Slow-fast oscillations of delayed feedback systems: theory and experimentLionel Weicker

Slow-fast oscillations of

delayed feedback systems:

theory and experiment

Lionel Weicker

Promoters:

Prof. Dr. Jan Danckaert

A thesis submitted in fulfilment of the requirements

for the award of the degree of Doctor in Science by

In this work, we investigate two classes of problems exhibiting a delayed feedback. The …rst family of problems concerns optoelectronic oscillators (OEOs). An OEO is capable of generating either a single tone microwave oscillation, as used in radar applications, or a broadband chaotic carrier typically intended for physical data encryption in high bit rate optical communications. The second class of problems concentrates on mathematical models for delayed coupled neurons. A new form of synchronization is observed where an oscillatory regime is an alternative to a stable steady state. Both OEOs and coupled neurons are currently hot topics and a common feature is that they admit slow-fast evolving oscillations. A large part of this work is devoted to a detailed analysis of this type of regime. Because we are dealing with nonlinear delay-di¤erential equations, classical asymptotic techniques need to be carefully revisited. In addition to theory, original experiments have been performed in collaboration with experimental partners. For the OEOs, we have visited the laboratories of L. Larger at the University of Franche-Comté (Besançon, France) and of D. J. Gauthier at Duke University (Durham, USA). The work on coupled excitable neurons bene…ted from electronic experiments done in the labs of the Applied Physics group at the Vrije Universiteit Brussel.

In particular, we want to highlight the observation and mathematical analysis of stable asymmetric time-periodic square-waves showing di¤erent plateau lengths for the same overall period induced by an OEO. A primary Hopf bifurcation of a basic state is the mechanism leading to these square-waves. A second phenomenon that has been explored both numerically and experimentally for the OEOs and the coupled neurons is the coexistence of several squawaves with distinct but re-lated period. For the OEOs, these square-waves can, in the limit of large delay, be related to several closely located primary Hopf bifurcations. The stability mecha-nism is similar to the Eckhaus scenario for spatially extended systems. For coupled

Dans ce travail, nous étudions deux types de problèmes à retard. Le premier traite des oscillateurs optoélectroniques (OOEs). Un OOE est un système bouclé per-mettant de délivrer une onde électromagnétique radio-fréquence de grande pureté spectrale et de faible bruit électronique. Le second problème traite du couplage retardé de neurones. Une nouvelle forme de synchronisation est observée où un régime oscillant est une alternative à un état stationnaire stable. Ces deux prob-lèmes présentent des oscillations de type slow-fast. Une grande partie de ma thèse est dévouée à l’analyse de ces régimes. Etant donné qu’il s’agit d’équations non-linéaires à retard, les techniques asymptotiques classiques ont dû être revues. En plus d’une étude théorique, des expériences ont été e¤ectuées. Le travail sur les OOEs a été rendu possible grâce aux invitations respectives de L. Larger dans son laboratoire à l’Université de Franche-Comté et de D.J. Gauthier à Duke Univer-sity. Le travail sur le couplage de neurones a béné…cié d’expériences réalisées par L. Keuninckx du groupe « Applied Physics » de la Vrije Universiteit Brussel.

Une contribution importante de cette thèse est à la fois l’analyse mathéma-tique mais aussi l’observation expérimentale d’ondes carrées stables asymétriques présentant des longueurs de plateau di¤érentes mais ayant la même période dans un OOE. Une bifurcation de Hopf primaire d’un état stationnaire est le mécanisme menant à ces régimes. Un deuxième phénomène qui a été à la fois observé pour l’OOE et pour les neurones couplés est la coexistence entre plusieurs ondes carrées ayant des périodes di¤érentes. Pour l’OOE, ces oscillations peuvent être reliées à plusieurs bifurcations de Hopf primaires qui sont proches les unes des autres à cause du grand délai. Le mécanisme de stabilité est similaire à celui de "Eckhaus" pour les systèmes spatialement étendus. Pour le couplage de cellules excitables, nous avons étudié des équations couplées de type FitzHugh-Nagumo (FHN) linéaires par morceaux et obtenu des résultats analytiques. Nous montrons que le mécanisme

In dit werk onderzoeken we twee klassen van problemen die voorkomen in niet lin-eaire systemen onderworpen aan tijdsvertraagde terugkoppeling (delayed feedback). Een eerste type heeft betrekking op opto-elektronische oscillatoren die enerzijds een eenvoudige oscillatie kunnen opwekken, zoals bv. gebruikt wordt in radar toepassin-gen, maar anderzijds ook chaotische signalen kunnen genereren, gebruikt voor o.a. fysische data encryptie in optische communicatie. Een tweede klasse van onder-zochte problemen zijn de wiskundige modellen met vertraagde gekoppelde neuronen, waarin we een nieuwe vorm van oscillaties hebben kunnen waarnemen. Zowel de modellen voor de opto-elektronische oscillatoren als voor de gekoppelde neuronen zijn nu het onderwerp van intensief onderzoek en hebben ook een gemeenschap-pelijk kenmerk, namelijk snel/trage evoluerende oscillaties (slow/fast oscillations), die een belangrijk aspect van dit werk vormen. Omdat we hier te maken hebben met niet-lineaire di¤erentiaalvergelijkingen met tijdsvertraging (delay) moeten de klassieke asymptotische technieken zorgvuldig herbekeken worden. In de keuze van de door ons onderzochte onderwerpen hebben we ons laten leiden door die syste-men waar originele experisyste-menten mogelijk zijn. De gekoppelde opto-elektronische oscillatoren hebben we kunnen bestuderen in de laboratoria van de Université de Franche-Comté (groep van L. Larger, Besançon - Frankrijk) en van Duke University (groep van D.J. Gauthier, Durham - USA). Experimenten op stelsels van gekop-pelde elektronische neuronen werden uitgevoerd in het laboratorium van de Applied Physics groep aan de Vrije Universiteit Brussel (België).

Een belangrijke bijdrage in dit werk is de observatie en wiskundige analyse van stabiele asymmetrische periodische blokgolven (square-waves) die verschillende plateaulengtes kunnen vertonen voor eenzelfde totale periode. Dit type van gol-ven, die ontstaan via een primaire Hopf bifurcatie vanuit een stabiele toestand, was nog nooit eerder teruggevonden in een systeem met tijdsvertraging. Een

Foremost, I would like to express my gratitude to the people who made this thesis possible: my promoters Prof. Erneux and Prof. Danckaert. Since the beginning of my PhD, they provided me a continuous support. I’m very grateful for their patience, motivation, enthusiasm, and knowledge. Their advices helped me a lot in every moment. Un remerciement tout particulier à Thomas Erneux qui m’a initié aux équations di¤érentielles à retard, qui m’a ensuite fait su¢ sament con…ance pour m’encadrer durant mon mémoire et en…n durant ma thèse de doctorat. Depuis le début, j’ai pu béné…cier de ses connaissances immenses et de son inventivité aussi bien d’un point de vue scienti…que que personnel. Les discussions que j’ai eu avec lui étaient toujours riches et m’ont apporté énormément.

Besides my mentors, I’d like to thank the rest of my thesis committee: Prof. Van der Sande, Dr. Gorza, Prof. Vounckx, Dr. Kozyre¤, Prof. Larger and Prof. Sciamanna who agreed to take out time from their busy schedules to review my thesis. Un grand merci à Gregory Kozyre¤ qui m’a suivi tout au long de ma thèse en tant que collègue et président de mon comité d’accompagnement.

My sincere thanks also go to Prof. Larger, and Prof. Gauthier for o¤ering me the opportunity to work at their respective universities on diverse exciting projects. Thanks to them, I’ve made my …rst steps in the …eld of experimental physics. It was a real pleasure to work under your supervisions.

David, it is an honor to have been able to work with you at Durham. Even if you were very busy, you always took time to help me with the experiments. Our collaboration has been very rewarding for me. I’m also grateful for your warmful welcome in the US.

Lars, it is a pleasure to have met you. Our discussions both professional and personal were always very interesting and helpful.

Christian, I really enjoyed sharing my o¢ ce with you during those 3 months. Thanks for all the vivid discussions that we had.

Thanks to all the other people with whom I had the chance to collaborate with: Otti D’Huys, Maxime Jacquot, and Yanne Chembo.

We often forget them but I’d like to thank the developpers of the following soft-wares: Dynamics Solver, Matlab, PSI-Plot, Gnuplot, and Scienti…c Word. Without your work, this thesis will probably not be the same.

