Theses and Dissertations from UMD

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New submissions to the thesis/dissertation collections are added automatically as they are received from the Graduate School. Currently, the Graduate School deposits all theses and dissertations from a given semester after the official graduation date. This means that there may be up to a 4 month delay in the appearance of a give thesis/dissertation in DRUM

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Subject: search.filters.subject.Nonlinear dynamics

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Now showing 1 - 5 of 5
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    Coupled Oscillator Arrays: Dynamics and Influence of Noise
    (2021) Alofi, Abdulrahman Mohammed; Balachandran, Balakumar; Mechanical Engineering; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    Coupled oscillator arrays can be used to model several natural systems and engineering systems including mechanical systems. In this dissertation work, the influence of noise on the dynamics of coupled mono-stable oscillators arrays is investigated by using numerical and experimental methods. This work is an extension of recent efforts, including those at the University of Maryland, on the use of noise to alter a nonlinear system's response. A chain of coupled oscillators is of interest for this work. This dissertation research is guided by the following questions: i) how can noise be used to create or quench spatial energy localization in a system of coupled, nonlinear oscillators? and ii) how can noise be used to move the energy localization from one oscillator to another? The coupled oscillator systems of interest were harmonically excited and found experimentally and numerically to have a multi-stability region (MR) in the respective frequency response curves. Relative to this region, it has been found that the influence of noise depends highly on the excitation frequency location in the MR. Near either end of the MR, the oscillator responses were found to be sensitive to noise addition in the input and it was observed that the change in system dynamics through movement amongst the stable branches in the deterministic system could be anticipated from the corresponding frequency response curves. The system response is found to be robust to the influence of noise as the excitation frequency is shifted toward the middle of the MR. Also, the effects of noise on different response modes of the coupled oscillators arrays were investigated. A method for predicting the behavior is based on so-called basins of attractions of high dimensional systems. Through the findings of this work, many unique noise influenced phenomena are found, including spatial movement of an energy localization to a neighboring oscillator, response movement gradually up the energy branches, and generation of energy cascades from a localized mode.
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    Extensions of the Kuramoto model: from spiking neurons to swarming drones
    (2020) Chandra, Sarthak; Girvan, Michelle; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    The Kuramoto model (KM) was initially proposed by Yoshiki Kuramoto in 1975 to model the dynamics of large populations of weakly coupled phase oscillators. Since then, the KM has proved to be a paradigmatic model, demonstrating dynamics that are complex enough to model a wide variety of nontrivial phenomena while remaining simple enough for detailed mathematical analyses. However, as a result of the mathematical simplifications in the construction of the model, the utility of the KM is somewhat restricted in its usual form. In this thesis we discuss extensions of the KM that allow it to be utilized in a wide variety of physical and biological problems. First, we discuss an extension of the KM that describes the dynamics of theta neurons, i.e., quadratic-integrate-and-fire neurons. In particular, we study networks of such neurons and derive a mean-field description of the collective neuronal dynamics and the effects of different network topologies on these dynamics. This mean-field description is achieved via an analytic dimensionality reduction of the network dynamics that allows for an efficient characterization of the system attractors and their dependence not only on the degree distribution but also on the degree correlations. Then, motivated by applications of the KM to the alignment of members in a two-dimensional swarm, we construct a Generalized Kuramoto Model (GKM) that extends the KM to arbitrary dimensions. Like the KM, the GKM in even dimensions continues to demonstrate a transition to coherence at a positive critical coupling strength. However, in odd dimensions the transition to coherence occurs discontinuously as the coupling strength is increased through 0. In contrast to the unique stable incoherent equilibrium for the KM, we find that for even dimensions larger than 2 the GKM displays a continuum of different possible pretransition incoherent equilibria, each with distinct stability properties, leading to a novel phenomenon, which we call `Instability-Mediated Resetting.' To aid the analysis of such systems, we construct an exact dimensionality reduction technique with applicability to not only the GKM, but also other similar systems with high-dimensional agents beyond the GKM.
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    Experimental study of entropy generation and synchronization in networks of coupled dynamical systems
