Browsing by Subject "Dynamical Systems"

Now showing 1 - 10 of 10
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    Conceptual Modeling of Adaptive Therapy Dosing for Chemotherapeutic Administration in Cancer Allows for the Direct Comparison of Continuous and Adaptive Dosing Regimes
    (2022-05) McGehee, Cordelia
    Adaptive therapy of cytotoxic (cell killing) chemotherapy has been proposed as a method to prolong progression-free survival in certain cancers when underlying cell-cell competition between sensitive and resistant cancer cells is present. Traditionally, cytotoxic chemotherapy dosing is administered at the maximal tolerated dose with the goal of rapidly shrinking tumor growth. In the case of a tumor where underlying intratumoral cell-cell competition between a drug sensitive and drug resistant population leads to competition for resources, it is hypothesized that maximally killing the sensitive cell population allows for competitive release of the resistant cell population and outgrowth of a chemotherapy resistant tumor. In adaptive therapy, chemotherapy is administered when a tumor reaches a certain upper threshold and then is discontinued when the tumor shrinks to a specified lower threshold. The purpose of this strategy is to use the sensitive cell population to inhibit the growth of the resistant cell population and increase the length of time to competitive release and outgrowth of the resistant cell population. In this thesis, a modified Lotka-Volterra competition model is explored across competition parameters in order to analytically address 1) the optimality of continuous fixed dose versus adaptive dosing schedules and 2) the role of drug dose and mechanism of action in the choice of dosing regime. Using this model, several novel results are shown. First, for certain parameters, complete tumor eradication can be achieved in the presence of a resistant subpopulation under adaptive cytotoxic or continuous antiproliferative (decreasing growth rate) dosing schedules. Second, in this parameter space, fixed dose antiproliferative dosing schedules are more robust than cytotoxic adaptive regimes to uncertainty in initial conditions. Third, in parameter spaces where eradication of the resistant cell population is not feasible, both fixed dose antiproliferative schedules and cytotoxic adaptive therapy schedules may result in delayed resistant cell outgrowth over maximum tolerated dose and are comparable in their benefits. Overall, these results indicate that both antiproliferative continuous fixed dose therapy and cytotoxic adap-tive therapy can be used for tumor management in the case of underlying intratumoral competition between chemotherapy sensitive and chemotherapy resistant cells.
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    Conley Index Theory for Multivalued Dynamical Systems and Piecewise-Continuous Differential Equations
    (2021-05) Thieme, Cameron
    Modern dynamical systems–particularly heuristic models–often take the form of piecewise- continuous differential equations. In order to better understand the behavior of these systems we generalize aspects of Conley index theory to this setting. Because nons- mooth models do not generally have unique solutions, this process involves organizing the solution set of the piecewise-continuous equation into a set-valued object called a multiflow. We prove several properties of this object, providing us with a foundation for extending Conley’s techniques. This framework allows us to define isolating neigh- borhoods and demonstrate that they are stable under perturbation. We also provide an attractor-repeller pair decomposition of compact invariant sets for multiflows which helps us to understand the limiting behavior of solutions in such sets. This decomposi- tion is shown to continue under perturbation. Because we assume very little structure in proving these results we are able to connect them to many different existing formu- lations of the Conley index for multivalued dynamical systems. Therefore we are able to identify isolating neighborhoods in a large class of differential inclusions, decompose the associated isolated invariant sets into an attractor-repeller pair, and provide the index of of the original isolated invariant set, the attractor, and the repeller; all of this information is stable under small perturbations. This process is carried out on a piecewise-continuous model from oceanography as an example.
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    Introducing the Picard Method for Approximating Solutions of Differential Equations with Neural Networks
    (2024-05) Frazier, Ty
    For decades, differential equations have been used to model various problems in the natural sciences and engineering. However, ordinary differential equations (ODEs) are usually not analytically solvable, so many numerical approaches have been developed to produce approximate solutions. More recently, it has been proposed that neural networks can learn solutions of ODEs and thus provide faster and more accurate numerical approximations. Here, we propose a novel approach of having neural networks learn solutions to ODEs via the Picard formulation. We show, through examples, that this approach produces approximations that are at least as reliable as earlier approaches.
