Systems of globally coupled phase oscillators can have robust attractors that are heteroclinic networks. Such a heteroclinic network is generated, where the phases cluster into three groups, within a specific regime of parameters when the phase oscillators are globally coupled using the function $g(\varphi) = -\sin(\varphi + \alpha) + r \sin(2\varphi + \beta)$. The resulting network switches between 30 partially synchronised states for a system of $N=5$ oscillators. Considering the states that are visited and the time spent at those states a spatio-temporal code can be generated for a given navigation around the network. We explore this phenomenon further by investigating the effect that noise has on the system, how this system can be used to generate a spatio-temporal code derived from specific inputs and how observation of a spatio-temporal code can be used to determine the inputs that were presented to the system to generate a given coding. We show that it is possible to find chaotic attractors for certain parameters and that it is possible to detail a genetic algorithm that can find the parameters required to generate a specific spatio-temporal code, even in the presence of noise. In closing we briefly explore the dynamics where $N>5$ and discuss this work in relation to winnerless competition.

Thesis advisor: Jan R. Engelbrecht / Thesis advisor: Renato E. Mirollo / Many phenomena in nature that involve ordering in time can be understood as collective behavior of coupled oscillators. One paradigm for studying a population of self-sustained oscillators is the Kuramoto model, where each oscillator is described by a phase variable, and interacts with other oscillators through trigonometric functions of phase differences. This dissertation studies $N$ identical Kuramoto oscillators in a general form \[ \dot{\theta}_{j}=A+B\cos\theta_{j}+C\sin\theta_{j}\qquad j=1,\dots,N, \] where coefficients $A$, $B$, and $C$ are symmetric functions of all oscillators $(\theta_{1},\dots,\theta_{N})$. Dynamics of this model live in group orbits of M\"obius transformations, which are low-dimensional manifolds in the full state space. When the system is a phase model (invariant under a global phase shift), trajectories in a group orbit can be identified as flows in the unit disk with an intrinsic hyperbolic metric. A simple criterion for such system to be a gradient flow is found, which leads to new classes of models that can be described by potential or Hamiltonian functions while exhibiting a large number of constants of motions. A generalization to extended phase models with non-identical couplings gives rise to richer structures of fixed points and bifurcations. When the coupling weights sum to zero, the system is simultaneously gradient and Hamiltonian. The flows mimic field lines of a two-dimensional electrostatic system consisting of equal amounts of positive and negative charges. Bifurcations on a partially synchronized subspace are discussed as well. / Thesis (PhD) — Boston College, 2017. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

Communication delays are common in many complex systems. It has been shown that these delays cannot be neglected when they are long enough compared to other timescales in the system. In systems of coupled phase oscillators discrete delays in the coupling give rise to effects such as multistability of steady states. However, variability in the communication times inherent to many processes suggests that the description with discrete delays maybe insufficient to capture all effects of delays. An interesting example of the effects of communication delays is found during embryonic development of vertebrates. A clock based on biochemical reactions inside cells provides the periodicity for the successive and robust formation of somites, the embryonic precursors of vertebrae, ribs and some skeletal muscle. Experiments show that these cellular clocks communicate in order to synchronize their behavior. However, in cellular systems, fluctuations and stochastic processes introduce a variability in the communication times. Here we account for such variability by considering the effects of distributed delays. Our approach takes into account entire intervals of past states, and weights them according to a delay distribution. We find that the stability of the fully synchronized steady state with zero phase lag does not depend on the shape of the delay distribution, but the dynamics when responding to small perturbations about this steady state do. Depending on the mean of the delay distribution, a change in its shape can enhance or reduce the ability of these systems to respond to small perturbations about the phase-locked steady state, as compared to a discrete delay with a value equal to this mean. For synchronized steady states with non-zero phase lag we find that the stability of the steady state can be altered by changing the shape of the delay distribution. We conclude that the response to a perturbation in systems of phase oscillators coupled with discrete delays has a sharper functional dependence on the mean delay than in systems with distributed delays in the coupling. The strong dependence of the coupling on the mean delay time is partially averaged out by distributed delays that take into account intervals of the past.

