Sharif Digital Repository

Periodic orbits are one of the basic objects of interest in Dynamical systems. And methods for detecting, counting and determining the periods of periodic orbits are of great interest. In spite of this interest, periodic orbits remain difficult to detect. The most celebrated existence result is the Poincare-Bendixson theorem. Unfortunately, this theorem is false in general for higher dimensional systems. The success of fixed point theory inspired the hope that similar topological techniques could be of use in proving the existence of periodic orbits. In this Thesis we have analyzed some topological and geometrical theories which could be of use in proving the existence of periodic orbits.... 

In modeling of biological systems and chemical reaction networks, analy-sis the existence of multiple equilibria is important. Our interest in using dynamical system in such analyses comes from biology. So, we will study the qualitative properties of a dynamical system, described by a system of ordinary differential equations. In this thesis we will discuss the question of how to decide when a general chemical reaction system is incapable of admitting multiple euilibria, regardless of parameter values such as reaction rate constants. We will relate different graphs to chemical reaction systems, and by means of its paths and circuits, we will introduce some sufficient conditions for... 

Primal discontinuous Galerkin (DG) methods are formulated to solve the transport equations for modeling migration and survival of viruses with kinetic and equilibrium adsorption in porous media. An entropy analysis is conducted to show that DG schemes are numerically stable and that the free energy of a DG approximation decreases with time in a manner similar to the exact solution. Combining results for free and attached virus concentrations, we establish optimal a priori error estimates for the coupled partial and ordinary differential equations of virus transport  

Using a Heegaard diagram for the pullback of a knot K ⊂ S3 in its cyclic double branched cover Σ2(K), we give a combinatorial proof for the invariance of the associated knot Floer homology over Z  

Invariant manifolds are important objects in the theory of dynamical systems. The stable manifold theorem is a very important theorem about this concept which proves the existence of stable and unstable manifolds in a wide range of dynamical systems. The importance of invariant manifolds encourages us to view their pictures. It helps us to understand their bahavior. For this purpose, at first we need to approximate the invariant manifold we want to visualize. There are several algorithms designed to approximate invariant manifolds. Those algorithms approximate a set of points on an invariant manifold and then provide an approximation of the manifold by the calculated points. Visualizing an... 

This thesis presents an investigation of the dynamics of two coupled non-identical FitzHugh–Nagumo neurons. It is known that signal transmission in coupled neurons is not instantaneous in general, and time delay is inevitable in signal transmission for real neurons. Therefore we consider the system of two coupled neurons with delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try to classify all possible dynamics which is fairly rich and we will study the excitability of the neurons. By bifurcation study of the system the coupling strength and delay-dependent stability regions are illustrated in the parameter plane, to describe typical... 

In order to generalize the Morse Index for fixed points of a flow, Conley introduced an invariant for isolated invariant sets under a flow which is known as Conley Index.However a similar generalization in discrete dynamical systems is more difficult. Category theory, shape theory and algebraic topology have been used in the generalization. In this thesis we propose a construction of Conley index for discrete dynamical systems, which goes along Conley lines as long as it was possible, but certain modifications has been done for convenience. After introducing the index, we will consider some applications: the generalized Morse inequalities and the Morse decomposition. At the end we will... 

Reservoir simulation, recognition of the past and present reservoir behavior and prediction of its behavior in the future, which forms one of the most important parts of the management of a reservoir. In fact, the reservoir is like a living creature and increasing the accuracy in predicting its behavior requires a precise model. Reservoirs have high physical dimensions that are associated with high physical changes. As a result, a large number of computational cells are needed to simulate the reservoirs, which results in a very large linear equation system. The numerical solution of these systems is very time consuming. Many observed examples indicate that 80 \% of the runtime is included.... 

In this thesis we employ finite elements and conjugate gradient methods to exibit a numerical solution for governing equations of borehole stability  

Scattering theory on locally symmetric space first was studied by Peter Lax and Ralph Phillips. They studied the and develop scattering theory on it. Without the Fourier analysis tools this theory was completed with a lot of computations. In this thesis first we try to construct Fourier analysis by definition Fourier transform then by means of new tools we prove very short version of scattering theory on locally symmetric space . In general case for studying scattering theory we need firs spectral decomposition of Hilbert space of square integrable functions locally symmetric spaces of rational rank one is achieved by quotient of symmetric space where G is Lie group and K is maximal... 

