# Browsing by Subject "nonlinear dynamics"

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Item Open Access Convexity, Concavity, and Human Agency in Large-scale Coastline Evolution(2014) Ells, Kenneth DanielCoherent, large-scale shapes and patterns are evident in many landscapes, and evolve according to climate and hydrological forces. For large-scale, sandy coastlines, these shapes depend on wave climate forcing. The wave climate is influenced by storm patterns, which are expected to change with the warming climate, and the associated changes in coastline shape are likely to increase rates of shoreline change in many places. Humans have historically responded to coastline change by manipulating various coastal processes, consequently affecting long-term, large-scale coastline shape change. Especially in the context of changing climate forcing and increasing human presence on the coast, the interaction of the human and climate-driven components of large-scale coastline evolution are becoming increasingly intertwined.

This dissertation explores how climate shapes coastlines, and how the effects of humans altering the landscape interact with the effects of a changing climate. Because the coastline is a spatially extended, nonlinear system, I use a simple numerical modeling approach to gain a basic theoretical understanding of its dynamics, incorporating simplified representations of the human components of coastline change in a previously developed model for the physical system.

Chapter 1 addresses how local shoreline stabilization affects the large scale morphology of a cuspate-cape type of coastline, and associated large-scale patterns of shoreline change, in the context of changing wave climate, comparing two fundamentally different approaches to shoreline stabilization: beach nourishment (in which sediment is added to a coastline at a long-term rate that counteracts the background erosion), and hard structures (including seawalls and groynes). The results show that although both approaches have surprisingly long-range effects with spatially heterogeneous distributions, the pattern of shoreline changes attributable to a single local stabilization effort contrast greatly, with nourishment producing less erosion when the stabilization-related shoreline change is summed alongshore.

Chapter 2 presents new basic understanding of the dynamics that produce a contrasting coastline type: convex headland-spit systems. Results show that the coastline shapes and spatially-uniform erosion rates emerge from two way influences between the headland and spit components, and how these interactions are mediated by wave climate, and the alongshore scale of the system. Chapter 2 also shows that one type of wave-climate change (altering the proportion of `high-angle' waves) leads to changes in coastline shape, while another type (altering wave-climate asymmetry) tends to reorient a coastline while preserving its shape.

Chapter 3 builds on chapter 2, by adding the effects of human shoreline stabilization along such a convex coastline. Results show that in the context of increasing costs for stabilization, abandonment of shoreline stabilization at one location triggers a cascade of abandonments and associated coastline-shape changes, and that both the qualitative spatial patterns and alongshore speed of the propagating cascades depends on the relationship between patterns of economic heterogeneity and the asymmetry of the wave-climate change--although alterations to the proportion of high-angle waves in the climate only affects the time scales for coupled morphologic/economic cascades.

Item Open Access Network Dynamics and Systems Biology(2009) Norrell, Johannes AdrieThe physics of complex systems has grown considerably as a field in recent decades, largely due to improved computational technology and increased availability of systems level data. One area in which physics is of growing relevance is molecular biology. A new field, systems biology, investigates features of biological systems as a whole, a strategy of particular importance for understanding emergent properties that result from a complex network of interactions. Due to the complicated nature of the systems under study, the physics of complex systems has a significant role to play in elucidating the collective behavior.

In this dissertation, we explore three problems in the physics of complex systems, motivated in part by systems biology. The first of these concerns the applicability of Boolean models as an approximation of continuous systems. Studies of gene regulatory networks have employed both continuous and Boolean models to analyze the system dynamics, and the two have been found produce similar results in the cases analyzed. We ask whether or not Boolean models can generically reproduce the qualitative attractor dynamics of networks of continuously valued elements. Using a combination of analytical techniques and numerical simulations, we find that continuous networks exhibit two effects -- an asymmetry between on and off states, and a decaying memory of events in each element's inputs -- that are absent from synchronously updated Boolean models. We show that in simple loops these effects produce exactly the attractors that one would predict with an analysis of the stability of Boolean attractors, but in slightly more complicated topologies, they can destabilize solutions that are stable in the Boolean approximation, and can stabilize new attractors.