This work is concerned with the spatio-temporal structures that emerge when non-identical, diffusively coupled oscillators synchronize. It contains analytical results and their confirmation through extensive computer simulations. We use the Kuramoto model which reduces general oscillatory systems to phase dynamics. The symmetry of the coupling plays an important role for the formation of patterns. We have studied the ordering influence of an asymmetry (non-isochronicity) in the phase coupling function on the phase profile in synchronization and the intricate interplay between this asymmetry and the frequency heterogeneity in the system. The thesis is divided into three main parts. Chapter 2 and 3 introduce the basic model of Kuramoto and conditions for stable synchronization. In Chapter 4 we characterize the phase profiles in synchronization for various special cases and in an exponential approximation of the phase coupling function, which allows for an analytical treatment. Finally, in the third part (Chapter 5) we study the influence of non-isochronicity on the synchronization frequency in continuous, reaction diffusion systems and discrete networks of oscillators. / Die vorliegende Arbeit beschäftigt sich in Theorie und Simulation mit den raum-zeitlichen Strukturen, die entstehen, wenn nicht-identische, diffusiv gekoppelte Oszillatoren synchronisieren. Wir greifen dabei auf die von Kuramoto hergeleiteten Phasengleichungen zurück. Eine entscheidene Rolle für die Musterbildung spielt die Symmetrie der Kopplung. Wir untersuchen den ordnenden Einfluss von Asymmetrie (Nichtisochronizität) in der Phasenkopplungsfunktion auf das Phasenprofil in Synchronisation und das Zusammenspiel zwischen dieser Asymmetrie und der Frequenzheterogenität im System. Die Arbeit gliedert sich in drei Hauptteile. Kapitel 2 und 3 beschäftigen sich mit den grundlegenden Gleichungen und den Bedingungen für stabile Synchronisation. Im Kapitel 4 charakterisieren wir die Phasenprofile in Synchronisation für verschiedene Spezialfälle sowie in der von uns eingeführten exponentiellen Approximation der Phasenkopplungsfunktion. Schliesslich untersuchen wir im dritten Teil (Kap.5) den Einfluss von Nichtisochronizität auf die Synchronisationsfrequenz in kontinuierlichen, oszillatorischen Reaktions-Diffusionssystemen und diskreten Netzwerken von Oszillatoren.

No description available.