Merci à mes collègues et amis du groupe de Physique des systèmes dynamiques: Mustapha Tlidi, Evgeny Viktorov, Gregory Kozyre¤, Gaetan Friart, Yassine Cha¢ , et Etienne Averlant pour les longues et stimulantes discussions que nous avons eu, ainsi que pour les litres de café (et de bière pour certains) ingurgités durant ces 4 années. Un tout grand merci à Etienne qui n’a jamais hésité à sacri…er un peu de son temps pour m’aider tout au long de ma thèse.

Je remercie Marie-France, Fabienne, et Delphine pour leur aide et leur bonne humeur au quotidien.

J’aimerais aussi remercier mes parents, Brigitte et Marc, de m’avoir laissé la liberté d’évoluer comme je le désirais tout en n’hésitant pas à me recadrer quand c’était nécessaire. Merci à mon entourage qui m’a soutenu durant ces années.

Abstract iii

Résumé v

Samenvatting vii

Acknowledgments ix

1 Introduction 1

1.1 Chronology of the thesis work (2010-2014) . . . 3

1.2 Organization of the thesis . . . 7

2 Strongly asymmetric square waves in an optoelectronic oscillator 9 2.1 Introduction . . . 10

2.2 Experimental setup . . . 11

2.2.1 Laser source . . . 13

2.2.2 Modulator . . . 13

2.2.3 Delay line . . . 15

2.2.4 Photodiode, …lter, and ampli…cation . . . 16

2.3 Model equations . . . 16

2.4.1 Real root of the characteristic equation . . . 20

2.4.2 Imaginary root of the characteristic equation . . . 22

2.4.3 First Hopf bifurcation for < 0 . . . 24

2.4.4 Complex root of the characteristic equation . . . 25

2.5 Experimental observations and numerical simulations . . . 28

2.6 Asymptotic analysis of the D-periodic square-wave oscillations . . 30

2.6.1 Slowly-varying plateaus . . . 34

2.6.2 The fast transition layers . . . 39

2.7 Comparison between numerical and analytical bifurcation diagrams 43 2.8 Coexistence between a stable steady state and D-periodic oscillations 44 2.9 5 kHz - 530 kHz band-pass …lter . . . 47

2.10 Summary . . . 51

3 Multi-rhythmicity in an OEO system 55 3.1 Introduction . . . 55

3.2 Experiments and simulations . . . 58

3.2.1 Case m > 0 . . . 58

3.2.2 Case m < 0 . . . 60

3.3 Linear stability of the plateaus . . . 64

3.4 Discussion . . . 65

4 Network of excitable cells 67 4.1 Introduction . . . 68

4.2 Theory . . . 71

4.2.1 Slowly-varying parts . . . 72

4.2.2 Transition layers . . . 73

4.2.3 Analysis near to the saddle-node bifurcation point . . . 76

4.3 Numerical simulations . . . 81

4.4 Experiments . . . 82

4.5 Generalization to n delay coupled cells . . . 85

5 From the OEO to the FHN model 93

5.1 Introduction . . . 93

5.2 Transformation of the OEO model . . . 94

5.3 Analytic analysis . . . 97

5.4 Summary . . . 99

6 Optoelectronic oscillator with 2 delays 101 6.1 Introduction . . . 101

6.2 Experimental setup . . . 102

6.3 Dual-delay dynamical model . . . 106

6.4 Eigenmodes . . . 108 6.5 Primary bifurcation . . . 110 6.6 Secondary bifurcation . . . 112 6.7 Conclusion . . . 114 7 Conclusions 117 7.1 Summary . . . 117 7.2 Future work . . . 119

7.2.1 Stabilization of metastable square waves . . . 120

7.2.2 The continuous FHN system . . . 120

A Determination of A, B, C, and D 125 References 127 Scienti…c activities 135 Peer-reviewed journal articles . . . 135

Contributions to international conferences . . . 136

1.1 Example of multistability in a OEO system . . . 3

1.2 Square-wave oscillations with di¤erent duty cycles in a system com-posed of 2 coupled OEOs . . . 4

1.3 Square-wave oscillations with di¤erent duty cycles observed in a sin-gle OEO . . . 4

1.4 Di¤erent regimes observed in an experimental OEO subject to two delays . . . 5

2.1 Schematic representation of an OEO . . . 12

2.2 Actual experimental set-up . . . 12

2.3 V ILD curve of the laser . . . 13

2.4 Actual semiconductor CW laser . . . 14

2.5 Schematic representation of a MZM . . . 14

2.6 Nonlinear function induced by the MZM . . . 15

2.7 Actual Mach-Zehnder electro-optic intensity modulator . . . 16

2.8 Fiber used for the experiment . . . 17

2.9 Electronic circuit designed in Besançon . . . 18

2.10 Real roots of the characteristic equation . . . 21

2.11 Relation between Hopf bifurcation points and frequencies . . . 22

2.13 Relation between Hopf bifurcation points and . . . 26 2.14 Comparison between experimental and numerical time series for the

slowly varying solutions . . . 29 2.15 Comparison between experimental and numerical time series for the

slowly varying solutions . . . 30 2.16 Comparison between experimental and numerical time series for the

D-periodic oscillations . . . 31

2.17 Comparison between experimental and numerical time series for the 2 D-periodic oscillations . . . 32

2.18 Construction of the D-periodic regimes . . . 33

2.19 Construction of the D-periodic regimes for other values of parameters 35

2.20 Analytical bifurcation diagram . . . 38 2.21 Comparison between analytical and numerical bifurcation diagram . 44 2.22 Di¤erence between Hopf bifurcation point and the critical value in

function of the phase shift . . . 45 2.23 Relation between Hopf bifurcation points and " . . . 45 2.24 Relation between Hopf bifurcation points and . . . 46 2.25 Comparison between experimental and numerical time series for the

D-periodic oscillations . . . 48

2.26 Comparison between experimental and numerical time series for the

D-periodic oscillations . . . 49

3.6 Map of the OEO model which provides approximations of the

ex-trema in function of . . . 64

4.1 Representation of a ring of n unidirectionally delay-coupled units. . 70

4.2 Numerical time serie obtained with two coupled delayed FHNs . . . 72

4.3 Blow up of the fast transition layers . . . 74

4.4 Blow up of the fast transition layers . . . 74

4.5 Analytical bifurcation diagrams . . . 76

4.6 Fast transition layers close to the limit point . . . 78

4.7 Limit point obtained analytically . . . 80

4.8 Bifurcation diagram depending on two parameters . . . 81

4.9 Numerical bifurcation diagram of 2 coupled FHNs . . . 82

4.10 Di¤erent regimes observed in coexistence for two delay-coupled FHN units . . . 83

4.11 FHN electronic circuit . . . 84

4.12 Experimental bifurcation diagram with a as the bifurcation parameter. 86 4.13 Time series representing di¤erent regimes of stable square-wave os-cillations with period close to n D=m . . . 87

4.14 Di¤erent regimes of stable square-wave oscillations with period close to n D=m represented in the phase-plane . . . 88

4.15 Bifurcation diagrams of di¤erent oscillatory regimes for 3 and 6 cou-pled units . . . 90

5.1 Stable square-wave oscillations with period close to the delay ob-tained numerically from the OEO model . . . 94

5.2 Stable square-wave oscillations with period close to the delay in the phase-plane obtained numerically from the OEO model . . . 95

5.3 Stable square-wave oscillations with period close to the delay ob-tained numerically from the FHN model . . . 97

5.4 Stable square-wave oscillations with period close to the delay in the phase-plane obtained numerically from the FHN model . . . 98

5.6 Comparison between theoretical bifurcation diagram obtained from the FHN model and numerical bifurcation diagram obtained from the OEO model . . . 99 6.1 Experimental rf spectra and time series for di¤erent values of the

Chapter

1

Introduction

Among the research directions developed in nonlinear physics since the beginning of this century, problems exhibiting a delayed feedback have attracted particular attention. This is testi…ed by the large number of monographs and special jour-nal issues that have appeared during the period 2009-2011 [1, 2, 3, 4, 5, 6, 7, 8]. These problems emerge in all scienti…c disciplines including population biology, physiology, engineering, and even the social sciences. The delay of the feedback can describe the maturation time from larva to adult insects, the time to measure information and react in automation and robotics, the …nite communication time between neurons, or the time to invest in a business cycle. All these systems are modeled mathematically by delay di¤erential equations known since the beginning of the twentieth century. Thanks to the evolution of the performances of our mod-ern computers and the development of speci…c integration softwares, we are able to systematically explore them. Old problems such as chatter instabilities in milling and drilling or optically bistable devices bene…t from accurate stability analyses and new areas of research have appeared such as the synchronization properties of neurons and their e¤ects on Parkinson disease.

