Classification of Hidden Dynamics in Discontinuous Dynamical Systems

2015 ◽  
Vol 14 (3) ◽  
pp. 1454-1477 ◽  
Author(s):  
Nicola Guglielmi ◽  
Ernst Hairer
Keyword(s):  
Dynamical Systems

Classification of AF Flows

2003 ◽  
Vol 46 (2) ◽  
pp. 164-177 ◽  
Author(s):  
Andrew J. Dean
Keyword(s):  
Dynamical Systems ◽  
Automorphism Group ◽  
Bratteli Diagram ◽  
Finite Dimensional

AbstractAn AF flow is a one-parameter automorphism group of an AF C*-algebra A such that there exists an increasing sequence of invariant finite dimensional sub-C*-algebras whose union is dense in A. In this paper, a classification of C*-dynamical systems of this form up to equivariant isomorphism is presented. Two pictures of the actions are given, one in terms of a modified Bratteli diagram/pathspace construction, and one in terms of a modified K0 functor.

scholarly journals Transient phenomena in ecology

Science ◽  
2018 ◽  
Vol 361 (6406) ◽  
pp. eaat6412 ◽  
Author(s):  
Alan Hastings ◽  
Karen C. Abbott ◽  
Kim Cuddington ◽  
Tessa Francis ◽  
Gabriel Gellner ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Systems Theory ◽  
Ecological Systems ◽  
Dynamical Systems Theory ◽  
Model Systems ◽  
Transient Dynamics ◽  
Transient Phenomena ◽  
Ecological Theories

The importance of transient dynamics in ecological systems and in the models that describe them has become increasingly recognized. However, previous work has typically treated each instance of these dynamics separately. We review both empirical examples and model systems, and outline a classification of transient dynamics based on ideas and concepts from dynamical systems theory. This classification provides ways to understand the likelihood of transients for particular systems, and to guide investigations to determine the timing of sudden switches in dynamics and other characteristics of transients. Implications for both management and underlying ecological theories emerge.

Reduction of Arbitrary Dynamical Systems by Wavelet Bases

2001 ◽  
Author(s):  
Roger Ghanem ◽  
Francesco Romeo
Keyword(s):  
Dynamical Systems ◽  
Reduction Process ◽  
Inner Product ◽  
Scaling Functions ◽  
Time Varying ◽  
Governing Equations ◽  
Wavelet Bases ◽  
Input And Output ◽  
Analytical Expressions

Abstract A procedure is developed for the identification and classification of nonlinear and time-varying dynamical systems based on measurements of their input and output. The procedure consists of reducing the governing equations with respect to a basis of scaling functions. Given the localizing properties of wavelets, the reduced system is well adapted to predicting local changes in time as well as changes that are localized to particular components of the system. The reduction process relies on traditional Galerkin techniques and recent analytical expressions for evaluating the inner product between scaling functions and their derivatives. Examples from a variety of dynamical systems are used to demonstrate the scope and limitations of the proposed method.

scholarly journals UHF-slicing and classification of nuclear C*-algebras

2014 ◽  
Vol 06 (04) ◽  
pp. 465-540 ◽  
Author(s):  
Karen R. Strung ◽  
Wilhelm Winter
Keyword(s):  
Dynamical Systems ◽  
Classification Theorem ◽  
Discrete Version ◽  
C Algebras ◽  
Minimal Dynamical Systems ◽  
Tracial States

In this paper we show that certain simple locally recursive subhomogeneous (RSH) C*-algebras are tracially approximately interval algebras after tensoring with the universal UHF algebra. This involves a linear algebraic encoding of the structure of the local RSH algebra allowing us to find a path through the algebra which looks like a discrete version of [0, 1] and exhausts most of the algebra. We produce an actual copy of the interval and use properties of C*-algebras tensored with UHF algebras to move the honest interval underneath the discrete version. It follows from our main result that such C*-algebras are classifiable by Elliott invariants. Our theorem requires finitely many tracial states that all induce the same state on the K0-group; in particular we do not require that projections separate tracial states. We apply our results to classify some examples of C*-algebras constructed by Elliott to exhaust the invariant. We also give an alternative way to classify examples of Lin and Matui of C*-algebras of minimal dynamical systems. In this way our result can be viewed as a first step towards removing the requirement that projections separate tracial states in the classification theorem for C*-algebras of minimal dynamical systems given by Toms and the second named author.

scholarly journals Classification of perturbations of the hydrogen atom by small static electric and magnetic fields

2007 ◽  
Vol 463 (2083) ◽  
pp. 1771-1790 ◽  
Author(s):  
K Efstathiou ◽  
D.A Sadovskií ◽  
B.I Zhilinskií
Keyword(s):  
Dynamical Systems ◽  
Hydrogen Atom ◽  
Magnetic Fields ◽  
Parameter Space ◽  
Field Limit ◽  
Electric And Magnetic Fields ◽  
Orthogonal Field

We consider perturbations of the hydrogen atom by sufficiently small homogeneous static electric and magnetic fields of all possible mutual orientations. Normalizing with regard to the Keplerian symmetry, we uncover resonances and conjecture that the parameter space of this family of dynamical systems is stratified into zones centred on the resonances. The 1 : 1 resonance corresponds to the orthogonal field limit, studied earlier by Cushman & Sadovskií (Cushman & Sadovskií 2000 Physica 142 , 166–196). We describe the structure of the 1 : 1 zone, where the system may have monodromy of different kinds, and consider briefly the 1 : 2 zone.

Cellular Automata

2017 ◽  
Vol 29 (1) ◽  
pp. 42-50 ◽  
Author(s):  
Rupali Bhardwaj ◽  
Anil Upadhyay
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Empirical Study ◽  
Time Evolution ◽  
Discrete Dynamical Systems ◽  
The State ◽  
Dimensional Lattice ◽  
Elementary Cellular Automata

Cellular automata (CA) are discrete dynamical systems consist of a regular finite grid of cell; each cell encapsulating an equal portion of the state, and arranged spatially in a regular fashion to form an n-dimensional lattice. A cellular automata is like computers, data represented by initial configurations which is processed by time evolution to produce output. This paper is an empirical study of elementary cellular automata which includes concepts of rule equivalence, evolution of cellular automata and classification of cellular automata. In addition, explanation of behaviour of cellular automata is revealed through example.

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