Excitability is one of the most important characteristics of a neuron. In 1948, Hodgkin identified three different types of excitability of neurons. Excitability an all of its types can be observed in Hodgkin-Huxley model of neuronal dynamics (H-H model) as a four-dimensional system of differential equations and in at least two dimensional reductions of H-H type models. By applying the theory of dynamical systems (e.g. bifurcation theory), one can give a mathematical definition of excitability.Excitability of the neuron is equivalent to that the neuronal model is near a bifurcation through which the state of the system approaches to a stable limit cycle. In this thesis, a two-dimensional... 

Canard, first observed in Van der Pol oscillations, is a typical phenomenon in oscillatory systems. Canard is also observed in many oscillatory systems such as electrical circuits and neurons. In many fields of science and engineering there are complex oscillations that exhibit canard for certain values of parameters. These three dimensional systems exhibit complex oscillatory behavior never observed in two dimensional dynamics. Some of these systems are chaotic for certain parameter values. It seems than in oscillatory systems canards can make complex behavior. Several methods such as the singular perturbation theory have been used to study this complexity. In this project, we study canard... 

In this thesis, we study a homological invariant in Knot theory, called Khovanov homology. The main property of this invariants is that it gives us the Jones polynomial, as its graded Euler characteristic. Besides, the functor (1+1) TQFT, from the category of closed one-manifolds to the category of vector spaces is employed in its construction. By making some changes to this functor and defining another functor and some other steps, the so-called Lee spectral sequence is derived which starts from Khovanov homology and converges to another homological invariant of links, called Lee-Khovanov homology. Computation of this homology is very simple. By using this spectral sequence, a numerical... 

The problem of learning using high volume of data as stream, has attracted much attention recently. In this thesis, the problem is modeled and analized using Online Convex Optimization tools [1], [2]. General performance bounds are stated and clarified in this framework [8]. Using the practical experience in Online Decision Making (e.g., predicting price in Stock Market), the need for a more flexible model, which adapts to changes in problem, is presented. In this thesis, after reviewing the literature and online convex optimization framework, we will define ”Concept Drift”, which describes changes in the dynamics of the problem and the statistical tools to detect it [13], [5]. And finally,... 

Application of optimal control in cancer modeling is studied through both linear and nonlinear modeling of the dynamics in ordinary differential equations. At the outset, a fairly straight-forward analysis of a linear model in presented. Through comparably simple machinery, this seminal work published at early 2000s covers some of most important techniques previously developed. The model here is infinite- dimensional, taking different number of gene amplifications into account. Thereafter by surveying recently published papers, the literature is reviewed and different lines of progress is followed, culminating in detailed study of a specific approach which is theoretically of interest.... 

Reservoir simulation models are profound tools in reservoir management. Number of grid blocks in a simulation model is much less than the number of grid blocks in geological model, therefore geological model must become upscaled in order to construct simulation model with convenient dimensions. This causes loosing of some details and introduces errors due to discretization and homogenization. One method of construction of an efficient computational grid is use of adaptive mesh refinement (AMR) methods. These methods refine computational grid in regions needing further resolution. An error indicator is used for determining regions candidate for refinement or coarsening. In present study... 

In this thesis, we review some models for mathematical modeling of neuronal circuits of brain and especially, the main aim of this thesis is chapter two that we review a rigorous model for cortical microcir-cuits which is called liquid state machine. The structure of this thesis is as follows, first in the introduction we give some prerequisites that are essential for understanding contents of this thesis and also, one of the first mathematical model for neural computations, attractor neural networks, has been mentioned. In chapter one, we introduce feedforward neural networks equipped with dynamical synapses. In fact in this chapter we review two experimental models for dynam-ical synapses... 

Let M be a connected, compact, Riemannian manifold. Geodesic flow is a flow on the unit tangent bundle of M . This flow can be studied in dynamics prespective. for example entropy or complexity of the geodesic flow. in this thesis we will follow methods of entropy estimation or computing for geodesic flow. we will follow the method of anthony manning and Ricardo Mañe for proving such result. Maning present two results linking the topological entropy of the geodesic flow on M. we expalin how he find exponential growth rate volume of balls in universal cover as a lower bound for topologycal entropy. another theorem , Mañe represent the equlity between exponential growth rate of avrage of... 

The numerical solution of differential equations in high dimensions has always been a challenge and has been associated with various computational difficulties. These equations appear naturally in a variety of problems such as financial mathematics, control, and physics, and their optimal solution with high accuracy and speed can open new windows on new applications. Conventional methods such as Finite element and finite difference method in high dimensions lose their efficiency, which is a barrier to fast and accurate calculation of these equations. In this dissertation first, we review some theoretical and practical aspects of deep neural networks and then we try to examine the recent...