scholarly journals The Existence and Structure of Rotational Systems in the Circle

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Jayakumar Ramanathan
Keyword(s):  
Dynamical System ◽  
Unit Circle ◽  
Closed Subset ◽  
Structure Theorem ◽  
Universal Cover ◽  
Invariant Sets ◽  
Continuous Transformation ◽  
Standard Orientation ◽  
Rotational Systems ◽  
Explicit Mention

By a rotational system, we mean a closed subset X of the circle, T=R/Z, together with a continuous transformation f:X→X with the requirements that the dynamical system (X,f) be minimal and that f respect the standard orientation of T. We show that infinite rotational systems (X,f), with the property that map f has finite preimages, are extensions of irrational rotations of the circle. Such systems have been studied when they arise as invariant subsets of certain specific mappings, F:T→T. Because our main result makes no explicit mention of a global transformation on T, we show that such a structure theorem holds for rotational systems that arise as invariant sets of any continuous transformation F:T→T with finite preimages. In particular, there are no explicit conditions on the degree of F. We then give a development of known results in the case where Fθ=d·θmod⁡1 for an integer d>1. The paper concludes with a construction of infinite rotational sets for mappings of the unit circle of degree larger than one whose lift to the universal cover is monotonic.

scholarly journals Personalized Immunotherapy Treatment Strategies for a Dynamical System of Chronic Myelogenous Leukemia

Cancers ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 2030
Author(s):  
Paul A. Valle ◽  
Luis N. Coria ◽  
Corina Plata
Keyword(s):  
Dynamical System ◽  
Cancer Cells ◽  
Chronic Myelogenous Leukemia ◽  
Sufficient Conditions ◽  
Treatment Strategies ◽  
Myelogenous Leukemia ◽  
Invariant Sets ◽  
Cellular Immunotherapy ◽  
Complete Eradication ◽  
Treatment Parameter

This paper is devoted to exploring personalized applications of cellular immunotherapy as a control strategy for the treatment of chronic myelogenous leukemia described by a dynamical system of three first-order ordinary differential equations. The latter was achieved by applying both the Localization of Compact Invariant Sets and Lyapunov’s stability theory. Combination of these two approaches allows us to establish sufficient conditions on the immunotherapy treatment parameter to ensure the complete eradication of the leukemia cancer cells. These conditions are given in terms of the system parameters and by performing several in silico experimentations, we formulated a protocol for the therapy application that completely eradicates the leukemia cancer cells population for different initial tumour concentrations. The formulated protocol does not dangerously increase the effector T cells population. Further, complete eradication is considered when solutions go below a finite critical value below which cancer cells cannot longer persist; i.e., one cancer cell. Numerical simulations are consistent with our analytical results.

Sets of determination and kernels of certain associated operators

2010 ◽  
Vol 53 (2) ◽  
pp. 503-510
Author(s):  
Arne Stray
Keyword(s):  
Unit Circle ◽  
Unit Disc ◽  
Closed Subset ◽  
Bounded Function ◽  
Harmonic Extension ◽  
Linear Map ◽  
Associated Operators

AbstractLet m be a measure supported on a relatively closed subset X of the unit disc. If f is a bounded function on the unit circle, let fm denote the restriction to X of the harmonic extension of f to the unit disc. We characterize those m such that the pre-adjoint of the linear map f → fm has a non-trivial kernel.

AN ERGODIC THEORY OF CONSCIOUSNESS

2009 ◽  
Vol 19 (04) ◽  
pp. 1397-1400 ◽  
Author(s):  
ABRAHAM BOYARSKY ◽  
PAWEŁ GÓRA
Keyword(s):  
Dynamical System ◽  
Ergodic Theory ◽  
Experimental Work ◽  
Conscious State ◽  
Theoretical Framework ◽  
System Model ◽  
Invariant Sets

It has been suggested that the properties of "integration" and "differentiation" are necessary for the emergence of consciousness. We present a dynamical system model that is based on these two conditions. The collection of neurons are partitioned into clusters on which we define a map that reflects the communication between clusters. Such a map displays the forward and backward circuitry between clusters in a probabilistic manner. The presence of "re-entry" guarantees that the map is sufficiently complex, that is, nonlinear and chaotic, to possess numerous invariant sets of clusters, which are referred to as agglomerations. We suggest that an agglomeration that is mixing characterizes a conscious state. The model establishes a theoretical framework that may structure and encourage experimental work.

scholarly journals Uniform Convergence of Cesaro Averages for Uniquely Ergodic C*-Dynamical Systems

Entropy ◽  
2018 ◽  
Vol 20 (12) ◽  
pp. 987 ◽  
Author(s):  
Francesco Fidaleo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Uniform Convergence ◽  
Unit Circle ◽  
Ergodic Theorem ◽  
Ergodic Theory ◽  
C Algebra ◽  
Cesaro Averages ◽  
Uniformly Convergent ◽  
Uniquely Ergodic

Consider a uniquely ergodic C * -dynamical system based on a unital *-endomorphism Φ of a C * -algebra. We prove the uniform convergence of Cesaro averages 1 n ∑ k = 0 n − 1 λ − n Φ ( a ) for all values λ in the unit circle, which are not eigenvalues corresponding to “measurable non-continuous” eigenfunctions. This result generalizes an analogous one, known in commutative ergodic theory, which turns out to be a combination of the Wiener–Wintner theorem and the uniformly convergent ergodic theorem of Krylov and Bogolioubov.

scholarly journals On the notion of the dimension with respect to a dynamical system

1984 ◽  
Vol 4 (3) ◽  
pp. 405-420 ◽  
Author(s):  
Ya. B. Pesin
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Invariant Sets ◽  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Common Features ◽  
Dynamical Properties ◽  
The One ◽  
Hyperbolic Sets