In the same time, laser physicists analyzing the e¤ects of a delayed optical or optoelectronic feedback realized that their devices could be used as benchmark systems to explore new dynamical phenomena directly caused by the delay [9]. Generally speaking, when the delay of the feedback surpasses a critical threshold, the steady state output of the system loses stability and regular or irregular time-dependent responses take place. In particular, slow-fast oscillations appear if the delay is su¢ ciently large. The main objective of this thesis is to discover their bifurcation origins.

Slow-fast oscillations may take the form of periodic square-waves as in the case of many optical devices or more complex forms consisting of successive slowly-varying pro…les connected by fast transition layers as in the case of biological excitable systems. Substantial numerical work already exists for such problems but we de-liberately choose to explore them in two other ways. First, we give priority to analytical approaches by developing singular perturbation techniques that take ad-vantage of the slow-fast nature of the solutions. These methods are not routine applications of standard methods known for ordinary di¤erential equations because the presence of a delayed state variable implies that some anticipation of the so-lution is needed. Second, all our problems have been selected by the opportunity of performing experimental studies. This became possible by collaborating with three di¤erent laboratories during the four years of my thesis. Together, experi-mental, numerical, and analytical approaches have signi…cantly contributed to our understanding of the role of the delayed feedback.

A delayed feedback system that allows systematic comparisons between theory and experiments is an optoelectronic oscillator (OEO). An OEO is a closed system capable of delivering a microwave electromagnetic wave of high spectral purity and of low electronic noise. OEOs were …rst designed by Yao and Maleki [10, 11] in 1994. They are typically composed of an electro-optic modulator, an optical-…ber delay line in a closed-loop resonating con…guration. The modulator and the optical …ber provide the nonlinearity and the delay, respectively. The interaction between nonlinearity and delay leads to complex dynamical behaviors ranging from ultrastable clocks to chaotic outputs. They can be modeled mathematically by relatively simple rate equations and systematic comparisons between theory and experiments have shown that these equations quantitatively describe a large variety of pulsating outputs [12].

β 1.0 1.5 2.0 2.5 3.0 x -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Figure 1.1: Numerical example of multistability in a OEO system. The bifurca-tion diagram is obtained by gradually increasing (black) and then by progressively decreasing (red) the bifurcation parameter . Redraw from Fig. 1.2 of Ref. [13].

1.1

Chronology of the thesis work (2010-2014)

2010

Initiated during my master thesis work, we …rst pursued our investigation on the coexistence of stable periodic solutions for OEOs exhibiting distinct frequencies. The problem being motivated by experiments done at the University of Maryland [14, 15], we investigated the bifurcation diagram in detail (see Fig. 1.1). This coexistence of periodic regimes will become a leitmotif during the course of this thesis as we examined other delayed feedback systems. A second problem dealing with two coupled OEOs bene…ted from a collaboration with O. D’Huys (Ph.D. Vrije Universiteit Brussel 2011 [16]). Square-wave regimes were found numerically with a total period close to twice the delay but with di¤erent plateau lengths. See Fig. 1.2. They are called duty cycles. It raised the important question of whether we could change these time intervals by changing a parameter keeping the total period …xed.

s 201000 201005 201010 201015 x1 a nd x2 -0. 4 -0. 2 0. 0 0. 2 0. 4 0. 6 0. 8

Figure 1.2: Square-wave oscillations with di¤erent duty cycles observed numerically in a model of two coupled OEOs. The oscillations of x1 (black) and x2 (red) are in

antiphase and exhibit di¤erent plateau lengths. Redraw from Fig. 1.3 of Ref. [13].

Figure 1.3: Square-wave oscillations observed in a single OEO with period close to the delay. The plateaus lengths can be adjusted by tuning a parameter. Redraw from Fig. 1 of Ref. [17].

duty lengths were controlled by changing one of the feedback parameters (see Fig. 1.3). We later provided an analytical description of these square-waves valid in the limit of large delays. The bifurcation origin remained however an open problem until 2013.

2011

FitzHugh-Nagumo (FHN) units, he found that stable slow-fast periodic oscillations may coexist with a stable steady state. There are no Hopf bifurcations and this new form of synchronization between cells results from the delay. This prompted a singular perturbation analysis of the coupled FHN system. We were then wondering if such oscillations could be observed experimentally in the presence of noise. This question led to a new collaboration with L. Keuninckx (Ph.D. Vrije Universiteit Brussel 2015) who designed an electronic circuit modeled mathematically by the same coupled FHN equations. Quantitative comparisons between experimental and theoretical bifurcation diagrams showed that the time-periodic attractor is robust with respect to noise. The details of the joint work as well as the extension of the analysis from 2 to n-coupled units are described in Chapter 4.

2012-2013

The coexistence of a stable limit-cycle and a stable steady state for the coupled FHN systems was explained by assuming a saddle-node bifurcation of limit-cycles [20]. Is this the same mechanism for the OEO exhibiting a stable periodic regime and a stable steady state for the same value of the feedback amplitude? In 2012, during the visit of the laboratory in Besançon, I was able to perform more detailed experiments that provided accurate comparisons between experiments and theory. We also used the software DDE-Biftool that allows the numerical continuation of periodic solution branches. We clearly demonstrated that the periodic regime with a period close to the delay emerges through a Hopf bifurcation mechanism. The Hopf bifurcation branch is …rst subcritical and unstable but folds back leading to a stable slow-fast oscillatory regime. The details of this work are given in Chapter 2. We also initiate fruitful discussions with Profs. D.J. Gauthier (Duke University), E. Schöll (TU Berlin), and their joint student D.P. Rosin (Ph.D. TU Berlin 2014). It turns out that we both found numerically the coexistence of stable periodic regimes with periods that are either integer fraction of the delay or odd integer fraction of twice the delay. The experimental veri…cation of this coexistence was planned and I visited the laboratory at Duke University to perform the experiments with D.P. Rosin. Excellent quantitative agreement was obtained and details of this collaboration are presented in Chapter 3.

2014

1.2

Organization of the thesis

In Chapter 2, we describe the experimental set-up of the OEO studied at the University of Franche-Comté. We determine the model equations and the primary Hopf bifurcations. We propose di¤erent comparisons between analytical, numerical, and experimental bifurcation diagrams.

We …rst consider the case of a periodic solution of period close to the delay but exhibiting di¤erent plateau lengths (di¤erent duty cycles). An analytical construc-tion of the limit-cycle soluconstruc-tion allows us to determine its bifurcaconstruc-tion diagram. The latter is compared to numerical and experimental bifurcation diagrams showing quantitative agreements.

Second, we wonder if a stable periodic solution may coexist with a stable steady state. To this end, we analyze the model equations analytically and numerically and …nd that this case is possible by controlling the bandpass …lter. Our theoretical predictions are then veri…ed experimentally.

In Chapter 3, we consider the experimental setup of the OEO designed at Duke University. We determine the model equations and concentrate on the case of stable periodic regimes coexisting for the same values of the parameters. Each solution is characterized by a period close to either an integer fraction of the delay or an odd fraction of twice the delay. The numerical solutions are compared to the experimental observations showing again excellent agreement. We relate these solutions to distinct primary Hopf bifurcations that stabilize at a …nite amplitude following a scenario similar to Eckhaus instability in spatially extended systems. The linear stability of the square-waves is substantiated analytically by determining stable …xed points of a map.

We next extend our previous analysis to n identical unidirectionally delay-coupled excitable units on a ring. We obtain analytical bifurcation equations for di¤erent oscillatory regimes. These regimes are slow-fast oscillations with di¤erent but related period. The period is close to an integer fraction of n times the delay. Numerical simulations con…rm the analytical predictions.

In Chapter 5, we discuss the similarities of the di¤erent regimes observed in the OEO and FHN systems. We show how to transform the OEO model to the FHN model under speci…c conditions. We obtain a simpler model for the OEO which is similar to the one studied in Chapter 4. The validity of the reduced model is veri…ed by numerical simulations of the reduced and original OEO models.

In Chapter 6, we investigate an OEO subject to two distinct delayed feedbacks. Its route to chaos starts with regular pulsating gigahertz oscillations that we in-vestigate both experimentally and theoretically. Of particular physical interest are the transitions to various crenelated fast time-periodic oscillations, prior to the onset of chaotic regimes. The two-delay problem is described mathematically by two coupled delay-di¤erential equations, which we analyze by using a multiple time scale method. We show that the interplay of a large delay and a relatively small delay is responsible for the onset of fast oscillations modulated by a slowly varying square-wave envelope. As the bifurcation parameter progressively increases, this envelope undergoes a sequence of bifurcations that corresponds to successive …xed points of a sine map.