Second, we investigate ensembles of large, random networks. Of particular interest is the transition between ordered and disordered dynamics, which is well characterized in Boolean systems. Networks at the transition point, called critical, exhibit many of the features of regulatory networks, and recent studies suggest that some specific regulatory networks are indeed near-critical. We ask whether certain statistical measures of the ensemble behavior of large continuous networks are reproduced by Boolean models. We find that, in spite of the lack of correspondence between attractors observed in smaller systems, the statistical characterization given by the continuous and Boolean models show close agreement, and the transition between order and disorder known in Boolean systems can occur in continuous systems as well. One effect that is not present in Boolean systems, the failure of information to propagate down chains of elements of arbitrary length, is present in a class of continuous networks. In these systems, a modified Boolean theory that takes into account the collective effect of propagation failure on chains throughout the network gives a good description of the observed behavior. We find that propagation failure pushes the system toward greater order, resulting in a partial or complete suppression of the disordered phase.

Finally, we explore a dynamical process of direct biological relevance: asymmetric cell division in A. thaliana. The long term goal is to develop a model for the process that accurately accounts for both wild type and mutant behavior. To contribute to this endeavor, we use confocal microscopy to image roots in a SHORTROOT inducible mutant. We compute correlation functions between the locations of asymmetrically divided cells, and we construct stochastic models based on a few simple assumptions that accurately predict the non-zero correlations. Our result shows that intracellular processes alone cannot be responsible for the observed divisions, and that an intercell signaling mechanism could account for the measured correlations.

Item Open Access On Improving the Predictable Accuracy of Reduced-order Models for Fluid Flows(2020) Lee, Michael WilliamThe proper orthogonal decomposition (POD) is a classic method to construct empirical, linear modal bases which are optimal in a mean L2 sense. A subset of these modes can form the basis of a dynamical reduced-order model (ROM) of a physical system, including nonlinear, chaotic systems like fluid flows. While these POD-based ROMs can accurately simulate complex fluid dynamics, a priori model accuracy and stability estimates are unreliable. The work presented in this dissertation focuses on improving the predictability and accuracy of POD-based fluid ROMs. This is accomplished by ensuring several kinematically significant flow characteristics -- both at large scales and small -- are satisfied within the truncated bases. Several new methods of constructing and employing modal bases within this context are developed and tested. Reduced-order models of periodic flows are shown to be predictably accurate with high confidence; the predictable accuracy of quasi-periodic and chaotic fluid flow ROMs is increased significantly relative to existing approaches.

Item Open Access Out of Equilibrium Superconducting States in Graphene Multiterminal Josephson Junctions(2022) Arnault, Ethan GreggMultiterminal Josephson junctions have attracted attention, driven by the promise that they may host synthetic topological phase of matter and provide insight into Floquet states. Indeed, the added complexity of the additional contacts in multiterminal Josephson junctions greatly expands its parameter space, allowing for unexpected results. This work sheds light onto the out of equilibrium superconducting states that can exist within a ballistic multiterminal Josephson junction. The application of a microwave excitation produces unexpected fractional Shapiro steps, which are a consequence of the multiterminal circuit network. The application of a finite voltage reveals a robust cos 2φ supercurrent along the multiplet biasing condition nV1=-mV2. This supercurrent is found to be born from the RCSJ equations and has a stability condition analogous to Kapitza’s pendulum. Finally, the injection of hot carriers poisons supercurrent contributions from the Andreev spectrum, revealing a continuum mediated supercurrent.

Item Open Access Single-track Vehicle Dynamics and Stability(2014) Lipp, Genevieve MarieThis work is concerned with the dynamics and stability of nonlinear systems that roll in a single track, including holonomic and nonholonomic systems. First the classic case of Euler's disk is introduced as an example of a nonholnomic system in three dimensions, and the methodology for deriving equations of motion that is used throughout this work is demonstrated, including use of Lagrange's equations, accommodating constraints with both Lagrange multipliers and with Gauss's Principle.