Communication delays are common in many complex systems. It has been shown that these delays cannot be neglected when they are long enough compared to other timescales in the system. In systems of coupled phase oscillators discrete delays in the coupling give rise to effects such as multistability of steady states. However, variability in the communication times inherent to many processes suggests that the description with discrete delays maybe insufficient to capture all effects of delays. An interesting example of the effects of communication delays is found during embryonic development of vertebrates. A clock based on biochemical reactions inside cells provides the periodicity for the successive and robust formation of somites, the embryonic precursors of vertebrae, ribs and some skeletal muscle. Experiments show that these cellular clocks communicate in order to synchronize their behavior. However, in cellular systems, fluctuations and stochastic processes introduce a variability in the communication times. Here we account for such variability by considering the effects of distributed delays. Our approach takes into account entire intervals of past states, and weights them according to a delay distribution. We find that the stability of the fully synchronized steady state with zero phase lag does not depend on the shape of the delay distribution, but the dynamics when responding to small perturbations about this steady state do. Depending on the mean of the delay distribution, a change in its shape can enhance or reduce the ability of these systems to respond to small perturbations about the phase-locked steady state, as compared to a discrete delay with a value equal to this mean. For synchronized steady states with non-zero phase lag we find that the stability of the steady state can be altered by changing the shape of the delay distribution. We conclude that the response to a perturbation in systems of phase oscillators coupled with discrete delays has a sharper functional dependence on the mean delay than in systems with distributed delays in the coupling. The strong dependence of the coupling on the mean delay time is partially averaged out by distributed delays that take into account intervals of the past.:Abstract i Acknowledgement iii I. INTRODUCTION 1. Coupled Phase Oscillators Enter the Stage 5 1.1. Adjusting rhythms – synchronization . . . . . . . . . . . . . . . . . . 5 1.2. Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3. Reducing variables – phase models . . . . . . . . . . . . . . . . . . . . 9 1.4. The Kuramoto order parameter . . . . . . . . . . . . . . . . . . . . . . 10 1.5. Who talks to whom – coupling topologies . . . . . . . . . . . . . . . . 12 2. Coupled Phase Oscillators with Delay in the Coupling 15 2.1. Communication needs time – coupling delays . . . . . . . . . . . . . . 15 2.1.1. Discrete delays consider one past time . . . . . . . . . . . . . . 16 2.1.2. Distributed delays consider multiple past times . . . . . . . . 17 2.2. Coupled phase oscillators with discrete delay . . . . . . . . . . . . . . 18 2.2.1. Phase locked steady states with no phase lags . . . . . . . . . 18 2.2.2. m-twist solutions: phase-locked steady states with phase lags 21 3. The Vertebrate Segmentation Clock – What Provides the Rhythm? 25 3.1. The clock and wavefront mechanism . . . . . . . . . . . . . . . . . . . 26 3.2. Cyclic gene expression on the cellular and the tissue level . . . . . . 27 3.3. Coupling by Delta-Notch signalling . . . . . . . . . . . . . . . . . . . . 29 3.4. The Delayed Coupling Theory . . . . . . . . . . . . . . . . . . . . . . . 30 3.5. Discrete delay is an approximation – is it sufficient? . . . . . . . . . 32 4. Outline of the Thesis 33 II. DISTRIBUTED DELAYS 5. Setting the Stage for Distributed Delays 37 5.1. Model equations with distributed delays . . . . . . . . . . . . . . . . . 37 5.2. How we include distributed delays . . . . . . . . . . . . . . . . . . . . 38 5.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6. The Phase-Locked Steady State Solution 41 6.1. Global frequency of phase-locked steady states . . . . . . . . . . . . . 41 6.2. Linear stability of the steady state . . . . . . . . . . . . . . . . . . . . 42 6.3. Linear dynamics of the perturbation – the characteristic equation . 43 6.4. Summary and application to the Delayed Coupling Theory . . . . . . 50 7. Dynamics Close to the Phase-Locked Steady State 53 7.1. The response to small perturbations . . . . . . . . . . . . . . . . . . . 53 7.2. Relation between order parameter and perturbation modes . . . . . 54 7.3. Perturbation dynamics in mean-field coupled systems . . . . . . . . 56 7.4. Nearest neighbour coupling with periodic boundary conditions . . . 62 7.4.1. How variance and skewness influence synchrony dynamics . 73 7.4.2. The dependence of synchrony dynamics on the number of oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.5. Synchrony dynamics in systems with arbitrary coupling topologies . 88 7.6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 8. The m-twist Steady State Solution on a Ring 95 8.1. Global frequency of m-twist steady states . . . . . . . . . . . . . . . . 95 8.2. Linear stability of m-twist steady states . . . . . . . . . . . . . . . . . 97 8.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9. Dynamics Approaching the m-twist Steady States 105 9.1. Relation between order parameter and perturbation modes . . . . . 105 9.2. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 10.Conclusions and Outlook 111 vi III. APPENDICES A. 119 A.1. Distribution composed of two adjacent boxcar functions . . . . . . . 119 A.2. The gamma distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 124 A.3. Distribution composed of two Dirac delta peaks . . . . . . . . . . . . 125 A.4. Gerschgorin’s circle theorem . . . . . . . . . . . . . . . . . . . . . . . . 127 A.5. The Lambert W function . . . . . . . . . . . . . . . . . . . . . . . . . . 127 A.6. Roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 B. Simulation methods 129