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    An Optimal Control Perspective on Externally Induced Tipping of Rigidly Shifting Systems
    (2024-02) Zhang, Grace
    The standard setting for rate-induced tipping involves fixing a particular parameterized family of smooth forcing functions and identifying a critical value of the rate parameter. In contrast, we consider a broad collection of all possible forcing functions, continuous but not necessarily smooth, and seek a general property possessed by those which effect tipping behavior. We focus on rigidly shifting asymptotically autonomous scalar systems x ̇ = f(x + λ(t)) and identify a nonsmooth choice of forcing function λ(t) which is an optimal tipping strategy in the sense that it utilizes the least possible maximum speed. Under a co-moving change of coordinates, the problem of finding this optimal λ(t) becomes dual to the problem of finding an additive control function that achieves basin escape with minimum fuel. We show the optimizer is a bang-bang control. The outcome is a lower bound on the speed |λ ̇(t)| that must be attained at least once in order to induce tipping. Its value depends on the total arclength ∞s∞|λ ̇(t)| dt of forcing, and may be interpreted as a safe threshold rate associated to each given arclength, such that if the speed of forcing remains everywhere slower than this, tipping cannot occur. The bound is tight in the sense that there exists a forcing function which induces tipping, possesses the required arclength, and never exceeds the threshold speed. Further, the threshold speed is a strictly decreasing function of arclength, thus capturing the abstract trade off between how fast and how far of a minimal disturbance characterizes tipping. While our results assume a scalar setting, the prospect of generalizing to n-dimensions is discussed and formulated as a conjecture. The control-theoretic construction used in deriving the above inspires a new theory of resilience, which is a slight modification of the intensity kof attraction framework of McGehee and Meyer. This is a family of resilience values parameterized by a number representing the allowable L1 norm of perturbations; in the limit as the integral-constraining parameter grows unbounded, these values approach the intensity of attraction. This integral-constrained intensity of attraction has the advantage of increased descriptiveness under scenarios where limited total resources are available for perturbing the system. We suggest it to be the natural choice for quantifying the resilience of a rigidly shifting system to externally-forced tipping.
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    Pattern formation in the wake of external mechanisms
    (2016-06) Goh, Ryan
    Pattern formation in nature has intrigued humans for centuries, if not millennia. In the past few decades researchers have become interested in harnessing these processes to engineer and manufacture self-organized and self-regulated devices at various length scales. Since many natural pattern forming processes nucleate or grow from a homogeneous unstable state, they typically create defects, caused by thermal and other inherent sources of noise, which can hamper effectiveness in applications. One successful experimental method for controlling the pattern forming process is to use an external mechanism which moves through a system, transforming it from a stable state to an unstable state from which the pattern forming dynamics can take hold. In this thesis, we rigorously study partial differential equations which model how such triggering mechanisms can select and control patterns. We first use dynamical systems techniques to study the case where a spatial trigger perturbs a pattern forming freely invading front in a scalar partial differential equation. We study such perturbations for the two generic types of scalar invasion fronts, known as pulled and pushed fronts, which roughly correspond to fronts which invade either through a linear or nonlinear mechanism. Our results give the existence of perturbed fronts and provide expansions in the speed of the triggering mechanism for the wavenumber perturbation of the pattern formed. With the hope of moving towards the more complicated geometries which can arise in two spatial dimensions, where many dynamical systems methods cannot be readily applied, we also develop a functional analytic method for the study of Hopf bifurcation in the presence of continuous spectrum. Our method, while still giving computable information about the bifurcating solution, is more direct than previously proposed methods. We develop this method in the context of a triggered Cahn-Hilliard equation, in one spatial dimension, which has been used to study many triggered pattern forming systems. Furthermore, we use these abstract results to characterize an explicit example and also use our method to give a simplified proof of the bifurcation of oscillatory shock solutions in viscous conservation laws.
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    Patterns Selected by Spatial Inhomogeneity
    (2019-05) Weinburd, Jasper
    This thesis studies patterns that form in environments with sharp spatial variation. In a uniform environment, spots or stripes typically form with a self-consistent width. This width is taken from an interval around a characteristic value determined by the system. With a dramatic spatial variation, our environments only allow patterns in half the spatial region. This sets up a region of patterns directly adjacent to an area where patterns are suppressed. We show that this environmental inhomogeneity significantly restricts the widths of patterns that may occur in a given system. That is, the length of the interval around the characteristic value is significantly reduced. We examine this phenomenon using a universal partial differential equation model. Reduction techniques from dynamical systems simplify our study to the behavior in a normal form equation (amplitude equation). A difficulty arrises at the location of the discontinuous inhomogeneity; results in the normal form equations on the left and right cannot be directly compared. We construct a transformation of variables that bridges this jump and allows a heteroclinic gluing argument. The explicit form of this transformation determines the widths of patterns that can occur in the inhomogeneous environment.