Next, a disk in two dimensions with an eccentric center of mass is explored. The disk is assumed to roll on a cubic curve, creating the possibility of well-escape behavior, which is examined analytically and numerically, showing regions of multi-periodicity and chaos. This theoretical system is compared to an experiment designed

to demonstrate the same behavior.

The remainder of the present document is concerned with the stability of a bicycle, both on flat ground, and on a type of trainer known as "rollers." The equations of motion are derived using Lagrange's equations with nonholonomic constraints, then the equations are linearized about a constant forward velocity, and a straight path, yielding a two degree of freedom system for the roll and steer angles. Stability is then determined for a variety of different parameters, exploring the roll of bicycle geometry and rider position, along with the effect of adding a steering torque, taking the form of different control laws.

Finally, the system is adapted to that of a bicycle on rollers, and the related equations of motion are derived and linearized. Notable differences with the classic bicycle case are detailed, a new eigenvalue behavior is presented, and configurations for optimal drum spacing are recommended.

Item Open Access The Lid-Driven Cavity's Many Bifurcations - A Study of How and Where They Occur(2017) Lee, MichaelComputational simulations of a two-dimensional incompressible regularized lid-driven cavity were performed and analyzed to identify the dynamic behavior of the flow through multiple bifurcations which ultimately result in chaotic flow. Pseudo-spectral numerical simulations were performed at Reynolds numbers from 1,000 to 25,000. Traditional as well as novel methods were implemented to characterize the system's behavior. The first critical Reynolds number, near 10,250, is found in agreement with existing literature. An additional bifurcation is observed near a Reynolds number of 15,500. The largest Lyapunov exponent was studied as a potential perspective on chaos characterization but its accurate computation was found to be prohibitive. Phase space and power spectrum analyses yielded comparable conclusions about the flow's progression to chaos. The flow's transition from quasi-periodicity to chaos between Reynolds numbers of 18,000 and 23,000 was observed to be gradual and of the form of a toroidal bifurcation. The concepts of frequency shredding and power capacity are introduced which, paired with an existing understanding of frequency entrainment, can help explain the system's progression through quasi-periodicity to chaos.

Item Open Access Wave Propagation in Nonlinear Systems of Coupled Oscillators(2014) Bernard, Brian PatrickMechanical oscillators form the primary structure of a wide variety of devices including energy harvesters and vibration absorbers, and also have parallel systems in electrical fields for signal processing. In the area of wave propagation, recent study in periodic chains have focused on active tuning methods to control bandgap regions, bands in the frequency response in which no propagating wave modes exist. In energy harvesting, several coupled systems have been proposed to enhance the peak power or bandwidth of a single harvester through arrays or dynamic magnification. Though there are applications in several fields, the work in this dissertation can all fit into the category of coupled non-linear oscillators. In each sub-field, this study demonstrates means to advance state of the art techniques by adding nonlinearity to a coupled system of linear oscillators, or by adding a coupled device to a nonlinear oscillator.

The first part of this dissertation develops the analytical methods for studying wave propagation in nonlinear systems. A framework for studying rotational systems is presented and used to design an testbed for wave propagation experiments using a chain of axially aligned pendulums. Standard analytical methods are also adapted to allow uncertainty analysis techniques to provide insight into the relative impact of variations in design parameters. Most analytical insight in these systems is derived from a linearlized model and assumes low amplitude oscillations. Additional study on the nonlinear system is performed to analyze the types of deviations from this behavior that would be expected as amplitudes increase and nonlinear effects become more prominent.

The second part of this dissertation describes and demonstrates the first means of passive control of bandgap regions in a periodic structure. By imposing an asymmetrical bistability to an oscillator in each unit cell, it is analytically shown that each potential well has different wave propagation behaviors. Experimental demonstrations are also provided to confirm the simulated results.

The final section performs analytical and numerical analysis of a new system design to improve the performance of a nonlinear energy harvester by adding an excited dynamic magnifier. It is shown that this addition results in higher peak power and wider bandwidth than the uncoupled harvester. Unlike standard dynamic magnifiers, this performance does not come at the expense of power efficiency, and unlike harvester arrays, does not require the added cost of multiple energy harvesters.