Dynamical Systems Commons^™

Stability Of Discrete Solitons In The Presence Of Parametric Driving, H. Susanto, Q. E. Hoq, Panos Kevrekidis

Panos Kevrekidis

In this Brief Report, we consider parametrically driven bright solitons in the vicinity of the anticontinuum limit. We illustrate the mechanism through which these solitons become unstable due to the collision of the phase mode with the continuous spectrum, or eigenvalues bifurcating thereof. We show how this mechanism typically leads to complete destruction of the bright solitary wave.

Radiationless Travelling Waves In Saturable Nonlinear Schrödinger Lattices, T. R. O. Melvin, A. R. Champneys, Panos Kevrekidis, J. Cuevas

Panos Kevrekidis

The long-standing problem of moving discrete solitary waves in nonlinear Schrödinger lattices is revisited. The context is photorefractive crystal lattices with saturable nonlinearity whose grand-canonical energy barrier vanishes for isolated coupling strength values. Genuinely localized traveling waves are computed as a function of the system parameters for the first time. The relevant solutions exist only for finite velocities.

The Elementary Theory Of Normed Linear Spaces And Linear Functionals, Suresh Eswarthasan

Honors Capstone Projects - All

Abstract not Included

A Remark On Conservative Diffeomorphisms, Jairo Bochi, Bassam R. Fayad, Enrique Pujals

Publications and Research

Abstract:

We show that a stably ergodic diffeomorphism can be C¹ approximated by a diffeomorphism having stably non-zero Lyapunov exponents.

Résumé:

On montre qu'un difféomorphisme stablement ergodique peut être C¹ approché par un difféomorphisme ayant des exposants de Lyapunov stablement non-nuls.

Decoupling Of The General Scalar Field Mode And The Solution Space For Bianchi Type I And V Cosmologies Coupled To Perfect Fluid Sources, T. Christodoulakis, Th. Grammenos, Ch. Helias, Panos Kevrekidis, A. Spanou

Panos Kevrekidis

The scalar field degree of freedom in Einstein’s plus matter field equations is decoupled for Bianchi type I and V general cosmological models. The source, apart from the minimally coupled scalar field with arbitrary potential V(Φ), is provided by a perfect fluid obeying a general equation of state p = p(ρ). The resulting ODE is, by an appropriate choice of final time gauge affiliated to the scalar field, reduced to first order, and then the system is completely integrated for arbitrary choices of the potential and the equation of state.

On 3+1 Dimensional Scalar Field Cosmologies, Panos Kevrekidis

Panos Kevrekidis

In this communication, we analyze the case of 3+1 dimensional scalar field cosmologies in the presence, as well as in the absence of spatial curvature, in isotropic, as well as in anisotropic settings. Our results extend those of Hawkins and Lidsey [Phys. Rev. D {\bf 66}, 023523 (2002)], by including the non-flat case. The Ermakov-Pinney methodology is developed in a general form, allowing through the converse results presented herein to use it as a tool for constructing new solutions to the original equations. As an example of this type a special blowup solution recently obtained in Christodoulakis {\it et al.} …

Geometric Field Stability And Normal Field Curvature Of Solution Sets Of Ordinary Differential Equations In Two Variables, Leslie L. Kerns

Theses, Dissertations and Capstones

The classical linearization approach to stability theory determines whether or not a system is stable in the vicinity of its equilibrium points. This classical approach partly depends on the validity of the linear approximation. The definition of stability developed in this article takes a different approach and uses a curvature function to assess the relative locations of solutions within a field of solutions (the underlying solution set of the ODE). The present approach involves calculations that directly yield stability information, without having to enter into the often lengthy eigenvalue-eigenvector method. The present results both complement and are compatible with the …

The Dynamics Of Newton's Method On Cubic Polynomials, Shannon N. Miller

Theses, Dissertations and Capstones

The field of dynamics is itself a huge part of many branches of science, including the motion of the planets and galaxies, changing weather patterns, and the growth and decline of populations. Consider a function f and pick x0 in the domain of f . If we iterate this function around the point x0, then we will have the sequence x0, f (x0), f (f (x0)), f (f (f (x0))), ..., which becomes our dynamical system. We are essentially interested in the end behavior of this system. Do …

Solving Higher Order Dynamic Equations On Time Scales As First Order Systems, Elizabeth R. Duke

Theses, Dissertations and Capstones

Time scales calculus seeks to unite two disparate worlds: that of differential, Newtonian calculus and the difference calculus. As such, in place of differential and difference equations, time scales calculus uses dynamic equations. Many theoretical results have been developed concerning solutions of dynamic equations. However, little work has been done in the arena of developing numerical methods for approximating these solutions. This thesis work takes a first step in obtaining numerical solutions of dynamic equations|a protocol for writing higher-order dynamic equations as systems of first-order equations. This process proves necessary in obtaining numerical solutions of differential equations since the Runge-Kutta …