In Fig 14 we show four stable periodic orbits that coexist under certain operating conditions Roughly speaking the larger period orbits exist outside the smaller period orbits These orbits share many other systematics which have been describei In Fig 15 we show an example of a chaotic time series taken for A N 13 The chaotic attractor based on the period2 orbit the periodl orbit has just collided with the period3 regular saddlei The perioddoubling accumulation inverse noisy periodhalving scenario described above is often interrupted by a crisis Grebogi and Ott of one type or another Boundary Crisis A regular saddle on a periodn branch in the boundary of the basin of attraction surrounding either the periodn node or one of its periodic or noisy periodic progeny collides with the attractori The basin is annihilated or enlargedi Internal Crisis A ip saddle of period n X 2k in the boundary of a basin surrounding a noisy perio n X 2k1 orbit collides with the attractor to produce a noisy periodhalving bifurcationi External Crisis A regular saddle of period 7 in the boundary of a period n R PW strange attractor collides with the attractor thereby annihi lating or enlarging the basin of attraction Figure 16a provides a schematic representation of the bifurcation diagram shown in Fig 13 The different kinds of bifurcations encountered in both experiments and simulations are indicated here These include both direct and inverse dd 4 quotf 39 er39 4 d bquot 39 39 and boundary and external crisesi As the laser operating parameters 1607w change the bifurcation diagram changes 8 CHAPTER 1 EXPERIMENTAL BACKGROUND Figure 14 Multiple basins of attraction can coexist over a broad range of parameter values The stable periodic orbits and the strange attractors within these basins have a characteristic organization The coexisting orbits shown above are7 from the inside to the outside period 2 bifurcated from a period 1 branch period 2 period 3 period 4 The two inner orbits are separated by an unstable period 2 orbit not shown all three are part of a snake s arbunits t periods Figure 15 This time series from a laser with rnodulated losses was taken at a value of A N 13 which is just beyond the collision crisis of the strange attractors based on the period 2 and period 3 orbits There is an alternation in this time series between noisy period 2 and noisy period 3 behavior 11 LASER WITH MODULATED LOSSES 9 a b c Figure 16 Schematics of three bifurcation diagrams for three different operating conditions of Eqs 12 As control parameters change the bifurcation diagram is modified Slow change in control parameter values deforms the bifurcation diagram from a to b to The sequence a to c shows the unfolding of the snake in the period 2 orbit The unstable period 2 orbit connecting the two lowest branches is invisible in a and b since only stable attractors are shown In each diagram the bifurcations are i saddle node A inverse saddle node inverse saddle node doesn t work a boundary crisis external crisis Period doubling bifurcations are indicated by a small vertical line separating stable orbits of periods differing by a factor of 2 Accumulation points are indicated by A Strange attractors based on period n orbits are indicated by the On In Fig 16b and c we show the schematics of bifurcation diagrams ob tained for slightly different values of these operating or control parameters In addition to the subharmonic orbits of period n created at increasing values of A Fig 13 there are orbits of period n that do not appear to belong to that series Newhouse series of subharmonics The clearest example is the period 2 orbit which bifurcates from period 1 at A N 08 Another is the period 3 orbit pair created in a saddle node bifurcation that occurs at A N 245 These bifurcations were seen in both experiments and simulations It was possible to trace the unstable orbits of period 2 01 g A S 085 and period 3 04 g A g 25 in simulations and find that these orbits are components of an orbit snake Alligood Alligood Sauer and Yorke This is a single orbit that folds back and forth on itself in direct and reverse saddle node bifurcations as A increases this is not unlike a Feynman diagram for hard scattering of an electron by a photon which scatters the electron backward in time creating a positron The unstable period 2 orbit 01 g T S 085 is part of a snake By changing operating conditions both snakes can be eliminated see Fig 16c As a result the subharmonic P2 is really nothing other than the period 2 orbit which bifurcates from the period 1 branch P1 Furthermore instead of having saddle node bifurcations creating four inequivalent period 3 orbits at A N 04 and A N 245 there is really only one pair of period 3 orbits the other pair being components of a snake Topological tools relative rotation rates Solari and Gilmore were first de veloped to determine which orbits might be equivalent or components of a 10 CHAPTER 1 EXPERIMENTAL BACKGROUND snake and which are not Components of a snake have the same topological invariants cf Chapter 4 These tools suggested that the Smale horseshoe mechanism was responsible for generating the nonlinear phenomena observed in both the experiments and the simulations This mechanism predicts that additional inequivalent subharmonics of period n can exist for n 2 5 Since the size of a basin of attraction decreases rapidly with n a search was made for additional inequivalent basins of attraction of period 5 Two additional stable period5 orbits besides P5 were located in simulations Their locations are shown in Fig 13 at A N 06 and A N 245 One was also located experimen tally The other may also have been seen but its basin was too small to be certain of its existence Bifurcation diagrams had been observed for a variety of physical systems at that time other lasers electric circuits a biological model and a bouncing balli Their bifurcation diagrams are similar but not identical to those shown above This raised the question of whether similar processes were governing