Chapter

2

Strongly asymmetric square waves in an

optoelectronic oscillator

Time-delayed systems are known to exhibit symmetric square-waves oscillating with a period close to twice the delay. Here, we show that strongly asymmetric square-waves of period close to the delay are pos-sible. The plateau lengths can be tuned by changing a control parame-ter. The problem is investigated experimentally and numerically using a simple bandpass optoelectronic delay oscillator modeled by nonlinear delay integro-di¤erential equations. An asymptotic approximation of the square-wave periodic solution valid in the large delay limit allows an analytical description of its main properties (extrema and square pulse durations). Additionally, we investigate the fast transition layers between plateaus and show how they contribute to the total period. We show that the square waves emerge from a Hopf bifurcation of the basic steady state and that they may coexist with stable low-frequency periodic oscillations for the same value of the control parameter.

We wonder if a stable periodic solution may coexist with a stable steady state. To this end, we analyze the model equations analytically and numerically. We …nd that this case is possible by controlling the bandpass …lter. Our theoretical predictions are then veri…ed experimen-tally.1

1_{Parts of the work presented in this chapter have been published in Refs. [17, 21].}

2.1

Introduction

A fundamental property of nonlinear dynamical systems controlled by a delayed feedback is their tendency to exhibit square-wave oscillations if the delay D is

su¢ ciently large. These oscillations typically consist of 2 D-periodic transitions

between two or more ‡at plateaus. The following scalar delay di¤erential equation (DDE),

"x0 = x + f (x(s 1); ); (2.1)

has been studied intensively for its square-wave solutions (the prime stands for the derivative with respect to the time s). Equation (2.1) arises in a variety of applications, for example, physiological control systems [22], the transmission of light through a ring cavity [23], and population biology [24]. Here the dimensionless time is s t= D, and " = D > 0 is a small parameter de…ned as the ratio of

the linear decay time to the delay D. f (x; ) denotes a nonlinear function of x,

and is a control parameter. Setting " to 0 reduces Eq. (2.1) to the following map relating xn= x(s)and xn 1= x(s 1)

xn = f (xn 1; ): (2.2)

The two …xed points of the period-2 solution of Eq. (2.2) are expected to approach each of the two plateaus of the square-wave solutions of Eq. (2.1) if " ! 0. Sig-ni…cant contributions to the asymptotic relations between the solutions of the map (2.2) and the solutions of the DDE (2.1) have been made by Chow and Mallet-Paret [25], Mallet-Mallet-Paret and Nussbaum [26], Chow et al. [27] and Hale and Huang [28]. Close to a Hopf bifurcation point = 0(") of the DDE (2.1), the oscillations

quickly change their shape from sinusoidal to square-waves as the amplitude of the solution increases [29]. The square-wave oscillations then consist of sharp transition layers of a size proportional to ", connecting the two plateaus, of lengths close to unity.

systems. The plateau lengths of the waveform can be changed gradually by tuning a control parameter, but the overall pattern remains D-periodic. We also study

the e¤ects of the bandpass …lter on the system and observe di¤erent coexisting regimes.

The chapter is organized as follows. In Section 2.2, we brie‡y describe the ex-perimental set-up of an OEO. From the set-up, we determine in Section 2.3 the model equations which consist of two DDEs. In Section 2.4, we study analytically the equations and investigate the di¤erent Hopf bifurcation points. In Section 2.5, we compare numerical and experimental regimes. Particular attention is given on

D-periodic oscillations. In Section 2.6, we investigate those oscillations

analyti-cally. In Section 2.7, we compare our results with numerical bifurcation diagram. We predict in Section 2.8 that a coexistence between those regimes and the stable steady state is possible by controlling the bandpass …lter. In Section 2.9, we verify experimentally our theoretical predictions. Finally, we discuss our results in Section 2.10.

2.2

Experimental setup

The experimental set-up of an OEO is schematically shown in Fig. 3.1. It consists of an integrated optics Mach-Zehnder modulator (MZM) used to provide the nonlinear transformation as an electronically tunable two-wave interference. A dc bias can be adjusted externally to …x the zero drive interference condition (parameter 0). The

MZM output is delayed in time by a 4:2-km …ber spool ( D ' 20 s), and then

detected by a photodiode. A self-made electronic circuitry performs ampli…cation as well as the dynamical limitation for the signal to be fed back on the MZM. The dynamics of the oscillation loop is thus controlled via its electronic Fourier …ltering feature, which is forced to a broadband bandpass …lter. The characteristic response time of the corresponding cascaded high pass and low pass …rst order …lters are approximately 6:63 ms ( H) and 0:3 s ( L), respectively. The normalized gain

Figure 2.1: Schematic representation of an optoelectronic oscillator.

I (mA)

0 20 40 60 80

V

0 1 2 3 4 5 6 7 8

Figure 2.3: V ILD curve of the laser. Threshold is about 14:4 mA.

2.2.1

Laser source

The laser used is a semiconductor CW laser. As shown in Fig. 3.1, it is the only component of the system which is outside the optoelectronic feedback loop. Most of the characteristics of the laser have no signi…cant in‡uence on the dynamic of the setup. Only the V ILD curve is important and plays a signi…cant role in the

dynamics of the OEO. This curve is represented in Fig. 2.3, where V is measured as a voltage obtained by the combination of a photodiode and a transimpedance ampli…er. We observe that the laser threshold is about 14:4 mA. The feedback gain of the loop is proportional to the power of the laser. Therefore, it is an important parameter that can be easily adjusted during the experiment. The actual laser is shown in Fig. 2.4.

2.2.2

Modulator

prop-Figure 2.4: The actual semiconductor CW laser used in the experiment. Optical input Optical output VR F VDC a) b) c) d) e)

Figure 2.5: Schematic representation of a MZM.

agates along two di¤erent paths of equal lengths (d). Then, the two optical signals are recombined (e) and lead to the output optical signal. The Pockels e¤ect (which consists in modifying the refractive index by applying an electric …eld) occurs in crystals that lack inversion symmetry, such as the Lithium Niobate (LiN bO3). It

Φ

0 -2.0 -1.0 0.0 1.0 2.0

c

os

2

(

Φ

0

)

0.0 0.2 0.4 0.6 0.8 1.0

Figure 2.6: Nonlinear function induced by the MZM when vRF = 0.

The optical output (p) of the modulator is given by p = P0cos2

vRF

2V ;RF

+ 0 ; (2.3)

where:

- P0 is the input optical power,

- V ;RF is the half-wave voltage (voltage that needs to be applied to produce an

additional phase shift of 180 degrees between the two waves),

- 0 = VDC= [2 (V )DC]is an o¤set phase tuned by an external voltage VDC applied

on the DC electrodes of the MZM.

Fixing 0 allows us to choose speci…c operating points. The nonlinear function

(2.3) with vRF = 0is shown in Fig. 2.6. For 0 = 0 mod , the dynamics is around

a constructive interference (maximum of the nonlinear transformation), whereas

0 = =2 mod , the dynamics is around a minimum (destructive interference).

The actual MZM is shown in Fig. 2.7. For the following of the thesis, we consider di¤erent cases depending on the sign of sin (2 0). Experimentalists are in general

more used to speak about positive and negative slope. A negative value of sin (2 0)

is associated to the positive slope (positive feedback) whereas a positive value of sin (2 0) is associated to the negative slope (negative feedback).

2.2.3

Delay line

Figure 2.7: The Mach-Zehnder electro-optic intensity modulator used in the exper-iment.

the optoelectronic feedback. For example, if we choose a relatively high bandwidth of 10 MHz, the delay will be large compared to the response time of the system and the chaotic attractor will exhibit a large dimension (a few thousands). On the other hand, if we choose a low bandwidth of, for example 10 kHz, the chaotic attractor dimension will be of a few units. The actual …ber is shown in Fig. 2.8.

2.2.4

Photodiode, …lter, and ampli…cation

The photodiode converts optical intensities (or power, in mW) into electrical am-plitude (current in mA, or voltage in Volts via Ohm’s law, when the current ‡ows through a resistor R). The signal is then …ltered with a bandpass …lter character-ized by two cut-o¤ frequencies (24 Hz and 530 kHz) to be …nally ampli…ed using operational ampli…ers and reinjected into the MZM. The experimental electronic circuit is shown in Fig. 2.9.

2.3

Model equations

The transfer function is de…ned as the ratio of the output voltage [U (i!)] over the input voltage [P (i!)]

