12 research outputs found
Bifurcations of the global stable set of a planar endomorphism near a cusp singularity
The dynamics of a system defined by an endomorphism is essentially different from that of a system defined by a diffeomorphism due to interaction of invariant objects with the so-called critical locus. A planar endomorphism typically folds the phase space along curves J0 where the Jacobian of the map is singular. The critical locus, denoted J1, is the image of J0. It is often only piecewise smooth due to the presence of isolated cusp points that are persistent under perturbation. We investigate what happens when the stable set Ws of a fixed point or periodic orbit interacts with J1 near such a cusp point C1. Our approach is in the spirit of bifurcation theory, and we classify the different unfoldings of the codimension-two singularity where the curve Ws is tangent to J1 exactly at C1. The analysis uses a local normal-form setup that identifies the possible local phase portraits. These local phase portraits give rise to different global manifestations of the behavior as organized by five different global bifurcation diagrams. © 2008 World Scientific Publishing Company
Weak and strong fillability of higher dimensional contact manifolds
For contact manifolds in dimension three, the notions of weak and strong
symplectic fillability and tightness are all known to be inequivalent. We
extend these facts to higher dimensions: in particular, we define a natural
generalization of weak fillings and prove that it is indeed weaker (at least in
dimension five),while also being obstructed by all known manifestations of
"overtwistedness". We also find the first examples of contact manifolds in all
dimensions that are not symplectically fillable but also cannot be called
overtwisted in any reasonable sense. These depend on a higher-dimensional
analogue of Giroux torsion, which we define via the existence in all dimensions
of exact symplectic manifolds with disconnected contact boundary.Comment: 68 pages, 5 figures. v2: Some attributions clarified, and other minor
edits. v3: exposition improved using referee's comments. Published by Invent.
Mat
Mathematical modeling with applications in biological systems, physiology, and neuroscience
Doctor of PhilosophyDepartment of MathematicsBacim AlaliDynamical systems modeling is used to describe different biological and physical systems as well as to predict the interactions between multiple components of a system over time. A dynamical system describes the evolution of a given system over time using a set of mathematical laws, typically described by differential equations. There are two main methods to model the dynamical behaviors of a system: continuous time modeling and discrete-time modeling. When the time between two measurements is negligible, the continuous time modeling governs the evolution of the system, however, when there is a gap between any two consecutive measurements, discrete-time system modeling comes into play. Differential equations are used to model continuous systems and iterated maps represent the generations in discrete-time systems. In this dissertation, we study some dynamical systems and present their applications to different problems in biological systems, physiology, and neuroscience.
In chapter one, we study the local dynamics of some interesting systems and show the local stable behavior of the system around its critical points. Moreover, we investigate the local dynamical behavior of different systems including the Hénon-Heiles system, the Duffing oscillator, and the Van der Pol equation. Furthermore, we discuss about the chaotic behavior of Hamiltonian systems using two different and new examples.
In chapter two, we consider some models in computational neuroscience. Due to the complexity of nerve systems, linear modeling methods are not sufficient to understand the various phenomena in neuroscience. We use nonlinear methods and models, which aim at capturing certain properties of the neurons and their complex dynamics. Specifically, we explore the interesting phenomenon of firing spikes and complex dynamics of the Morris-Lecar model. We consider a set of parameters such that the model exhibits a wide range of phenomenon. We investigate the influences of injected current and temperature on the spiking dynamics of Morris-Lecar model. In addition, we study bifurcations, and computational properties of this neuron model. Moreover, we provide a bound for the membrane potential and a certain voltage value or threshold for firing the spikes. Studying the two co-dimension bifurcations demonstrates more complicated behaviors for this single neuron model. Furthermore, we describe the phenomenon of neural bursting and investigate the dynamics of Morris-Lecar model as a square-wave burster, elliptic burster and parabolic burster.
Pharmacokinetic models are mathematical models, which provide insights into the interaction of chemicals with certain biological processes. In chapter three, we consider the process of drug and nanoparticle (NPs) distribution throughout the body. We use a tricompartmental model to study the perfusion of NPs in tissues and a six-compartmental model to study drug distribution in different body organs. We perform global sensitivity analysis by LHS Monte Carlo method using Partial Rank Correlation Coefficient (PRCC). We identify the key parameters that contribute most significantly to the absorption and distribution of drugs and NPs in different organs in the body.
