Posts by Collection

Limit and Morse Sets for Deterministic Hybrid Systems

Published in Qualitative Theory of Dynamical Systems, 2012

The term “hybrid system” refers to a continuous time dynamical system that undergoes Markovian perturbations at discrete time intervals. In this paper, we find that under the right formulation, a hybrid system can be treated as a dynamical system on a compact space. This allows us to study its limit sets. We examine the Morse decompositions of hybrid systems, find a sufficient condition for the existence of a non-trivial Morse decomposition, and study the Morse sets of such a decomposition. Finally, we consider the case in which the Markovian perturbations are small, showing that trajectories in a hybrid system with small perturbations behave similarly to those of the unperturbed dynamical system.

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A Behavioral Characterization of Discrete Time Dynamical Systems over Directed Graphs

Published in Qualitative Theory of Dynamical Systems, 2014

Using a directed graph, a Markov chain can be treated as a dynamical system over a compact space of bi-infinite sequences, with a flow given by the left shift of a sequence. In this paper, we show that the Morse sets of the finest Morse decomposition on this space can be related to communicating classes of the directed graph by considering lifting the communicating classes to the shift space. Finally, we prove that the flow restricted to these Morse sets is chaotic.

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Modelling acute myeloid leukaemia in a continuum of differentiation states

Published in Letters of Biomathematics, 2018

Here we present a mathematical model of movement in an abstract space representing states of cellular differentiation. We motivate this work with recent examples that demonstrate a continuum of cellular differentiation using single cell RNA sequencing data to characterize cellular states in a high-dimensional space, which is then mapped into R^2 or R^3 with dimension reduction techniques. We represent trajectories in the differentiation space as a graph, and model directed and random movement on the graph with partial differential equations. We hypothesize that flow in this space can be used to model normal and abnormal differentiation processes. We present a mathematical model of hematopoeisis parameterized with publicly available single cell RNA-Seq data and use it to simulate the pathogenesis of acute myeloid leukemia (AML). The model predicts the emergence of cells in novel intermediate states of differentiation consistent with immunophenotypic characterizations of a mouse model of AML.

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Chain Recurrence in Graph Determined Hybrid Systems

Published in Journal of Dynamics and Control Systems, 2019

In this paper, we consider a family of dynamical systems on the same compact metric space. We then consider the dynamics given when the given flow shifts between these different flows at regular time intervals. We further require that shifts be allowed by a given directed graph. We then define a type of set, called a chain set, that exhibits many similar properties to chain transitive sets of flows. By considering the dynamics as a skew product flow, we are able to demonstrate that chain sets can be lifted to a chain transitive set if the given graph is complete.

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Search for invariant sets of the generalized tent map

Published in Journal of Difference Equations and Applications, 2022

This paper describes a predictive control method to search for unstable periodic orbits of the generalized tent map. The invariant set containing periodic orbits is a repelling set with a complicated Cantor-like structure. Therefore, a simple local stabilization of the orbit may not be enough to find a periodic orbit, due to the small measure of the basin of attraction. It is shown that for certain values of the control parameter, both the local behavior and the global behavior of solutions change in the controlled system. In particular, the invariant set enlarges to become an interval or the entire real axis. The computational particularities of using the control system are considered, and necessary conditions for the orbit to be periodic are given. The question of local asymptotic stability of subcycles of the controlled systems stable cycles is fully investigated and some statistical properties of the subset of the classical Cantor middle thirds set that is determined by the periodic points of the generalized tent map are described.

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Math 350 - Foundations for Theoretical Mathematics

Undergraduate course, CSUSM, Department, 2022

Fall 22 Teaching: Bridge course between computation-driven mathematics and theoretical mathematics. Designed to familiarize the student with the language and process of rigorous mathematical thought, speech, and writings through the introduction of typical and important examples from algebra, analysis, combinatorics, and geometry. Covers elementary logic, methods of proof, mathematical induction, sets, relations, equivalence relations, functions, inverse functions, binary operations and cardinality. Mathematics majors are encouraged to take this course as early as possible.

Math 448 - Mathematical Models and Methods in Biology

Undergraduate course, CSUSM, Department, 2022

Introduces mathematical models in Biology and their analysis. Both one dimensional models, including the Malthusian Model and the logistic model, and multi-dimensional models, including structured population models and predator-prey models, are studied, as are matrix models for base substitution in DNA, phylogenetic trees, and sequence alignment. Mathematical concepts and methods to formulate and analyze these models include limits, derivatives, matrix algebra, eigenvectors, probability theory, and dynamic programming. Software is used to simulate models and visualize the numerical results.