Browsing by Subject "Dynamical systems"
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Item A dynamical system study of the Hodgkin-Huxley Model of nerve membrane(2012-08) Bumbaugh, Julie; Roeger, Lih-Ing W.; Jang, SophiaEquations of the Hodgkin and Huxley model of action potentials in squid giant axons have been studied and described in this thesis. Explanation of the development of the space clamp model that won its authors the Nobel Prize for Physiology and Medicine in 1963 is provided since it is still one of the key mathematical models in the study of neural communications today. A numerical approach was used to obtain solutions to their model. The computer code used will be provided. More modern approaches to analyzing these models will be explored and discussed as a study of dynamical systems. A reduced Hodgkin-Huxley model will be explained and analyzed using modern methods. A set of problems will be included for those who wish to be guided through parts of the development of the models.Item Iterative milestoning(2016-12) Bello Rivas, Juan Manuel; Elber, Ron; Engquist, Bjorn; Makarov, Dmitrii E; Rodin, Gregory J; Zariphopoulou, ThaleiaComputer simulation of matter using Molecular Dynamics (MD) is a staple in the field of Molecular Biophysics. MD yields results suitable for comparison with laboratory experiments and, in addition, it serves as a computational microscope by providing insight into a variety of molecular mechanisms. However, some of the most interesting problems pertaining to the investigation of biomolecules remain outside of the scope of MD due to the long time scales at which they occur. Milestoning is a method that addresses the long time simulation of biomolecular systems without giving up the fully-atomistic spatial resolution necessary to understand biological processes such as signalling and biochemical reactions. The method works by partitioning the phase space of the system into regions whose boundaries are called milestones. The dynamics of the system restricted to the milestones defines a stochastic process whose transition probabilities and exit times can be efficiently computed by numerical simulation. By calculating the transition probabilities and exit times of this process, we can obtain global thermodynamic and kinetic properties of the original system such as its stationary probability, free energy, and reaction rates. The calculation of these properties would be unfeasible for many systems of interest if we were to approach the problem by plain MD simulation. The success of milestoning computations relies on certain modeling assumptions. In this dissertation we introduce an iterative variant of the Milestoning method that relaxes the assumptions required by the original method and can be applied in the non-equilibrium setting. The new method works by iteratively approximating the transition probabilities and exit times until convergence is attained. In addition to a detailed description of the method, we give various pedagogical examples, showcase its practical applications to molecular systems, and provide an alternative formulation of the method in terms of boundary value problems.Item On the role of invariant objects in applications of dynamical systems(2012-05) Blazevski, Daniel, 1984-; Llave, Rafael de la; Chen, Thomas; Koch, Hans; Morrison, Phil; Ocampo, Cesar; Pavlovic, Natasa; Vasseur, AlexisIn this dissertation, we demonstrate the importance of invariant objects in many areas of applied research. The areas of application we consider are chemistry, celestial mechanics and aerospace engineering, plasma physics, and coupled map lattices. In the context of chemical reactions, stable and unstable manifolds of fixed points separate regions of phase space that lead to a certain outcome of the reaction. We study how these regions change under the influence of exposing the molecules to a laser. In celestial mechanics and aerospace engineering, we compute periodic orbits and their stable and unstable manifolds for a object of negligible mass (e.g. a satellite or spacecraft) under the presence of Jupiter and two of its moons, Europa and Ganymede. The periodic orbits serve as convenient spot to place a satellite for observation purposes, and computing their stable and unstable manifolds have been used in constructing low-energy transfers between the two moons. In plasma physics, an important and practical problem is to study barriers for heat transport in magnetically confined plasma undergoing fusion. We compute barriers for which heat cannot pass through. However, such barriers break down and lead to robust partial barriers. In this latter case, heat can flow across the barrier, but at a very slow rate. Finally, infinite dimensional coupled map lattice systems are considered in a wide variety of areas, most notably in statistical mechanics, neuroscience, and in the discretization of PDEs. We assume that the interaction amont the lattice sites decays with the distance of the sites, and assume the existence of an invariant whiskered torus that is localized near a collection of lattice sites. We prove that the torus has invariant stable and unstable manifolds that are also localized near the torus. This is an important step in understanding the global dynamics of such systems and opens the door to new possible results, most notably studying the problem of energy transfer between the sites.Item Toward seamless multiscale computations(2013-05) Lee, Yoonsang, active 2013; Engquist, Björn, 1945-Efficient and robust numerical simulation of multiscale problems encountered in science and engineering is a formidable challenge. Full resolution of multiscale problems using direct numerical simulations requires enormous amounts of computational time and resources. This thesis develops seamless multiscale methods for ordinary and partial differential equations under the framework of the heterogeneous multiscale method (HMM). The first part of the thesis is devoted to the development of seamless multiscale integrators for ordinary differential equations. The first method, which we call backward-forward HMM (BFHMM), uses splitting and on-the-fly filtering techniques to capture slow variables of highly oscillatory problems without any a priori information. The second method, denoted by variable step size HMM (VSHMM), as the name implies, uses variable mesoscopic step sizes for the unperturbed equation, which gives computational efficiency and higher accuracy. VSHMM can be applied to dissipative problems as well as highly oscillatory problems, while BFHMM has some difficulties when applied to the dissipative case. The effect of variable time stepping is analyzed and the two methods are tested numerically. Multi-spatial problems and numerical methods are discussed in the second part. Seamless heterogeneous multiscale methods (SHMM) for partial differential equations, especially the parabolic case without scale separation are proposed. SHMM is developed first for the multiscale heat equation with a continuum of scales in the diffusion coefficient. This seamless method uses a hierarchy of local grids to capture effects from each scale and uses filtering in Fourier space to impose an artificial scale gap. SHMM is then applied to advection enhanced diffusion problems under incompressible turbulent velocity fields.