In chapter four, we study two infectious disease models and use nonlinear optimization and optimal control theory to help in identifying strategies for transmission control and forecasting the spread of infectious diseases. We analyze the effect of vaccination on the disease transmission in these models. Moreover, we perform global sensitivity analysis to investigate the key parameters in these models.
In chapter five, we investigate the complex dynamics of two-species Ricker-type discrete-time competitive model. We perform local stability analysis for the fixed points of the system and discuss about its persistence for boundary fixed points. This system inherits properties of the dynamics of a one-dimensional Ricker model such as the cascade of period-doubling bifurcation, periodic windows, and chaos. We explore the existence of chaos for the equilibrium points for a specific case of this system using Marotto theorem and show the existence of snap-back repeller.
In chapter six, we study the problem of chaos synchronization in certain discrete-time dynamical systems. We introduce a drive-response discrete-time dynamical system, which is coupled using convex link function. We investigate a synchronization threshold, after which, the drive-response system uncouples and loses its synchronized behaviors. We apply this method to the synchronized cycles of the Ricker model and show that this model displays a rich cascade of complex dynamics from a stable fixed point and cascade of period-doubling bifurcation to chaos. We numerically verify the effectiveness of the proposed scheme and demonstrate how this type of coupling affects the synchronization of the system.
In chapter seven, we study the synchronized cycles of a generalized Nicholson-Bailey model. This model demonstrates a rich cascade of complex dynamics from a stable fixed point to periodic orbits, quasi periodic orbits and chaos. We introduce a coupling of these two chaotic systems with different initial conditions and show how they synchronize over a short time. We investigate the qualitative behavior of Generalized Nicholson-Bailey model and its synchronized model using time series analysis and its long-time dynamics by using its bifurcation diagram
Stein fillings of contact structures supported by planar open books
In this thesis we study topology of symplectic fillings of contact manifolds supported by planar open books. We obtain results regarding geography of the symplectic fillings of these contact manifolds. Specifically, we prove that if a contact manifold (M,ξ) is supported by a planar open book, then Euler characteristic and signature of any Stein filling of (M,ξ) is bounded. We also prove a similar finiteness result for contact manifolds supported by spinal open books with planar pages. Moving beyond the geography of Stein fillings, we classify fillings of some lens spaces. In addition, we classify Stein fillings of an infinite family of contact 3-manifolds up to diffeomorphism. Some contact 3-manifolds in this family can be obtained by Legendrian surgeries on (S³, ξ std) along certain Legendrian 2-bridge knots. We also classify Stein fillings, up to symplectic deformation, of an infinite family of contact 3-manifolds which can be obtained by Legendrian surgeries on (S³, ξ std) along certain Legendrian twist knots. As a corollary, we obtain a classification of Stein fillings of an infinite family of contact hyperbolic 3-manifolds up to symplectic deformation.Ph.D
Number Theory, Analysis and Geometry: In Memory of Serge Lang
Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future.
In the spirit of Lang’s vast contribution to mathematics, this memorial volume contains articles by prominent mathematicians in a variety of areas, namely number theory, analysis and geometry, representing Lang’s own breadth of interests. A special introduction by John Tate includes a brief and engaging account of Serge Lang’s life
Number Theory, Analysis and Geometry: In Memory of Serge Lang
Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future.
In the spirit of Lang’s vast contribution to mathematics, this memorial volume contains articles by prominent mathematicians in a variety of areas, namely number theory, analysis and geometry, representing Lang’s own breadth of interests. A special introduction by John Tate includes a brief and engaging account of Serge Lang’s life
SIMULATING SEISMIC WAVE PROPAGATION IN TWO-DIMENSIONAL MEDIA USING DISCONTINUOUS SPECTRAL ELEMENT METHODS
We introduce a discontinuous spectral element method for simulating seismic wave in 2- dimensional elastic media. The methods combine the flexibility of a discontinuous finite
element method with the accuracy of a spectral method. The elastodynamic equations are discretized using high-degree of Lagrange interpolants and integration over an element is
accomplished based upon the Gauss-Lobatto-Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix and the use of discontinuous finite element method makes the calculation can be done locally in each element. Thus, the algorithm is simplified drastically. We validated the results of one-dimensional problem by comparing them with finite-difference time-domain method and exact solution. The comparisons show excellent agreement