scholarly journals Dynamics and spectral theory of quasi-periodic Schrödinger-type operators

2016 ◽  
Vol 37 (8) ◽  
pp. 2353-2393 ◽  
Author(s):  
C. A. MARX ◽  
S. JITOMIRSKAYA
Keyword(s):  
Dynamical Systems ◽  
Spectral Theory ◽  
Point Of View ◽  
Schrödinger Type Operators

We survey the theory of quasi-periodic Schrödinger-type operators, focusing on the advances made since the early 2000s by adopting a dynamical systems point of view.

scholarly journals On some kinds of dense orbits in general nonautonomous dynamical systems

2020 ◽  
Vol 7 (1) ◽  
pp. 163-175
Author(s):  
Mehdi Pourbarat
Keyword(s):  
Dynamical Systems ◽  
Initial Conditions ◽  
Point Of View ◽  
Topological Conjugacy ◽  
Topological Transitivity ◽  
Nonautonomous Systems ◽  
Nonautonomous Dynamical Systems ◽  
Sensitive Dependence ◽  
Dense Orbits

AbstractWe study the theory of universality for the nonautonomous dynamical systems from topological point of view related to hypercyclicity. The conditions are provided in a way that Birkhoff transitivity theorem can be extended. In the context of generalized linear nonautonomous systems, we show that either one of the topological transitivity or hypercyclicity give sensitive dependence on initial conditions. Meanwhile, some examples are presented for topological transitivity, hypercyclicity and topological conjugacy.

SURVEY Towards a global view of dynamical systems, for the C1-topology

2011 ◽  
Vol 31 (4) ◽  
pp. 959-993 ◽  
Author(s):  
C. BONATTI
Keyword(s):  
Dynamical Systems ◽  
Compact Manifold ◽  
Global Structure ◽  
Point Of View ◽  
List Type ◽  
Global View ◽  
Local Phenomenon ◽  
Generic Dynamics

AbstractThis paper suggests a program for getting a global view of the dynamics of diffeomorphisms, from the point of view of the C1-topology. More precisely, given any compact manifold M, one splits Diff1(M) into disjoint C1-open regions whose union is C1-dense, and conjectures state that each of these open sets and their complements is characterized by the presence of: •either a robust local phenomenon;•or a global structure forbidding this local phenomenon. Other conjectures state that some of these regions are empty. This set of conjectures draws a global view of the dynamics, putting in evidence the coherence of the numerous recent results on C1-generic dynamics.

scholarly journals Solvability conditions for some non-Fredholm operators

2010 ◽  
Vol 54 (1) ◽  
pp. 249-271 ◽  
Author(s):  
Vitali Vougalter ◽  
Vitaly Volpert
Keyword(s):  
Spectral Theory ◽  
Elliptic Equations ◽  
Scattering Theory ◽  
Adjoint Equation ◽  
Fredholm Operators ◽  
Solvability Conditions ◽  
Non Fredholm Operators ◽  
Schrödinger Type Operators

AbstractWe obtain solvability conditions for some elliptic equations involving non-Fredholm operators with the methods of spectral theory and scattering theory for Schrödinger-type operators. One of the main results of the paper concerns solvability conditions for the equation –Δu+V(x)u–au=fwhere a ≥ 0. The conditions are formulated in terms of orthogonality of the functionfto the solutions of the homogeneous adjoint equation.

scholarly journals On the spectral theory of a tapered rod

2002 ◽  
Vol 132 (3) ◽  
pp. 729-764 ◽  
Author(s):  
C. A. Stuart
Keyword(s):  
Function Space ◽  
Spectral Theory ◽  
Spectral Properties ◽  
Boundary Value ◽  
Point Of View ◽  
Point Boundary ◽  
Cross Sectional ◽  
Mechanical Interpretation ◽  
The One ◽  
Image Position

This paper establishes the main features of the spectral theory for the singular two-point boundary-value problem and which models the buckling of a rod whose cross-sectional area decays to zero at one end. The degree of tapering is related to the rate at which the coefficient A tends to zero as s approaches 0. We say that there is tapering of order p ≥ 0 when A ∈ C([0, 1]) with A(s) > 0 for s ∈ (0, 1] and there is a constant L ∈ (0, ∞) such that lims→0A(s)/sp = L. A rigorous spectral theory involves relating (1)−(3) to the spectrum of a linear operator in a function space and then investigating the spectrum of that operator. We do this in two different (but, as we show, equivalent) settings, each of which is natural from a certain point of view. The main conclusion is that the spectral properties of the problem for tapering of order p = 2 are very different from what occurs for p < 2. For p = 2, there is a non-trivial essential spectrum and possibly no eigenvalues, whereas for p < 2, the whole spectrum consists of a sequence of simple eigenvalues. Establishing the details of this spectral theory is an important step in the study of the corresponding nonlinear model. The first function space that we choose is the one best suited to the mechanical interpretation of the problem and the one that is used for treating the nonlinear problem. However, we relate this formulation in a precise way to the usual L2 setting that is most common when dealing with boundary-value problems.

scholarly journals Coherent structures and chaotic advection in three dimensions

2010 ◽  
Vol 654 ◽  
pp. 1-4 ◽  
Author(s):  
STEPHEN WIGGINS
Keyword(s):  
Dynamical Systems ◽  
Fluid Mechanics ◽  
Coherent Structures ◽  
Three Dimensional ◽  
Point Of View ◽  
Three Dimensions ◽  
Chaotic Advection ◽  
Periodic Flow ◽  
Two Dimensional ◽  
Time Periodic

In the 1980s the incorporation of ideas from dynamical systems theory into theoretical fluid mechanics, reinforced by elegant experiments, fundamentally changed the way in which we view and analyse Lagrangian transport. The majority of work along these lines was restricted to two-dimensional flows and the generalization of the dynamical systems point of view to fully three-dimensional flows has seen less progress. This situation may now change with the work of Pouransari et al. (J. Fluid Mech., this issue, vol. 654, 2010, pp. 5–34) who study transport in a three-dimensional time-periodic flow and show that completely new types of dynamical systems structures and consequently, coherent structures, form a geometrical template governing transport.

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