Numerical Analysis of Linear and Nonlinear Schrodinger Equations on Quantum Graphs

A wide variety of problems in quantum mechanics can be modeled with Schrodinger-type equations on quantum graphs. More specifically, graphs are useful for simplifying models of physical systems that feature nano-scaled branching structures. Since analytical solutions can only be found for a few trivial cases, it is necessary to consider how to accurately solve this type of problem numerically. While a wide variety of tools exist to solve these types of partial differential equations on lines, this is not well studied in the case of graphs - one critical difference between the two cases being that graphs require more complicated boundary conditions. This paper utilizes a new approach to include the boundary conditions in the discretized operator that preserves high levels of accuracy. Thus, a proper time evolution scheme must work in conjunction with a spatial operator that has incorporated boundary conditions and preserve the accuracy of our spatial component, and this is accomplished by implementing methods from differential algebraic equations. We study the numerical computation of linear and nonlinear states for Schrodinger equations on graphs, as well as numerically compute their linear stability properties. The latter result has implications for the future study of relative periodic orbits on graphs. All numerical components are being adapted into a MATLAB software package called QGLAB jointly with Roy Goodman on Github.Doctor of Philosoph

The direct scattering study of the parametrically driven nonlinear Schrödinger equation

Includes abstract. Includes bibliographical references

Complexity, Emergent Systems and Complex Biological Systems:\ud Complex Systems Theory and Biodynamics. [Edited book by I.C. Baianu, with listed contributors (2011)]

An overview is presented of System dynamics, the study of the behaviour of complex systems, Dynamical system in mathematics Dynamic programming in computer science and control theory, Complex systems biology, Neurodynamics and Psychodynamics.\u

Computational dynamical systems analysis : Bogdanov-Takens points and an economic model

The subject of this thesis is the bifurcation analysis of dynamical systems (ordinary differential equations and iterated maps). A primary aim is to study the branch of homoclinic solutions that emerges from a Bogdanov-Takens point. The problem of approximating such branch has been studied intensively but neither an exact solution was ever found nor a higher-order approximation has been obtained. We use the classical ``blow-up'' technique to reduce an appropriate normal form near a Bogdanov-Takens bifurcation in a generic smooth autonomous ordinary differential equations to a perturbed Hamiltonian system. With a regular perturbation method and a generalization of the Lindstedt-Poincare' perturbation method, we derive two explicit third-order corrections of the unperturbed homoclinic orbit and parameter value. We prove that both methods lead to the same homoclinic parameter value as the classical Melnikov technique and the branching method. We show that the regular perturbation method leads to a ``parasitic turn'' near the saddle point while the Lindstedt-Poincare' solution does not have this turn, making it more suitable for numerical implementation. To obtain the normal form on the center manifold, we apply the standard parameter dependent center manifold reduction combined with the normalization, using the Fredholm solvability of the homological equation. By systematically solving all linear systems appearing from the homological equation, we correct the parameter transformation existing in the literature. The generic homoclinic predictors are applied to explicitly compute the homoclinic solutions in the Gray-Scott kinetic model. The actual implementation of both predictors in the MATLAB continuation package MatCont and five numerical examples illustrating its efficiency are discussed. Besides, the thesis discusses the possibility to use the derived homoclinic predictor of generic ordinary differential equations to continue the branches of homoclinic tangencies in the Bogdanov-Takens map. The second part of this thesis is devoted to the application of bifurcation theory to analyze the dynamic and chaotic behaviors of a nonlinear economic model. The thesis studies the monopoly model with cubic price and quadratic marginal cost functions. We present fundamental corrections to the earlier studies of the model and a complete discussion of the existence of cycles of period 4. A numerical continuation method is used to compute branches of solutions of period 5, 10, 13 and 17 and to determine the stability regions of these solutions. General formulas for solutions of period 4 are derived analytically. We show that the solutions of period 4 are never linearly asymptotically stable. A nonlinear stability criterion is combined with basin of attraction analysis and simulation to determine the stability region of the 4-cycles. This corrects the erroneous linear stability analysis in previous studies of the model. The chaotic and periodic behavior of the monopoly model are further analyzed by computing the largest Lyapunov exponents, and this confirms the above mentioned results

Mathematical and Numerical Aspects of Dynamical System Analysis

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”