H (i!) U (i!) P (i!);

where U (i!) VRF(i!) =G (G is the gain associated to the RF-ampli…er) and

Figure 2.8: Fiber used in the experiment. input optical power of the photodiode is given by

p (t) = P0cos2[ vRF(t D) = (2V ;RF) + 0] :

If we consider a second order bandpass …lter, the transfer function reads as H (i!) U (i!) P (i!) = K 1 (1 + i! L) i! H (1 + i! H) ; (2.4)

where K stands for the photodiode conversion e¢ ciency and L ( H) for the

char-acteristic time of the low (high) pass …lter [ L=H = 1= 2 fL=H where fL=H is the

cuto¤ low/high-pass frequency]. Equation (2.4) can be rewritten as KP (i!) = (1 + i! L) U (i!) + 1 i! H U (i!) + L H U (i!) : Taking the inverse FT of this equation gives

In terms of vRF(t), it leads to vRF(t) + L dvRF dt (t) = KGP0cos 2_{[ v} RF(t D) = (2V ;RF) + 0] 1 H Z t t0 vRF(s) ds L H vRF(t) : (2.5)

Since L= H is very small compared to 1 in our con…guration, we neglect it.

Nor-malizing the amplitude with x ( vRF) = (2V ;RF) and the time s t= D, we

obtain the following dimensionless DDEs "dx ds = x y + cos 2_{(x (s 1) +} 0) cos2( 0) ; (2.6) dy ds = x; (2.7) where GKP0 2V ;rf ; 0 Vdc 2V ;dc ; " L= D; D= H; y = Z t t0 vRF(s) ds:

The arbitrary time t0 is Eq. (2.5) has been chosen so that our equation admits

a steady state for x = 0 which implies the cos2₍

0) term. The values of the

experimental parameters used during my visit of the laboratory of Prof. Larger in 2012 are documented in Tab. 2.1. For most of the work, we will use these values. We also plan to investigate cases of interest appearing for di¤erent values of the parameters which we will then specify.

2.4

Steady state and Hopf bifurcation

Parameter Value Meaning

L 0:3 s Characteristic time of the low-pass …lter

D 20 s Delay due to the optical …ber

H 6:63 ms Characteristic time of the high-pass …lter

0 ::: 4:76 Gain of the feedback

0 =2 ::: =2 External voltage applied to the MZM

" 0:015 L= D

0:003 D= H

Table 2.1: Values of the experimental parameters.

then use Eq. (2.7) in order to eliminate y. We then obtain the following second-order DDE for x

"x00= x0 x sin [2x (s 1) + 2 0] x0(s 1) : (2.8)

The linearized equation for x = 0 is

"x00= x0 x sin (2 0) x0(s 1) : (2.9)

We wish to determine all the solutions of the linearized equation. Equation (2.9) is linear which suggests to seek an exponential solution of the form

x = exp ( s) : (2.10)

Substituting Eq. (2.10) into Eq. (2.9) leads to the following characteristic equation for the growth rate

" 2+ + + exp ( ) = 0; (2.11)

where sin (2 0) is a single parameter that combines and 0. Equation

(2.11) is transcendental and admits several roots. In the following sections, we separate the case real, imaginary, and complex. Stability means that Re ( ) < 0 for all eigenvalues. A purely imaginary eigenvalue is a necessary condition for a Hopf bifurcation point.

2.4.1

Real root of the characteristic equation

-2.0 -1.5 -1.0 -0.5 0.0 -0.4 -0.2 0.0 0.2 0.4 σr γ γc2 γc1 γc3

Figure 2.10: Real roots of the characteristic Eq. (2.12). We used the values of the experimental parameters listed in Tab. 2.1.

Using the experimental values listed in Tab. 2.1, the study of (2.12) leads to two negative values of r for c1 < < 0 and two positive values r for < c2. See

Fig. 2.10. To obtain _c1, _c2, and _c3, we determine the values of r satisfying the

condition d =d r = 0. Equivalently, we need to …nd the roots of

" 3_r+ (1 + ") 2_r+ r = 0: (2.13)

Equation (2.13) admits three real roots: 0:055887, 0:052888, and 67:663667. Inserting these values into Eq. (2.12) leads to

c1 = 0:894091; c2 = 1:114952;

c3 = 0:

We conclude that the steady state of the OEO system is always unstable if < _c2. At = _c1, = _c2, and = _c3, complex eigenvalues emerge. This can be demonstrated by inserting = r + i i into (2.11) and by separating real and

imaginary parts. By then taking carefully the limit i ! 0, we obtain Eqs. (2.12)

-1.08 -1.06 -1.04 -1.02 -1.00 0 5 10 15 20 25 30 1.00 1.02 1.04 1.06 1.08 1.10 σi γ

Figure 2.11: versus i obtained from Eqs. (2.14)-(2.15) and solved by the

di-chotomy method. We used the values of the experimental parameters listed in Tab. 2.1.

2.4.2

Imaginary root of the characteristic equation

From Eq. (2.11) with = i i, we …nd, after separating real and imaginary parts,

the two conditions

0 = " 2_i + itan ( i) ; (2.14)

= 1

cos ( i)

: (2.15)

These relations are necessary conditions for a Hopf bifurcation point. By using the dichotomy method, we obtain the solutions shown in Fig. 2.11. For < 0, we observe that the frequencies of the di¤erent Hopf bifurcations are either close to 0 or close to a multiple of 2 . For > 0, the frequencies of the Hopf bifurcations are close to (2n + 1) , where n is an integer. In the following subsection, we …nd approximations of the di¤erent Hopf bifurcation points and frequencies in the limit " and tending to 0.

Approximations of the Hopf frequencies and Hopf bifurcation points We have seen in the previous section that the di¤erent Hopf frequencies are close to either 0, 2 n or (2n + 1) . We next analyze the three cases.

A. i close to 0( < 0). We consider Eqs. (2.14) and (2.15), and assume that iH is small and use the fact that " = O ( ) is also small. We obtain

iH0 '

p

= 0:0548; (2.16)

H0 ' 1

n Analytic iHn Dichotomy iHn Analytic Hn Dichotomy Hn

1 6:1894 6:1911 1:0044 1:0043

2 12:3781 12:3829 1:0177 1:0171

3 18:5670 18:5779 1:0399 1:0381

4 24:7559 24:7770 1:0710 1:0668

Table 2.2: Comparison between the Hopf bifurcation frequencies and points for di¤erent values of n obtained from the approximations [Eqs. (2.19)-(2.20)] and the solutions of Eqs. (2.14)-(2.15) by using the dichotomy method.

We note a good agreement with the values obtained from Eqs. (2.14) and (2.15) which are 0:0543 and 1:0015 for iH and H, respectively.

B. i close to a multiple of 2 ( < 0). We introduce iH = 2 n + ! in Eqs.

(2.14) and (2.15) where ! is assumed O (") and n is a positive integer. It leads to 0 = (1 + ") !2+ ("4 n + 2 n) ! + " (2 n)2;

H ' 1 +

!2

2 :

The solutions of the quadratic equation for ! in the limit " ! 0 are !+ =

2 n 2 n", (2.18)

! = 2 n

2 n,

where !+ is valid for n = O (1) and ! is not valid because we assume ! = O (").

We have the following approximations for the solutions of Eqs. (2.14) and (2.15)

iHn ' 2 n + 2 n 2 "; (2.19) Hn ' " 1 + 1 2 2 n 2 n" 2# : (2.20)

n Analytic iHn Dichotomy iHn Analytic Hn Dichotomy Hn

0 3:0954 3:0961 1:0011 1:0010

1 9:2837 9:2867 1:0099 1:0096

2 15:4725 15:48 1:0277 1:0266

3 21:6614 21:6769 1:0544 1:0515

Table 2.3: Comparison between the Hopf bifurcation frequencies and points for di¤erent values of n obtained from the approximations [Eqs. (2.21)-(2.22)] and the solutions of Eqs. (2.14)-(2.15) by using the dichotomy method.

C. i close to an odd multiple of ( > 0). We introduce iH = (2n + 1) +

! in Eqs. (2.14) and (2.15) where ! is assumed O (") and n is an integer. It leads to

0 = (1 + ") !2+ (2n + 1) (1 + 2") ! + " (2n + 1)2 2;

H ' 1 +

!2

2 :

The solutions of the quadratic equation for ! in the limit " ! 0 are !+ =

(2n + 1) (2n + 1) ",

! = (2n + 1)

(2n + 1) .