    (2015) Hagerstrom, Aaron Morgan; Roy, Rajarshi; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    This thesis describes work in two areas: unpredictability in a system with both single photon detection and chaos, and synchronization in networks of coupled systems. The unpredictability of physical systems can depend on the scale at which they are observed. For example, single photons incident on a detector arrive at random times, but slow intensity variations can be observed by counting many photons over large time windows. We describe an experiment in which a weak optical signal is modulated by feedback from a single-photon detector. We demonstrate that at low photon rates, the photon arrivals are described by Poisson noise, while at high photon rates, the intensity of the light shows a deterministic chaotic modulation. Furthermore, we show that measurements of the entropy rate can resolve both noise and chaos, and describe the relevance of this observation to random number generation. We also describe an experimental system that allows for the study of large networks of coupled dynamical systems. This system uses a spatial light modulator (SLM) and a camera in a feedback configuration, and has an optical nonlinearity due to the relationship between a spatially-dependent phase shift applied by the modulator, and the intensity measured by the camera. This system is a powerful platform for studying synchronization in large networks of coupled systems, because it can iterate many (~1000) maps in parallel, and can be configured with any desired coupling matrix. We use this system to study cluster synchronization. It is often observed that networks synchronize in clusters. We show that the clusters that form can be predicted based on the symmetries of the network, and their stability can be analyzed. This analysis is supported by experimental measurements. The SLM feedback system can also exhibit chimera states. These states were first seen in models of large populations of coupled phase oscillators. Their defining feature is the coexistence of large populations of synchronized and unsynchronized oscillators in spite of symmetrical coupling configurations and homogeneous oscillators. Our SLM feedback system provided one of the first two experimental examples of chimera states. We observed chimeras in coupled map lattices. These states evolve in discrete time, and in this context, ``coherence'' and ``incoherence'' refer to smooth and discontinuous regions of a spatial pattern.
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    Phase Transitions in Complex Network Dynamics
    (2014) Squires, Shane Anthony; Girvan, Michelle; Ott, Edward; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    Two phase transitions in complex networks are analyzed. The first of these is a percolation transition, in which the network develops a macroscopic connected component as edges are added to it. Recent work has shown that if edges are added "competitively" to an undirected network, the onset of percolation is abrupt or "explosive." A new variant of explosive percolation is introduced here for directed networks, whose critical behavior is explored using numerical simulations and finite-size scaling theory. This process is also characterized by a very rapid percolation transition, but it is not as sudden as in undirected networks. The second phase transition considered here is the emergence of instability in Boolean networks, a class of dynamical systems that are widely used to model gene regulation. The dynamics, which are determined by the network topology and a set of update rules, may be either stable or unstable, meaning that small perturbations to the state of the network either die out or grow to become macroscopic. Here, this transition is analytically mapped onto a well-studied percolation problem, which can be used to predict the average steady-state distance between perturbed and unperturbed trajectories. This map applies to specific Boolean networks with few restrictions on network topology, but can only be applied to two commonly used types of update rules. Finally, a method is introduced for predicting the stability of Boolean networks with a much broader range of update rules. The network is assumed to have a given complex topology, subject only to a locally tree-like condition, and the update rules may be correlated with topological features of the network. While past work has addressed the separate effects of topology and update rules on stability, the present results are the first widely applicable approach to studying how these effects interact. Numerical simulations agree with the theory and show that such correlations between topology and update rules can have profound effects on the qualitative behavior of these systems.
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    The Effects of Coupling Delay and Amplitude / Phase Interaction on Large Coupled Oscillator Networks
    (2012) Lee, Wai Shing; Ott, Edward; Antonsen, Thomas M.; Electrical Engineering; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    The interaction of many coupled dynamical units is a theme across many scientific disciplines. A useful framework for beginning to understanding such phenomena is the coupled oscillator network description. In this dissertation, we study a few problems related to this. The first part of the dissertation studies generic effects of heterogeneous interaction delays on the dynamics of large systems of coupled oscillators. Here, we modify the Kuramoto model (phase oscillator model) to incorporate a distribution of interaction delays. Corresponding to the continuum limit, we focus on the reduced dynamics on an invariant manifold of the original system, and derive governing equations for the system, which we use to study stability of the incoherent state and the dynamical transitional behavior from stable incoherent states to stable coherent states. We find that spread in the distribution function of delays can greatly alter the system dynamics. The second part of this dissertation is a sequel to the first part. Here, we consider systems of many spatially distributed phase oscillators that interact with their neighbors, and each oscillator can have a different natural frequency, and a different response time to the signals it receives from other oscillators in its neighborhood. By first reducing the microscopic dynamics to a macroscopic partial-differential-equation description, we then numerically find that finite oscillator response time leads to many interesting spatio-temporal dynamical behaviors, and we study interactions and evolutionary behaviors of these spatio-temporal patterns. The last part of this dissertation addresses the behavior of large systems of heterogeneous, globally coupled oscillators each of which is described by the generic Landau-Stuart equation, which incorporates both phase and amplitude dynamics. Our first goal is to investigate the effect of a spread in the amplitude growth parameter of the oscillators and that of a homogeneous nonlinear frequency shift. Both of these effects are of potential relevance to recently reported experiments. Our second goal is to gain further understanding of the observation that, at large coupling strength, a simple constant-amplitude sinusoidal oscillation is always a solution for the dynamics of the global order parameter when the system has constant nonlinear characteristics.