Made available in DSpace on 2015-05-14T12:14:15Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 705049 bytes, checksum: 4794f7004746261efe9996d47989dd1f (MD5) Previous issue date: 2014-10-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this thesis, we presented models of two state stochastic systems which interact through a global coupling, in a way that each population unit contributes to the state transition rates of the other units. We presented two models of global coupling in which is possible to observe a phase transition of a regime with units equally distributed on the two states to a phase where there is an agglomeration of units in one of the states. In the first coupling model this transition occurs in a continuous way as we increase the coupling parameter. Through a mean field approximation we shown that this phase transition occurs due to a subcritical pitchfork bifurcation where one of the phases is associated to a monostable regime (units equally distributed in the two states) and the other phase to a symmetric bistable regime (majority of the units agglomerated in one of the states). On the other hand the other model presents a discontinuous phase transition as we increase the coupling parameter, the mean field approach shows that this phase transition occurs due a supercritical pitchfork bifurcation where we have a monostable regime and a tristable regime presenting symmetry in relation to the central potential well, as the coupling parameter is increased the central stability reduces while the two other states becomes more stable. It was shown that for both coupling models, when we have a finite number of oscillators the system presents a multiplicative noise structure. This noise structure turns the stable states obtained with the mean field approximation on metastable states, also the fluctuations due to a finite number of units breaks the symmetry in the multistable regimes, this symmetry break occurs due to the asymmetric intensity of the fluctuations. We also obtained a Fokker-Planck equation for this system and the probability distribution of the number of units in each state, from this distribution it was possible to build a phase diagram for the phase transition from themonostable regime to the regime that presents multistability. This transition is characterized in terms of the coupling parameter and the number of units in the system. / Nesta tese, apresentamos modelos de sistemas estocásticos de dois níveis que interagem através de um acoplamento global, de forma que o estado ocupado por cada unidade da população influi na taxa de transição de estado das demais. Apresentamos dois modelos de acoplamento global onde é possível observar uma transição de fase de um regime onde as unidades estão distribuídas igualmente entre os dois estados para uma fase onde há a aglomeração de unidades em um dos estados. Em um dos modelos de acoplamento essa transição ocorre de forma continua com o parâmetro de acoplamento. Através de uma aproximação de campo médio mostramos que essa transição de fase ocorre devido a uma bifurcação de forquilha sub-crítica onde uma das fases ´e associada a um regime monopolista (unidades igualmente divididas entre os dois estados) e a outra fase a um regime bioestavel simétrico (maior parte das unidades aglomeradas em um dos estados). J´a o outro modelo apresenta uma transic¸ ao de fase descont´ınua com o par ametro de acoplamento. A abordagem de campo m´edio revela que essa transic¸ ao de fase ocorre atrav´es de uma bifurcac¸ ao de forquilha supercr´ıtica onde temos um regimemonoest´avel e um regime triest´avel apresentando simetria com relac¸ ao ao poc¸o de potencial central e a medida que o par ametro de acoplamento ´e aumentado a estabilidade central diminui enquanto os outros dois estados se tornam mais est´aveis. Foi mostrado que para ambos os modelos de acoplamento, quando temos uma quantidade finita de osciladores o sistema apresenta uma estrutura de ru´ıdo multiplicativo. Essa estrutura de ru´ıdo torna os estados est´aveis obtidos com a aproximac¸ ao de campo m´edio em estados metaest´aveis. Tamb´em foi mostrado que as flutuac¸ oes devido a quantidade finita de unidades quebra a simetria nos regimes com multiestabilidade, essa quebra de simetria ocorre devido a assimetrias da intensidade das flutuac¸ oes. Obtemos tamb´em uma equac¸ ao de Fokker-Planck para esse sistema. A soluc¸ ao da equac¸ ao de Fokker-Planck nos d´a a distribuic¸ ao de probabilidade da quantidade de unidades em cada estado. Essa distribuic¸ ao torna poss´ıvel a construc¸ ao de um diagrama de fases para a transic¸ ao de fase dos regimes monoest ´aveis para os regimes que apresentam multiestabilidade. Essa transic¸ ao ´e caracterizada em termos do par ametro de acoplamento e da quantidade de unidades do sistema.

No description available.

No description available.