!+ is valid for n = O (1) but ! is not valid because we assume ! = O ("). We

have the following approximations for the solutions of Eqs. (2.14) and (2.15)

iHn ' (2n + 1) + (2n + 1) (2n + 1) "; (2.21) Hn ' 1 + 1 2 (2n + 1) (2n + 1) " 2 : (2.22)

In Tab. 2.3, we show a comparison between the solutions obtained from Eqs. (2.14)-(2.15) and our approximations [Eqs. (2.21)-(2.22)]. We observe a good agreement. We note that the …rst bifurcation to appear corresponds to oscillations of frequency close to .

2.4.3

First Hopf bifurcation for

< 0

Depending on the values of and/or ", the …rst Hopf bifurcation is either H0

-2.0 -1.8 -1.6 -1.4 -1.2 -1.0 0.00 0.05 0.10 0.15 0.20 0.25 ε γ γH1 γH0 -1.010 -1.005 -1.000 -0.995 0.000 0.005 0.010 0.015 0.020 γH1 γH0 U U U U U S S

Figure 2.12: Hopf bifurcation lines as functions of " obtained from Eqs. (2.14)-(2.15). The …gure on the right is a blow up of the diagram near = 1. The dashed line corresponds to the value of " considered in our experiments. S and U mean stable and unstable, respectively.

Figures 2.12 and 2.13 represent H0 and H1, as functions of " ( …xed) and as

functions of (" …xed), respectively. They have been obtained numerically from Eqs. (2.14)-(2.15).

In Fig. 2.12, is …xed at = 0:003. We note that for " < 0:0086 oscillations of period close to 2 emerge …rst. For " > 0:0086, oscillations with a low frequency emerge …rst. The case where " = 0:0086 is a special case where we have a double Hopf bifurcation point. This particular case has been studied in my master thesis [13].

In Fig. 2.13, " is …xed to " = 0:015. We note that for > 0:0085 oscillations of period close to 2 emerge …rst. For < 0:0085, oscillations with a low frequency emerge …rst. The case where = 0:0085 is a special case where we have a double Hopf bifurcation point.

2.4.4

Complex root of the characteristic equation

Inserting = r+ i i into Eq (2.11) leads to two coupled transcendental equations

for rand i. An exact analytical solution for rand i is not possible and we shall

look for an approximation for " and small. We distinct three cases depending on the frequency of oscillations.

-1.15 -1.10 -1.05 -1.00 0.00 0.05 0.10 0.15 0.20 0.25 -1.005 -1.003 -1.001 -0.999 0.000 0.002 0.004 0.006 0.008 0.010 γH0 γH1 γH0 γH1 δ γ S S U U U U U

Figure 2.13: Hopf bifurcation lines as function of " obtained from Eqs. (2.14)-(2.15). The …gure on the right is a blow up of the diagram near = 1. The dashed line corresponds to the values of considered in our experiments. S and U mean stable and unstable, respectively.

We …rst expand our parameters as

= " 1; (2.23)

= 1 " ₁+ :::. (2.24)

We then seek a solution for of the form

= "1=2 1+ " 2+ "3=2 3+ :::. (2.25)

These scalings are suggested by the asymptotic approximations (2.16) and (2.17) of the Hopf frequencies. Introducing (2.23)-(2.25) into Eq. (2.11) leads to the following expressions for 1 and 2

O (") : 1 = i

p

1;

O "3=2 : 2 2 = 1 1=2:

The …rst equation provides a correction to the frequency. The second equation corresponds to the leading approximation of the real part given by

Re( 2) =

1

2 1

1

2 : (2.26)

In terms of the original parameters, the leading expression of is

= ip 1

2 1 + + 2 + O "

3=2

B. Oscillations of frequency close to 2 ( < 0) We …rst expand our parameters as

= " 1; (2.27)

= 1 "2 ₁+ :::. (2.28)

We then seek a solution for of the form

= 2 i + " 1+ "2 2+ :::. (2.29)

These scalings are suggested by the asymptotic approximations (2.19) and (2.20) of the Hopf frequencies. Introducing (2.27)-(2.29) into Eq. (2.11) leads to the following corrections for

O (") : 1 = 1 2 2 i; O "2 : 2 = 1 + 2 1 2 2 1 2 i 2 1:

The …rst equation provides a correction to the frequency. The second equation leads to the real part of 2 given by

Re ( 2) = 1 1 2 1 2 2 2 : (2.30)

In terms of the original parameters, the leading expression of is = i 2 + 2 2 " + O " 2 _{( + 1)} 1 2 2 2 " 2 + O "3 : The stability of the steady state is determined by the real part of . The stable and unstable domains are shown in Fig. 2.12 and Fig. 2.13 and were guided by (2.26) and (2.30).

C. Oscillations of frequency close to ( > 0) We …rst expand our parameters as

= " 1; (2.31)

= 1 + "2 ₁+ :::. (2.32)

We then seek a solution for of the form

These expansions are motivated by the analysis of the Hopf bifurcation points. Introducing (2.31)-(2.33) into Eq. (2.11) leads to the following equations for 1

and 2 O (") : 0 = 1 i 1+ 2; 1 = 1 i; O "2 : 0 = i 2 1 2 i 2 2 1+ 1 i 12 i; 2 = 2 1 2 + 2 1_{i +} 1 2 1:

The real part of 2 is

Re ( 2) = 1 1 2 1 2 :

In terms of the original parameters, the leading expression of is = i + " + O "2 + 1 1

2 "

2

+ O "3 :

2.5

Experimental observations and numerical

sim-ulations

In this section, we compare experimental observations with numerical simulations of Eqs. (2.6)-(2.7). Furthermore, we relate our results to our analytical predictions. The values of the parameters are …xed to " = 0:015, = 0:003, and 0 ' 0:714.

With this value of 0, is negative and we look for the dynamical response of the

system for di¤erent values of .

As predicted by the analysis, we observe a stable steady state for low values of ( < 1). Increasing leads to oscillations which emerge from a Hopf bifurcation [ H ' 1:015 (from experiments), H ' 1:01 (from simulations), H ' 1:01 (from

0 2 4 6 8 10

V

-0.4 -0.2 0.0 0.2 0 200 400 600

x

-0.8 -0.4 0.0 0.4 0.8 0 4 8 12

V

-0.4 -0.2 0.0 0.2 0.4 0 200 400 600

x

-0.5 0.0 0.5

t (ms)

0 10 20 30

V

-0.5 0.0 0.5

s

0 500 1000 1500

x

-1.0 -0.5 0.0 0.5 1.0 (a) (d) (b) (e) (c) (f)

t (ms)

0 10 20 30 40

V

-0.8 -0.4 0.0 0.4

s

0 500 1000 1500

x

-1.0 0.0 1.0

Figure 2.15: Left: Experimental time serie for = 1:79. Right: Numerical time series obtained from Eqs. (2.6)-(2.7) for = 1:79. We used the values of the experimental parameters listed in Tab. 2.1 with 0 = 0:714.

Guided by our analysis of the Hopf bifurcation, we look for another kind of regime coexisting with the slowly varying oscillations. It consists in oscillations of period close to the delay. To this end, we experimentally excite the system by injecting a signal at the desired frequency. Numerically, we …x the initial functions as x(s) = cos (2 s) and y (s) = 0 ( 1 < s < 0). See Fig 2.16a for experimental time series and Fig 2.16b for numerical solution. We next examine how those oscillations are changing with . The di¤erent time series obtained by increasing are shown in Figs. 2.16 and 2.17. As increases, we observe that the length of the small plateau increases whereas the time interval of the long plateau decreases. The period remains close to the delay. For higher values of , period doubling occurs ( _{' 1:88 from the experiment and ' 2:15 from the numerical simulations),} followed by a cascade of bifurcation leading to chaos. See Fig. 2.17.

From the numerical simulations, we note that the model equations reproduce the di¤erent behaviors found experimentally. Of particular interest are the oscillations of period close to the delay. To have a better understanding of the mechanism that produces these oscillations, we will use the fact that " and are two small parameters.

