Effect of Distributed Delays in Systems of Coupled Phase Oscillators

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 12

Gaussian approximations for stochastic systems with delay: chemical Langevin equation and application to a Brusselator system

We present a heuristic derivation of Gaussian approximations for stochastic chemical reaction systems with distributed delay. In particular we derive the corresponding chemical Langevin equation. Due to the non-Markovian character of the underlying dynamics these equations are integro-differential equations, and the noise in the Gaussian approximation is coloured. Following on from the chemical Langevin equation a further reduction leads to the linear-noise approximation. We apply the formalism to a delay variant of the celebrated Brusselator model, and show how it can be used to characterise noise-driven quasi-cycles, as well as noise-triggered spiking. We find surprisingly intricate dependence of the typical frequency of quasi-cycles on the delay period.Comment: 14 pages, 9 figure

Probability of local bifurcation type from a fixed point: A random matrix perspective

Results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems (real, random matrices), time-delay dynamical systems (companion matrices), and a set of mappings known for their properties as universal approximators (neural networks). The eigenvalue spectra is considered both numerically and analytically using previous work of Edelman et. al. Based upon the numerical evidence, various conjectures are presented. The conclusion is that in many circumstances, most bifurcations from fixed points of large dynamical systems will be due to complex eigenvalues. Nevertheless, surprising situations are presented for which the aforementioned conclusion is not general, e.g. real random matrices with Gaussian elements with a large positive mean and finite variance.Comment: 21 pages, 19 figure

Derivation of Delay Equation Climate Models Using the Mori-Zwanzig Formalism

Models incorporating delay have been frequently used to understand climate variability phenomena, but often the delay is introduced through an ad-hoc physical reasoning, such as the propagation time of waves. In this paper, the Mori-Zwanzig formalism is introduced as a way to systematically derive delay models from systems of partial differential equations and hence provides a better justification for using these delay-type models. The Mori-Zwanzig technique gives a formal rewriting of the system using a projection onto a set of resolved variables, where the rewritten system contains a memory term. The computation of this memory term requires solving the orthogonal dynamics equation, which represents the unresolved dynamics. For nonlinear systems, it is often not possible to obtain an analytical solution to the orthogonal dynamics and an approximate solution needs to be found. Here, we demonstrate the Mori-Zwanzig technique for a two-strip model of the El Nino Southern Oscillation (ENSO) and explore methods to solve the orthogonal dynamics. The resulting nonlinear delay model contains an additional term compared to previously proposed ad-hoc conceptual models. This new term leads to a larger ENSO period, which is closer to that seen in observations.Comment: Submitted to Proceedings of the Royal Society A, 25 pages, 10 figure

Capturing pattern bi-stability dynamics in delay-coupled swarms

Swarms of large numbers of agents appear in many biological and engineering fields. Dynamic bi-stability of co-existing spatio-temporal patterns has been observed in many models of large population swarms. However, many reduced models for analysis, such as mean-field (MF), do not capture the bifurcation structure of bi-stable behavior. Here, we develop a new model for the dynamics of a large population swarm with delayed coupling. The additional physics predicts how individual particle dynamics affects the motion of the entire swarm. Specifically, (1) we correct the center of mass propulsion physics accounting for the particles velocity distribution; (2) we show that the model we develop is able to capture the pattern bi-stability displayed by the full swarm model.Comment: 6 pages 4 figure

Investigation of the complex dynamics and regime control in Pierce diode with the delay feedback

In this paper the dynamics of Pierce diode with overcritical current under the influence of delay feedback is investigated. The system without feedback demonstrates complex behaviour including chaotic regimes. The possibility of oscillation regime control depending on the delay feedback parameter values is shown. Also the paper describes construction of a finite-dimensional model of electron beam behaviour, which is based on the Galerkin approximation by linear modes expansion. The dynamics of the model is close to the one given by the distributed model.Comment: 18 pages, 6 figures, published in Int. J. Electronics. 91, 1 (2004) 1-1

Galerkin approximations for the optimal control of nonlinear delay differential equations

Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE's solution. Optimal controls computed from the Pontryagin's maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation.Comment: 29 pages. This is a sequel of the arXiv preprint arXiv:1704.0042

Networked PID control design : a pseudo-probabilistic robust approach

Networked Control Systems (NCS) are feedback/feed-forward control systems where control components (sensors, actuators and controllers) are distributed across a common communication network. In NCS, there exist network-induced random delays in each channel. This paper proposes a method to compensate the effects of these delays for the design and tuning of PID controllers. The control design is formulated as a constrained optimization problem and the controller stability and robustness criteria are incorporated as design constraints. The design is based on a polytopic description of the system using a Poisson pdf distribution of the delay. Simulation results are presented to demonstrate the performance of the proposed method