Non-linear Dynamics Commons^™

A Differential Equation–Based Epidemiological Model Of Post-Operative Chronic Pain In Scoliosis Patients With Data-Driven Analysis, Paige Zhu, Padmanabhan Seshaiyer

CODEE Journal

Chronic post-surgical pain (CPSP) is a common and often overlooked complication following surgical correction of idiopathic scoliosis, impacting long-term patient wellbeing despite improvements in surgical outcomes. This project introduces a novel epidemiological framework to model the progression of CPSP using a compartmental structure. By applying a coupled system of nonlinear differential equations, we simulate pain trajectories over time and assess the effectiveness of surgical interventions. The model is implemented for a single-cohort population and extended to a two-cohort design to compare outcomes between Posterior Spinal Fusion (PSIF) and Vertebral Body Tethering (VBT) procedures. Further stratification by patient age enables us …

A Professional Development Course On Data-Driven Dynamical Systems At A Primarily Undergraduate Institution: Part A - Scientific Content, Alessandro M. Selvitella, Jeffrey R. Anderson

CODEE Journal

In the age of data-driven decision making, ordinary differential equations (ODEs) remain a powerful and interpretable framework for modeling dynamic processes, especially when integrated with modern tools from statistical learning and data-driven dynamical systems. Yet, general undergraduate and graduate curricula do not typically address key opportunities in data-driven dynamical systems.

This first paper in a series focuses on the mathematical and methodological core of a professional development course first developed in the academic year 2025-2026 at a Primarily Undergraduate Institution, Purdue University Fort Wayne. The curriculum developed in this course emphasized how regression, regularization, and sparse identification can be used …

Accuracy Of Parameter Estimation For A Simple Gene Regulatory Network Model Is Sensitive To Network Motif, Number Of Parameters Estimated, And Magnitude And Direction Of Regulatory Relationships, Nikki C. Chun, Kam Dahlquist

Honors Thesis

A gene regulatory network (GRN) is a set of transcription factors that regulate the expression of genes encoding other transcription factors. The dynamics of a GRN explain how gene expression changes over time. GRNmap is a MATLAB software package that uses ordinary differential equations to model dynamics of small-scale GRNs. We used the program to estimate production rates, expression thresholds, and regulatory weights for each transcription factor in three related literature-derived GRNs based on yeast cold shock microarray data previously collected in the Dahlquist Lab. We noticed large differences in estimated weight values when 1-2% of the expression values were …

Dynamic Homeostasis In Relaxation And Bursting Oscillations, Christopher J. Ryzowicz

Biology and Medicine Through Mathematics Conference

No abstract provided.

Modeling, Control Analysis, And Parameter Estimation Of Epidemic Dynamics Using The Unscented Kalman Filter, Muhammad Imran, Saira Batool, Brett Mckinney

Biology and Medicine Through Mathematics Conference

No abstract provided.

A Two-Phase Perspective On A Related-Rates Paradox, Charlie Vazquez Acosta

Honors Capstones

Related-rates problems are a standard topic in first-year calculus and have appeared in textbooks for over 150 years. These problems are used to teach implicit differentiation and the relationship between changing quantities. Common examples include the falling ladder, the fishing bobber, the melting snowball, and the leaking conical tank. In each of these problems, a quantity is changing at a constant rate, and students are asked to find the rate of change of another related quantity. While the computations themselves are usually straightforward, the standard models lead to unrealistic results near the end of the motion. For example, the falling …

Phenological Overlap In Obligate Plant-Pollinator Mutualism, Austin J. Carlson

2026 Spring Honors Capstones Projects

Plant-pollinator mutualisms require temporal overlap between flowering and pollinator activity, so climate-driven timing shifts can weaken the interaction and, in severe cases, destabilize the system. This work investigates how reduced overlap affects persistence in an obligate plant-pollinator pair using a coupled differential equation model in which a phenological overlap factor scales the saturating mutualistic benefit. Simplification with a constant overlap enables closed-form equilibrium and stability analysis, revealing that below a critical overlap threshold, coexistence is no longer maintained. Rescaling reduces the parameter space from ten quantities to seven dimensionless groups, and sensitivity analysis identifies the degree of species dependence and …

Safe Control Design For Quadruped Locomotion In Unstructured Environments Using Linear Transfer Operators, Sriram Sundar Krishnamoorthy Shankara Narayanan