2.6

Asymptotic analysis of the

D

-periodic

square-wave oscillations

0 20 40 60

V

-0.20 -0.10 0.00 0.10 0.0 1.0 2.0 3.0

x

-0.2 0.0 0.2 0 20 40 60

V

-0.2 0.0 0.2 0.0 1.0 2.0 3.0

x

-0.2 0.2

t (µs)

0 20 40 60

V

-0.4 -0.2 0.0 0.2 0.4

s

0.0 1.0 2.0 3.0

x

-0.8 -0.4 0.0 0.4 0.8 (a) (d) (b) (e) (c) (f)

Figure 2.16: Left: Experimental time series for = 1:03(a); = 1:08(b); = 1:24 (c). Right: Numerical time serie obtained from Eqs. (2.6)-(2.7) for = 1:03 (d); = 1:08 (e); = 1:24 (f). We used the values of the experimental parameters listed in Tab. 2.1 with 0 = 0:714. Initial functions for the numerical simulations

0 20 40 60 80 100

V

-0.8 -0.4 0.0 0.4 0 1 2 3 4 5

x

-1.0 0.0 1.0

t (µs)

s

0 20 40 60 80 100

V

-0.5 0.0 0.5 0 1 2 3 4 5

x

-1.0 0.0 1.0 (a) (b) (c) (d)

Figure 2.17: Left: Experimental time series for = 1:9 (a); = 2:2 (b). Right: Numerical time serie obtained from Eqs. (2.6)-(2.7) for = 2:3(c); = 2:5(d). We used the values of the experimental parameters listed in Tab. 2.1 with 0 = 0:714.

x -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 δy -0.04 -0.02 0.00 0.02 y0 x01 x03 x02 x 0 s0+εr εr εr s₀ 1 - s₀ x₀₂ x₀₁− 1+2εr s (a) (b)

Figure 2.18: (a) Numerical square-wave solution of Eqs. (2.6) and (2.7) during one period. We used the values of the experimental parameters listed in Tab. 2.1 with = 1:2, and 0 = 0:714. The two plateaus of the square-wave solution are of

length s0 and 1 s0;respectively. The fast transition layers contributes to the total

period by two corrections of size "r: (b) The periodic solution is shown in red in the phase plane (x; y). The S-shaped line is the function (2.39). The dot is the unique steady state (x; y) = (0; 0). The values of y0 = 0:0143, x01 = 0:584;

lengths for 0 < 0. See Fig. 2.18. The slow-fast time behavior of the solution is due

to the small value of ": As we shall later demonstrate, the relatively small change of y compared to x (see Fig. 2.18b) is the result of the small value of : Furthermore, the asymmetry of the square-wave oscillations (s0 < 1=2 in Fig. 2.18a) is related

to the deviation 0+ =4. In this section, we consider the values of the parameters

listed in Tab. 2.1 but also = 8:43 10 3 and a smaller value " = 2 10 4. The smaller " is used in order to better illustrate the slow-fast behavior of the periodic solution and check the validity of our assumptions as we construct the solution. From Fig. 2.12, we note that the 1-periodic Hopf bifurcation is the second primary Hopf bifurcation for " = 0:015 while it is the …rst for " = 2 10 4_.

We propose to construct the square-wave solution in the limit " ! 0: Speci…cally, we seek a P -periodic solution satisfying the condition

x(s P ) = x(s); (2.34)

where the period P is given by

P = 1 + 2"r; (2.35)

and r = O(1). As shown in Fig. 2.18a, the solution consists of two slowly varying plateaus connected by fast transition layers. We anticipate the analysis of the transition layers (see Section 2.6.2) by assuming that the contribution from these layers to the period P is the same ("r). In Sections 2.6.1 and 2.6.2, we analyze the slow and fast parts of the solution, respectively.

2.6.1

Slowly-varying plateaus

The leading approximation is obtained by setting " = 0 in Eqs. (2.6)-(2.7). The reduced equations with (2.34) and (2.35) are

y0 = x; (2.36)

0 = x y + cos2(x + 0) cos2( 0) ; (2.37)

x(s 1) = x(s): (2.38)

From Eq. (2.37), we determine y = y(x) as

y = 1 x + cos2(x + 0) cos2( 0) : (2.39)

pro-x -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 δy -0.06 -0.03 0.00 0.03 y0 x -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0 s0 1 s y δ-1y₀ x₁₁ x₁₂ (a) (b)

Figure 2.19: Numerical square-wave solution of Eqs. (2.6) and (2.7) during one period. The values of the parameters are the same as for Fig. 2.18a except that " = 2 10 4 _{is much smaller and = 8:43 10} 3_{. (a) x(s) exhibits sharp jumps}

at times s = 0, s = s0; and s = 1 while y remains continuous at those points. We

also determine x1j ' (x x0j)= (red) and note that x11 and x12 are continuous at

times 0, s0, 1. (b) The square-wave periodic solution is shown in the phase-plane

(x; y) (red). Comparing with Figure 2.18(b), we note that y = y0 is now located

vided > 1: The evolution of x and y along the left and right branches corresponds to the evolution along the plateaus of the square-wave periodic solution. They can be determined by inserting (2.39) into the left-hand side of Eq. (2.36) and by solv-ing the resultsolv-ing …rst-order equation for x. However, this solution is complicated and we may …nd simple analytical expressions by taking advantage of the small value of . Speci…cally, we seek a perturbation solution of Eqs. (2.36) and (2.37) of the form

y = 1y0(s) + y1j(s) + :::; (2.40)

x = x0j(s) + x1j(s) + :::; (2.41)

where j = 1 or 2 refer to the time domains 0 < s < s0 and s0 < s < 1;respectively

(see Fig. 2.19a).

Inserting (2.40) and (2.41) into Eqs. (2.36) and (2.37) and equating to zero the coe¢ cients of each power of leads to a sequence of problems for the unknowns functions y0; y1j; x0j;and x1j. The leading order problem is O(1) and is given by

y₀0 = 0; (2.42)

x0j y0+ cos2(x0j + 0) cos2( 0) = 0: (2.43)

Equation (2.42) implies that y0 is a constant. We already know that for a …nite

range of values of y0, Eq. (2.43) admits more than one root (see Fig. 2.18b).

The solutions corresponding to the left and right branches are denoted by x01< 0

and x02 > 0, respectively. We do not know the values of y0 and analyze the O( )

problem for y1j(s) and x1j(s). It is given by

0 s < s0 y0₁₁ = x01; (2.44) x11 y11 2 sin(2x01+ 2 0)x11 = 0; (2.45) s0 s < 1 y0₁₂ = x02; (2.46) x12 y12 2 sin(2x02+ 2 0)x12 = 0: (2.47)

Figure 2.19 suggests the following initial conditions for y11 and y12

y11(0) = y1M and y12(s0) = y1m; (2.48)

respectively. The solution of Eqs. (2.44)-(2.48) is then y11 = y1M + x01s; (2.49) y12 = y1m+ x02(s s0); (2.50) x11 = y11 1 + 2 sin(2x01+ 2 0) ; (2.51) x12 = y12 1 + 2 sin(2x02+ 2 0) : (2.52)

Continuity of y11 and y12 at times s = s0 and 1 leads to the conditions

y1M+ x01s0 = y1m; (2.53)

y1m+ x02(1 s0) = y1M; (2.54)

which are two equations for y1M y1m. A solution of Eqs. (2.53) and (2.54) is

possible only if

x01s0 + x02(1 s0) = 0: (2.55)

As for y11 and y12, we next assume that the corrections x11 and x12 are equal at

s = s0 and s = 1 (see Fig. 2.19a). From (2.51) and (2.52), we then obtain the

condition

sin(2x01+ 2 0) = sin(2x02+ 2 0); (2.56)

or equivalently,

cos(x01+ x02+ 2 0) sin(x01 x02) = 0: (2.57)

Equation (2.57) admits multiple solutions. We speci…cally look for a solution of Eq. (2.57) which satis…es the perfect square-wave condition x01 = x02 if 0 =

=4: This solution is given by

x01+ x02+ 2 0 = =2: (2.58)

Using (2.43), (2.58) allows one to determine y0; x01 and x02. Substracting Eq.

(2.43) with x01 and Eq. (2.43) with x02 gives

(x01 x02) sin(x01+ x02+ 2 0) sin(x01 x02) = 0: (2.59)

Using (2.58) then allows one to eliminate x02 in Eq. (2.59). We …nd

(2x01+ 2 0+ =2) + sin(2x01+ 2 0+ =2) = 0: (2.60)

Equation (2.60) provides the solution for x01= x01( )in the implicit form

= 2x01+ 2 0+ =2 sin(2x01+ 2 0+ =2)

β

0.8 1.0 1.2 1.4

s

0.0 1.0 2.0 3.0 -0.8 0.0 0.8

β

0.8 1.0 1.2 1.4 0.0 0.1 0.2 0.3 0.4 0.5

s

₀

x

(a) (b) (c) s₀ 1-s₀ x₀₁ x₀₂

Figure 2.20: Analytical bifurcation diagram of the square-waves. (a) The numeri-cally computed square-wave is shown for = 1:2 and the values of the parameters are the same as in Fig. 2.18. (b) Its extrema are in good agreement with the an-alytical predictions obtained from the parametric solution (2.61)-(2.63) (with x01

as the parameter). (c) The plateau lengths are s0 and 1 s0; respectively, and the

…gure shows s0.