the description of this large variety of physical systems During these analyses it became clear that the standard tools for analyzing chaotic data estimates ofthe spectrum ofLyapunov exponents and estimates of the various fractal dimensionsiwere not suf cient for a satisfying understanding of the stretching and squeezing processes that occur in phase space and which are responsible for generating chaotic behavior In the laser we found many coexisting basins of attraction some containing a periodic attractor others containing a strange attractor The rapid alternation between periodic and chaotic behavior as control parameters eg A and 9 were changed meant that Lyapunov exponents and fractal dimensions depended on the basins and varied at least as rapidly For this reason we sought to develop additional tools for the analysis of data generated by dynamical systems that exhibit chaotic behavior The objective was to develop measures that were invariant under control parameter changesi 12 Objectives of a New Analysis Procedure In view of the experiments just described and the data that they generated we hoped to develop a procedure for analyzing data that achieved a number of objectives These included an ability to answer the following questions 1 Is it possible to develop a procedure for understanding dynamical systems and their evolution perestroikas as the control parameters eg 160 m 7 or A Q 61 62 change to i Is it possible to identify a dynamical system by means of topological in variants following suggestions proposed by Poincare 3 Can selection rules be constructed under which it is possible to determine the order in which periodic orbits can be created andor annihilated by standard bifurcations Or when different orbits might belong to a single snake I 9quot PREVIEW OF RESULTS 11 g i ls it possible to determine when two strange attractors are a equivalent in the sense that one can be transformed into the other without creating or annihilating orbits or b adiabatically equivalent one can be deformed into the other by changing parameters to create or annihilate only a small number of orbit pairs below any period or c inequivalent there is no way to transform one into the other 13 Preview of Results A new topological analysis procedure was developed in response to the questions asked of the data initiallyi These questions are summarized in Section 12 The remarkable result is that there is now a positive and constructive answer to the question How can I look at experimental data such as shown in Fig 12 or 15 and extract useful information let alone information about stretching and squeezing let alone a small set of integers is new analysis procedure answered more questions than were asked orig inallyi It also raised a great many additional questions This is one of the ways we know that we are on the right track The results of this new topological analysis procedure are presented through out this book Below we provide a succinct preview of the major accomplish ments of this topological analysis tooli o It is possible to classify lowdimensional strange attractorsi These are strange attractors that exist in threedimensional spaces 0 This classi cation is topological in nature This classi cation exists at two levels a macroscopic level and a micro scopic level It is discrete at both levels Thus there exists a doubly discrete classi cation for lowdimensional strange attractorsi This doubly discrete classi cation depends in an essential way on the un stable periodic orbits which are embedded in strange attractorsi 0 At the macro level the classi cation is by means of a geometric structure that describes the topological organization of all the unstable periodic orbits that exist in a hyperbolic strange attractor This geometric structure is called variously a twodimensional branched manifold motholder or template 0 Branched manifolds can be identi ed by a set of integers Thus at the macro level the classi cation is discrete 0 At the micro level the class cation is by means of a set of orbits in a nonhyperbolic strange attractor whose existence implies the presence of all the other orbits that can be found in the nonhyperbolic strange attractor This subset of orbits is called a basis set of orbits CHAPTER 1 EXPERIMENTAL BACKGROUND To any given period a basis set of orbits is also discretei As control parameters change the basis set of orbits changes The changes that are allowed are limited by topological arguments Each different sequence of basis sets describing the transition from the laminar to the hyperbolic limit describes a different route to chaos Each different route to chaos is a different path in a forcing diagram shown in Fig 98 During this transition the underlying branched manifold is robust lt gen erally does not change Large changes in control parameter values can cause changes in the un derlying branched manifold These changes occur by adding branches to or removing branches from the branched manifold The branch changes that are allowed are also limited by topological and continuity arguments The information required for this doubly discrete classi cation of strange attractors can be extracted from experimental data The data requirements are not heavy Data sets of limited length are required The data need not be exceptionally cleani Only a modest signaltonoise level is required The analysis method degrades gracefully with noise Speci cally as the noise level degrades the data it becomes more dif cult to identify the higherperiod orbits which are the least important for this analysis The most important orbits those of lowest period persist longest with increasing noisei As a result Murphy is on vacation77 author of the famous law The data analysis method comes endowed with a rejection criterion The branched manifold identi es the stretching and squeezing mechanisms that generate chaotic behaviori Thus this doubly discrete classi cation describes how to model the dy namics77
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