All Dissertations

Deploying quadruped robots in unstructured, obstacle-rich environments requires control and planning methods that remain safe and reliable despite complex terrain geometry, limited sensing, and inevitable modeling errors. This thesis develops operator-theoretic tools for safe control design of robotic systems using linear transfer operators, with a focus on quadruped locomotion in unstructured environments. The central goal is to develop a unified operator-theoretic framework for safe control design based on the Perron–Frobenius (P–F) and Koopman operators. In particular, the thesis leverages \emph{density functions} to develop safe navigation frameworks in the dual space of densities. In the operator-theoretic perspective, the P–F operator governs …

Modeling Bitcoin Dynamics Using Differential Equations, Boone M. Fleenor

Departmental Honors & Graduate Capstone Projects

In this thesis, we develop and analyze two nonlinear systems of ordinary differential equations to model Bitcoin price dynamics. Analytical techniques are used to obtain exact or approximate solutions where possible. Then, numerical simulations using a fourth-order Runge–Kutta method are employed to explore system behavior beyond analytically tractable regimes. Finally, model outputs are compared to historical Bitcoin price data using normalized and resampled time series. These results suggest that deterministic models can provide meaningful insight into the structural behavior of Bitcoin markets, while highlighting the need for stochastic or time-dependent extensions for more realistic modeling.

A Stability Analysis Of The Phase-Lock Equations, Brian M. Sunguza

UNF Graduate Theses and Dissertations

Ginzburg and Landau have provided a set of equations that relate superconductivity to magnetic fields. Through a transformation process, Zhan has derived what are now called the phase-lock equations. A stability analysis of the spatially-independent phase-lock equations is the purpose of this presentation. This simplification is significant since it allowed for the analytical determination of equilibria, their stability, and the influence of a periodic forcing function. Through the use of an original code, numerical simulations are shown to corroborate the analytical results described above.

This analysis includes novel Lyapunov functions that allowed for the analytical determination of the instability region. …

Construction And Data-Driven Analysis Of A Stochastic, Individual-Based Opioid Epidemiology Network Model, Leigh Bennett Pearcy, Owen Queen, Vincent Jodoin, Suzanne Lenhart, Christopher Strickland

Mathematical Modelling and Numerical Simulation with Applications

While substance use epidemiology has been an active area of mathematical research in recent years, the social and mental processes that are involved in the development of substance use disorders have presented challenges to advancing the epidemiological theory and how they differ from the contraction of pathogenic disease. Such distinction is especially pertinent in the context of the current United States opioid epidemic and its intersection with the recent COVID-19 pandemic, as both prescription drugs and social influence play major roles in the development of opioid use disorder. In this paper, we construct a stochastic network model capturing how individual …

Modeling Synaptic Dysfunction As Neural Contagion: A Graph-Based Sedr Framework For Simulating Signal Spread, Michelle Marfo, Dr. Padmanabhan Seshaiyer, Alonso Ogueda-Oliva

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Immune Dysregulation In Covid-19: Mathematical Modeling Of The Within-Host Dynamics, Pagnapech Ngoun, Nicolas Alvarez, Ayesh Awad, Hwayeon Ryu

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Metapopulation Model For Oyster Restoration, Leah Shaw

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Understanding The Spread Of Black Sigatoka Disease: A Deterministic And Stochastic Modeling Approach, Bernard Asamoah Afful, Luis F. Gordillo

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Global Stability And Bifurcation Analysis Of A Predator-Prey Model Involving Allee Effect And Monod-Haldane Functional Response, Resmawan Resmawan, Agus Suryanto, Isnani Darti, Hasan S. Panigoro

Mathematical Modelling and Numerical Simulation with Applications

In this paper, the complexity of the dynamic behavior of the interaction between prey and predator is studied. The predator-prey relationship involves Allee effects and Monod-Haldane functional response. The constructed model has been shown to have validity in several respects, including the existence and uniqueness of the solution, as well as its non-negativity and boundedness. Three equilibrium points, namely trivial, axial, and coexistence points, are found, including their global dynamics using the Lyapunov function together with the LaSalle's invariance principle. The effect of the predation conversion rate causes changes in the dynamic behavior of predators and prey, which is characterized …

Dynamics And Optimal Intervention Strategies In A Shigellosis Transmission Model, Mehmet Gümüs, Shewafera Wondimagegnhu Teklu, Kemal Türk