AbstractFor the invariant sets of dynamical systems a new notion of dimension-the so-called dimension with respect to a dynamical system-is introduced. It has some common features with the general topological notion of the dimension, but it also reflects the dynamical properties of the system. In the one-dimensional case it coincides with the Hausdorff dimension. For multi-dimensional hyperbolic sets formulae for the calculation of our dimension are obtained. These results are generalizations of Manning's results obtained by him for the Hausdorff dimension in the two-dimensional case.

scholarly journals Non-autonomous invariant sets and attractors: Random dynamical system

2020 ◽  
Vol 25 (4) ◽  
pp. 17-23
Author(s):  
Mohamedsh Imran ◽  
Ihsan Jabbar Kadhim
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Global Attractors ◽  
Random Dynamical Systems ◽  
Invariant Sets ◽  
Random Dynamical System ◽  
Pullback Attractor ◽  
Pullback Attractors ◽  
Deterministic Dynamical System

 In this paper the concepts of pullback attractor ,pullback absorbing family in (deterministic) dynamical system are defined in (random) dynamical systems. Also some main result such as (existence) of pullback attractors ,upper semi-continuous of pullback attractors and uniform and global attractors are proved in random dynamical system .

Isomorphism Classes of Solenoidal Algebras I

1993 ◽  
Vol 36 (4) ◽  
pp. 414-418 ◽  
Author(s):  
Berndt Brenken
Keyword(s):  
Dynamical System ◽  
Growth Rate ◽  
Unit Circle ◽  
Compact Space ◽  
Periodic Points ◽  
Crossed Product ◽  
Roots Of Unity ◽  
Isomorphism Classes ◽  
Crossed Product Algebra ◽  
Product Algebra

AbstractEach g ∊ ℤ[x] defines a homeomorphism of a compact space We investigate the isomorphism classes of the C*-crossed product algebra Bg associated with the dynamical system An isomorphism invariant Eg of the algebra Bg is shown to determine the algebra Bg up to * or * anti-isomorphism if degree g ≤ 1 and 1 is not a root of g or if degree g = 2 and g is irreducible. It is also observed that the entropy of the dynamical system is equal to the growth rate of the periodic points if g has no roots of unity as zeros. This slightly extends the previously known equality of these two quantities under the assumption that g has no zeros on the unit circle.

scholarly journals Bounded Motions of the Dynamical Systems Described by Differential Inclusions

2009 ◽  
Vol 2009 ◽  
pp. 1-9
Author(s):  
Nihal Ege ◽  
Khalik G. Guseinov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Sufficient Conditions ◽  
Invariant Sets ◽  
Control Vector ◽  
Upper Semicontinuous ◽  
Right Hand ◽  
The Right

The boundedness of the motions of the dynamical system described by a differential inclusion with control vector is studied. It is assumed that the right-hand side of the differential inclusion is upper semicontinuous. Using positionally weakly invariant sets, sufficient conditions for boundedness of the motions of a dynamical system are given. These conditions have infinitesimal form and are expressed by the Hamiltonian of the dynamical system.

scholarly journals Dynamical informational structures characterize the different human brain states of wakefulness and deep sleep

2019 ◽  
Author(s):  
J. A. Galadí ◽  
S. Silva Pereira ◽  
Y. S. Perl ◽  
M.L. Kringelbach ◽  
I. Gayte ◽  
...  
Keyword(s):  
Dynamical System ◽  
Human Brain ◽  
Directed Graphs ◽  
Global Attractors ◽  
Mathematical Formalism ◽  
Invariant Sets ◽  
Time Variability ◽  
Brain State ◽  
Deep Sleep ◽  
Brain States

ABSTRACTThe dynamical activity of the human brain describes an extremely complex energy landscape changing over time and its characterisation is central unsolved problem in neuroscience. We propose a novel mathematical formalism for characterizing how the landscape of attractors sustained by a dynamical system evolves in time. This mathematical formalism is used to distinguish quantitatively and rigorously between the different human brain states of wakefulness and deep sleep. In particular, by using a whole-brain dynamical ansatz integrating the underlying anatomical structure with the local node dynamics based on a Lotka-Volterra description, we compute analytically the global attractors of this cooperative system and their associated directed graphs, here called the informational structures. The informational structure of the global attractor of a dynamical system describes precisely the past and future behaviour in terms of a directed graph composed of invariant sets (nodes) and their corresponding connections (links). We characterize a brain state by the time variability of these informational structures. This theoretical framework is potentially highly relevant for developing reliable biomarkers of patients with e.g. neuropsychiatric disorders or different levels of coma.

scholarly journals Linking Combinatorial and Classical Dynamics: Conley Index and Morse Decompositions

2020 ◽  
Vol 20 (5) ◽  
pp. 967-1012
Author(s):  
Bogdan Batko ◽  
Tomasz Kaczynski ◽  
Marian Mrozek ◽  
Thomas Wanner
Keyword(s):  
Dynamical System ◽  
Vector Field ◽  
Simplicial Complex ◽  
Geometric Realization ◽  
Invariant Sets ◽  
Multivalued Map ◽  
Classical Dynamics ◽  
Morse Decompositions ◽  
Multivalued Dynamical System ◽  
Combinatorial Vector Field

Abstract We prove that every combinatorial dynamical system in the sense of Forman, defined on a family of simplices of a simplicial complex, gives rise to a multivalued dynamical system F on the geometric realization of the simplicial complex. Moreover, F may be chosen in such a way that the isolated invariant sets, Conley indices, Morse decompositions and Conley–Morse graphs of the combinatorial vector field give rise to isomorphic objects in the multivalued map case.

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