We obtain x02 and s0 by using (2.58) and (2.55)

x02 = =2 2 0 x01; (2.62)

s0 =

x02

x02 x01

0: (2.63)

In Fig. 2.20, we compare our approximations with the numerical solution obtained for = 1:2. In Section 2.7, we numerically analyze the bifurcation diagram of the possible stable solutions and show good agreement with our approximations. Moreover, using a continuous integration method, we show that the square-wave oscillations emerge from a Hopf bifurcation.

The plateaus of the square wave are x = x01 < 0 and x = x02 > 0, in …rst

approximation. They are de…ned as two roots of Eq. (2.43) for a …xed y0. Figure

2.18b suggests that there is a third root of x0j. In the following, we determine this

third root and formulate an expression for y0. Equations for x01 and x02 are given

by Eqs. (2.58) and (2.60). From Eq. (2.60) we determine cos(2x01+ 2 )as

cos(2x01+ 2 0) = 2x01+ 2 0 +

2: (2.64)

From (2.43) with j = 1, we formulate an expression for y0 given by

y0 = x01+

Using (2.64), it leads to y0 = x01+ 1 2(2x01+ 2 0+ 2) 2cos(2 0); = 0+ 4 2 cos(2 0): (2.65)

In order to …nd the third root of Eq. (2.43), we introduce (2.65) into Eq. (2.43) and obtain

0+

4 = x0j + 2 cos(2x0j + 2 0): This equation admits the solution

x03= ( 0+

4): Using Eq. (2.58), we then obtain the relation

x01+ x02= 2x03;

or equivalently,

x02 x03 = x03 x01:

The two extreme roots are at equal distance from the central root x03. This

sym-metry property has important consequences. In particular, the two fast transitions layers admit the same equation and they contribute in the same way to the cor-rection of the period. In the next section, we study analytically the fast transition layers.

2.6.2

The fast transition layers

The plateaus of the square wave are connected by fast transition layers on time intervals proportional to ". See Fig. 2.18a.

Jump down at s = 0

We …rst consider the fast transition layer at s = 0 and introduce the inner variable

1 s" 1. The leading-order transition layer equations for y = Y1( 1) and x =

X1( 1) are then given by

dY1

d ₁ = 0; (2.66)

dX1

where we have used the periodicity condition

x(s 1) = x(s P + 2"r) = x(s + 2"r) = X1( 1+ 2r): (2.68)

Equation (2.66) implies that Y1 is a constant. It needs to match the constant

determined in our analysis of the slowly varying plateaus i.e., Y1 = y0 1. Using

the expression of y0 given by (2.65), Eq. (2.67) can be rewritten as

dX1 d ₁ = X1 4 + 2sin h 2X1( 1+ 2r) + 2 + 2 i : (2.69)

This equation can be reformulated in a simpler form by introducing the deviation z1 X1 x03= X1+ + =4. From Eq. (2.69), we obtain

dz1

d ₁ = z1+ 2sin [2z1( 1 + 2r)] : (2.70) The boundary conditions for the jump down transition are X1( 1) = x02 and

X1(1) = x01. In terms of z1, they take the simpler form

z1( 1) = a and z1(1) = a; (2.71)

where

a x02 x03> 0: (2.72)

Jump up at s = s0+ "r

We next consider the transition layer near s = s0 + "r and introduce the inner

variable ₂ (s s0 "r)" 1. The leading-order transition layer equations for

y = Y2( 2) and x = X2( 2) are given by

dY2 d ₂ = 0; (2.73) dX2 d ₂ ( 2 + s0+ "r " ) = X2( 2+ s0+ "r " ) Y2( 2 + s0+ "r " ) + 2 cos(2X2( 2 +s0+"r" + 2r) + 2 ) cos(2 ) : (2.74)

where we have used the periodicity condition

x(s 1) = x(s P + 2"r) = x(s + 2"r) = X2( 2+

s0+ "r

The constant solution for Y2 is again matching the value obtained from the analysis

of slowly varying plateaus i.e., Y2 = y0 1. Using the expression of y0 given by

(2.65), Eq. (2.74) simpli…es as dX2 d ₂ ( 2 + s0+ "r " ) = X2( 2+ s0+ "r " ) 4 + 2 sin 2X2( 2+ s0+ "r " + 2r) + 2 + 2 : (2.76) Introducing the deviation z2 X2 x03= X2+ + =4, Eq. (2.76) becomes

dz2 d ₂( 2+ s0+ "r " ) = z2( 2 + s0+ "r " ) + 2sin 2z2( 2+ s0+ "r " + 2r) : (2.77) We next note the following relations between the two inner variables

2 = 1

s0+ "r

" : (2.78)

Inserting (2.78) into Eq. (2.77), we formulate an equation for z2( 1)of the form

dz2

d ₁( 1) = z2( 1) + 2 sin [2z2( 1+ 2r)] : (2.79) The boundary conditions for the second transition layer now are

z2( 1) = a and z2(1) = a; (2.80)

where a is de…ned by (2.72). We realize that Eqs. (2.79) and (2.80) are the same as Eqs. (2.70) and (2.71) except that the boundary conditions have been interchanged. This implies that the solution of Eqs. (2.79) and (2.80) is related to the solution of Eqs. (2.70) and (2.71) by

z2( 1) = z1( 1): (2.81)

In conclusion, we found the same delay di¤erential equation for the two fast transition layers. It is given by

dz

d = z + 2 sin [2z( + 2r)] ; (2.82)

where we have omitted the subscript 1 for z1 and 1. We next proceed as in [25].

We note that by rescaling time as =2r, Eq. (2.82) can be rewritten as a DDE with delay 1 and parameter r

dz

d = 2r z 2 sin(2z( 1)) ; (2.84)

z( _{1) = a and z(1) = a:} (2.85)

z = a are both critical points of Eq. (2.84). This means that we are looking for a heteroclinic orbit for some value of r, that is, a trajectory joining these critical points as _{! 1: The delay parameter r is unknown a priory, and must be} determined as part of the solution. We cannot solve the problem analytically for arbitrary because it is a nonlinear DDE.

Correction to the period

In this section, we solve Eq. (2.84) and (2.85) for close to 1. Our objective is to demonstrate that there is indeed a unique value of r such that Eq. (2.84) and (2.85) admit a solution. To this end, we introduce a small parameter de…ned by

p

( 1)=b; (2.86)

where b = 1 if ? 1: We then expand the solution z and parameter r in power series of

z = Z1( ) + 2Z2( ) + :::; (2.87)

r = r0+ r1+ :::; (2.88)

where . The motivation for introducing (2.86) comes from the fact that a x02 x03 =

q

3

2( 1), in …rst approximation as ! 1, which implies that

the amplitude of the solution scales like p 1: After introducing (2.86)-(2.88) into (2.82), we equate to zero the coe¢ cients of each power of . The leading-order problem is O( ) and is given by

(1 2r0)

dZ1

d = 0: (2.89)

In order to have a non constant solution for z1, we require that r0 = 1=2. The next

problem is O( 2_{)and is given by}

with the boundary conditions Z1( 1) = p 3b=2 and Z1(1) = p 3b=2: (2.91)

We choose b = 1 and note that the damped Hamiltonian Eq. (2.90) has a unique solution z1 =

p

3=2 tanh( ) if r1 = 0. We conclude that we have found an

analytical expression for the transition layer solution provided > 1 and r = 1

2+ O [( 1)] : (2.92)

We have determined numerically the period of the square-wave oscillations with a high precision. The values of , ", and 0 are 0:003, 0:015, 1:2, and 0:714,

respectively. _{Numerically, we …nd P ' 1:015. From (2.35), we then compute} 2"r = 0:015which implies r = 0:5. The numerical value of r is in perfect agreement with the analytical value r = 0:5 given in (2.92).

2.7

Comparison between numerical and

analyti-cal bifurcation diagrams

In this section, we consider as our bifurcation parameter and compare our ana-lytical bifurcation diagram shown in Fig. 2.20b with a full numerical bifurcation diagram.

In Fig. 2.21a, we show the extrema of the 1-periodic solutions obtained numer-ically in black and the solutions of Eq. (2.62) in red. We note a good agreement between the analytical and numerical diagrams. Figure 2.21b is obtained using a continuation method (DDE-Biftool) and highlights the fact that the 1-periodic os-cillations emerge from a Hopf bifurcation at = _{2 ' 1:0146 as an unstable branch} but are stable after the branch folds back.