Mathematical Modelling and Numerical Simulation with Applications

This research explores how water treatment contributes to limiting the transmission of Shigellosis, an infection caused by bacteria from the Shigella genus. A mathematical framework is formulated to evaluate the influence of protective strategies and water purification on the spread of the disease. To confirm the model's biological relevance, its well-posedness is investigated. The basic reproduction number $(\mathfrak{R}_0)$, a critical indicator of disease behavior, is derived using the matrix operator method. Findings indicate that if $\mathfrak{R}_01$, the infection persists, with the endemic equilibrium exhibiting local asymptotic stability. A comprehensive cost-effectiveness analysis reveals that combining environmental protection with water treatment represents …

Derivation Of Adjoint Based Error Estimates For Nonlinear Ordinary Differential Equations With Application To Multistage Sir Models With Demographics, Daniel Alcala

Mathematics & Statistics ETDs

Ordinary Differential Equations (ODEs) are central to the mathematical modeling of various real-world phenomena, from mechanical systems governed by Newton’s laws to epidemic dynamics described by SIR-type ODEs. Since many ODEs do not admit closed-form analytic solutions, we approximate them numerically (e.g., with Euler’s, Runge–Kutta, or other such methods). This raises the key question: How accurate are these numerical solutions? In particular, reliably estimating the error in some quantity of interest (QoI) at time T without having an exact solution is of great scientific interest.

The first main contribution of this thesis is the development and analysis of adjoint-based error …

Evolutionary Dynamics Of Artificial Agents: Exploration And Learning In Games, Brian Mintz

Dartmouth College Ph.D Dissertations

The natural world abounds with examples of complex behavior in humans and many other species. Evolutionary game theory is a powerful mathematical framework to understand the origins of many such behaviors like cooperation. Since these behaviors are often selected against initially, understanding why they are so widespread has been a longstanding question. Rather than assuming agents' rationality, like in traditional game theory, this approach studies the mutation and selection of strategies themselves. However most behavior is neither perfectly rational nor entirely determined by genetics. This dissertation works to bridge the gap between these two perspectives by analyzing models where individuals …

Stability Insights From Modeling Chronic Myelogenous Leukemia, Giovani Thai

Master's Theses

This thesis centers around a model for chronic myelogenous leukemia (CML) as it behaves under imatinib treatment, a common medication for CML patients, and the anti-leukemia immune response. The dynamics are represented with a system of nonlinear delay-differential equations first constructed by Kim et al. in 2008, capturing population changes of T-cells and various CML growth stages. We investigate stability in both the clinical and mathematical sense. Through numerical simulations, we computationally incorporate a supplementary treatment plan to determine its effectiveness in aiding immune response and medication in achieving remission and full elimination. The primary goal is to conduct a …

Dual Quaternions For Gravity Recovery Missions, Ryan Kinzie

Doctoral Dissertations and Master's Theses

A dual quaternion-based modeling, state estimation and control approach is introduced as a better alternative to the traditional methods which are currently utilized for gravity recovery missions. The proposed modeling and control approach was verified against and compared to the tangent bundle to Special Euclidean Group 3 through MATLAB simulations. The dual quaternion-based approach shows superior performance over traditional linearized and uncoupled methodologies, in both modeling accuracy of spacecraft translational position, and the ability to control the pose of a test mass relative to its host spacecraft. Utilizing data products from the Gravity Recovery and Climate Experiment Follow-On mission, a …

Horizontal Infiltration Of Water Through Porous Snow As A Gravity Current, Anthony Cheng

Dartmouth College Master’s Theses

On the surface of the Greenland ice sheet or around the margins of the Antarctic ice shelf, water infiltrates porous ice. It is important to understand this infiltration process since water populating the pore space of ice directly impacts the density, porosity, and wetness of ice. These properties influence the mechanics and tensile strength of ice, as greater amounts of infiltration result in faster or more widespread deformation events, which may lead to adverse climatic effects such as sea level rise and ocean current disruption. While studies have considered the thermodynamics and fluid mechanics of water vertically percolating through snow …

A Mathematical Model On The Temporal Dynamics Of Aviation Competitive Pricing, Tichaona Chikore,, Farai Nyabadza,