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    Predicting the first ice-free summer in the Arctic using a daily scale Budyko-type Energy Balance Model
    (2020-07) Baek, Somyi
    Arctic ice extent serves as one of the best proxy indicators of climate change. The annual Arctic ice extent average has declined at an alarming and continual rate for the last 50 years. The Arctic ice is without a doubt becoming vulnerable and fragile in both thickness and resilience, and scientists around the world worry about witnessing an ice-free summer in the imminent future. Most climate scientists predict that the first ice-free Arctic summer will happen before or around 2050, including the researchers of the Intergovernmental Panel on Climate Change (IPCC). The majority of these forecasts are made with ocean-atmosphere general circulation models (GCMs), highly complex in their input and output, which often renders traditional mathematical analysis difficult. This thesis aims to develop a mathematically tractable and analyzable model of the Arctic sea ice extent using the structure of Budyko-type energy balance models (EBM) and a previous master's student Wen Xing's model, with the goal of making a deterministic prediction for the first ice-free Arctic summer. The new model, which will be called the Daily Budyko model, extends the original annual average Budyko-type EBM to the daily scale and recalibrates model parameters for the domain of the northern polar region. In addition, new model equations are constructed using the main idea of proportional growth in Xing's model. The Daily Budyko model predicts September of 2051 for the first ice-free Arctic summer, which is in line with the IPCC's forecast of around 2050.
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    Proceedings of the 22nd International Symposium on Mathematical Theory of Networks and Systems
    (2016-07)
    Conference Proceedings and Presentations for the 22nd International Symposium on Mathematical Theory of Networks and Systems
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    Smooth and Nonsmooth Bifurcations in Welander's Ocean Convection Model
    (2016-05) Leifeld, Juliann
    Welander's model is a conceptual ocean convection model, that describes ocean convection with a few, simple dynamical equations. Welander's goal in formulating his model was to show that internally driven oscillations could be caused solely by switching between strongly convective and nonconvective states. Because of the conceptual importance of the switching mechanism, Welander created two versions of the model, one with a smooth transition between convective states, and one with an abrupt nonsmooth transition. He was able to numerically find a periodic orbit in both versions of the model. The climactic import of the model is in the idea that oscillations can be internally driven, but the model also has interesting mathematical import. Welander's implicit assumption that the nonsmooth model is easier to analyze is mathematically suspect, and a rigorous comparison between the smooth and nonsmooth models is not immediately clear. In this work, we introduce the model with scientific context, and complete a rigorous nonsmooth analysis of the model. We find one known nonsmooth bifurcation analogous to a supercritical Hopf bifurcation, but we also find a bifurcation that has not been previously described. Finally, we compare the smooth and nonsmooth models, paying close attention to the dangers inherent in such a comparison.
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    What sustains behavioral changes? A dynamical systems approach to improving theories of change in physical exercise
    (2019-08) Lenne, Richard
    Health behaviors, such as physical exercise, are associated with chronic diseases that top the list of all-cause mortality. Yet, the most healthful lifestyle changes people can (and often want to) make, also tend to be the most challenging to sustain. This dissertation explores how modeling behavior as a dynamical system could improve understanding of psychological processes that sustain behavioral changes. I focus on two classes of processes—motivational and habitual—that may be most pertinent to sustaining changes in physical exercise. A model based on prior theorizing is constructed and simulated (Study 1), and observational data are analyzed (Study 2). Intensive longitudinal data are collected from healthy US-based Fitbit users who recently initiated an increase in exercise. Participants are prospectively observed for two months during which measures of motivation and habit are assessed three days per week, and exercise-as-usual is passively tracked via Fitbit. I find that within-person increases in the automaticity with which exercise is performed in a given week is associated with increases in time spent exercising. Furthermore, differences in the trajectory of automaticity and satisfaction with exercise over time may differentiate those who successfully maintain increases in exercise and those who do not. Results are placed in the context of contemporary theories of behavior change maintenance and suggestions for improvement are forwarded.