Journal of Aviation/Aerospace Education & Research

This study investigates the competitive dynamics of airport pricing using U.S. airport data to validate the findings. It employs linear and nonlinear ordinary differential equation models to analyze the influence of competitive interactions and internal factors on pricing decisions. The methodology involves parameter estimation via optimization techniques and quantile regression to capture heterogeneity across market segments. Mathematical analysis and simulation results show that if competitive coupling coefficients are low then there is weak competitive influence on pricing, with airports’ pricing largely driven by internal factors. Also, if the adjustment rates exhibit consistency across airports then internal dynamics are dominant in …

The Inverse Scattering Transform For The Nonlinear Schrödinger Equation, Ivan Casas-Rocha

Honors Undergraduate Theses

The Nonlinear Schrödinger (NLS) Equation, iψ_t + 1/2 ψ_xx ± |ψ|²ψ = 0, is a nonlinear partial differential equation which is used to model several physical phenomena including nonlinear effects inside optical fibers and the formation of rogue waves in shallow water. It is particu- larly difficult to study solutions to this equation due to the nonlinearity, and the nonlinearity leads to incredibly interesting solutions not found in linear PDEs such as solitons. In this thesis, we highlight two methods of obtaining solutions to the (NLS) equation: the Inverse Scattering Transform and the Dressing Method. Furthermore, …

Individual And Collective Properties Of Tunable Photochemical Belousov-Zhabotinsky Micro-Reactors, Kudakwashe Benedict Shumba

Graduate Theses, Dissertations, and Problem Reports (ETD)

Cell-like model chemical systems are powerful tools that can be used to explore the role of intercellular coupling on population level behaviors in communities of biological cells. Firstly, we present a new method for fabricating such micro-reactors using the photosensitive Belousov–Zhabotinsky (BZ) reaction system employed in silica microparticles. These BZ micro-reactors have a tunable response to photochemical coupling, varying from a fully excitatory response to a fully inhibitory response. Their response can be tuned through variations in either the reactive mixture or, on an individual micro-reactor level, by changes in the synthesis temperature used during the fabrication of the silica …

Dynamics Of Prey-Prey Mutualism With Varying Carrying Capacity And Herd Behavior In The Presence Of Predator, Felix Dela Djokoto

UNF Graduate Theses and Dissertations

This study investigates the dynamics of mutualistic relationships between two prey species in the presence of a predator, by taking into account herding in one species along with the influence of carrying capacities between two prey species. These mutualistic dynamics are presented by constructing two mathematical models, namely an indirect and direct mutualism model. As the strength of the symbiotic relationship increases the indirect model goes through transcritical bifurcation to Hopf bifurcation whereas in the direct mutualism model goes through saddle-node bifurcation to Hopf bifurcation.

Mathematical Modelling Of Disease Outbreak, Favour Christian, Matthew Molloy

SURE Journal: Science Undergraduate Research Experience Journal

Establishing a model framework for more research necessitates a thorough understanding of the causes, distribution, prevalence, and evolution of infectious illnesses. The main mathematical concept used in this modelling simulation is ordinary differential equations (ODEs). The purpose of this study was to investigate the significance of the many criteria linked to a zombie virus spread. The zombie framework provides an accessible and relatively simple representation of the nature of infectious disease spread, allowing for tractable assumptions and the development of more complex situations.

The models are designed around a zombie outbreak in which the zombie virus is spread through a …

Modelling Saccharomyces Cerevisiae For The Production Of Fermented Beverages, Paul A. Valle Dr., Yolocuauhtli Salazar Dr., Luis N. Coria Dr., Oscar N. Soto Dr., Jesus B. Paez Dr.

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Numerical Issues For A Non-Autonomous Logistic Model, Marina Mancuso, Kaitlyn M. Martinez, Carrie Manore, Fabio Milner

CODEE Journal

The user-friendly aspects of standardized, built-in numerical solvers in
computational software aid in the simulations of many problems solved using
differential equations. The tendency to trust output from built-in numerical
solvers may stem from their ease-of-use or the user’s unfamiliarity with the
inner workings of the numerical methods. Here, we show a case where the
most frequently used and trusted built-in numerical methods in Python’s
SciPy library produce incorrect, inconsistent, and even unstable approxima-
tions for a the non-autonomous logistic equation, which is used to model
biological phenomena across a variety of disciplines. Some of the most com-
monly used …

Modeling Fast Information And Slow(Er) Disease Spreading: A Geometric Analysis, Iulia Martina Bulai, Mattia Sensi, Sara Sottile

Biology and Medicine Through Mathematics Conference

No abstract provided.