One-dimensional dynamical systems

2021 ◽  
Vol 76 (5) ◽  
pp. 821-881
Author(s):  
L. S. Efremova ◽  
E. N. Makhrova
Keyword(s):  
Closed Interval ◽  
Sufficient Conditions ◽  
Finite Graph ◽  
One Dimensional ◽  
Finite Graphs ◽  
Continuous Maps ◽  
Periodic Behaviour ◽  
Similarities And Differences ◽  
Necessary And Sufficient ◽  
Sharkovsky's Theorem

Abstract The survey is devoted to the topological dynamics of maps defined on one-dimensional continua such as a closed interval, a circle, finite graphs (for instance, finite trees), or dendrites (locally connected continua without subsets homeomorphic to a circle). Connections between the periodic behaviour of trajectories, the existence of a horseshoe and homoclinic trajectories, and the positivity of topological entropy are investigated. Necessary and sufficient conditions for entropy chaos in continuous maps of an interval, a circle, or a finite graph, and sufficient conditions for entropy chaos in continuous maps of dendrites are presented. Reasons for similarities and differences between the properties of maps defined on the continua under consideration are analyzed. Extensions of Sharkovsky’s theorem to certain discontinuous maps of a line or an interval and continuous maps on a plane are considered. Bibliography: 207 titles.

Matchings in Arbitrary Graphs

1971 ◽  
Vol 69 (3) ◽  
pp. 401-407 ◽  
Author(s):  
R. A. Brualdi
Keyword(s):  
Perfect Matching ◽  
Sufficient Conditions ◽  
Finite Graph ◽  
Maximum Cardinality ◽  
Original Proof ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Arbitrary Graphs ◽  
Necessary And Sufficient

1. Tutte(10) has given necessary and sufficient conditions in order that a finite graph have a perfect matching. A different proof was given by Gallai(4). Berge(1) (and Ore (7)) generalized Tutte's result by determining the maximum cardinality of a matching in a finite graph. In his original proof Tutte used the method of skew symmetric determinants (or pfaffians) while Gallai and Berge used the much exploited method of alternating paths. Another proof of Berge's theorem, along with an efficient algorithm for constructing a matching of maximum cardinality, was given by Edmonds (2). In another paper (12) Tutte extended his conditions for a perfect matching to locally finite graphs.

Matchings in Countable Graphs

1977 ◽  
Vol 29 (1) ◽  
pp. 165-168 ◽  
Author(s):  
K. Steffens
Keyword(s):  
Perfect Matching ◽  
Sufficient Conditions ◽  
Finite Graph ◽  
Necessary And Sufficient Conditions ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Arbitrary Graphs ◽  
Necessary And Sufficient

Tutte [9] has given necessary and sufficient conditions for a finite graph to have a perfect matching. Different proofs are given by Brualdi [1] and Gallai [2; 3]. The shortest proof of Tutte's theorem is due to Lovasz [5]. In another paper [10] Tutte extended his conditions for a perfect matching to locally finite graphs. In [4] Kaluza gave a condition on arbitrary graphs which is entirely different from Tutte's.

On the Necessary and Sufficient Conditions for the Stability of Linear Generalized Ordinary Differential, Linear Impulsive and Linear Difference Systems

2009 ◽  
Vol 16 (4) ◽  
pp. 597-616
Author(s):  
Shota Akhalaia ◽  
Malkhaz Ashordia ◽  
Nestan Kekelia
Keyword(s):  
Linear System ◽  
Difference Equations ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Difference Systems ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Generalized Ordinary Differential Equations ◽  
Lyapunov Sense

Abstract Necessary and sufficient conditions are established for the stability in the Lyapunov sense of solutions of a linear system of generalized ordinary differential equations 𝑑𝑥(𝑡) = 𝑑𝐴(𝑡) · 𝑥(𝑡) + 𝑑𝑓(𝑡), where and are, respectively, matrix- and vector-functions with bounded total variation components on every closed interval from . The results are realized for the linear systems of impulsive, ordinary differential and difference equations.

GENERALIZED DECIMATION INVARIANT SEQUENCES IN DIMENSION N

2002 ◽  
Vol 12 (04) ◽  
pp. 709-737 ◽  
Author(s):  
A. BARBÉ ◽  
F. VON HAESELER
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
One Dimensional ◽  
Necessary And Sufficient

We generalize the concept of one-dimensional decimation invariant sequences, i.e. sequences which are invariant under a specific rescaling, to dimension N. After discussing the elementary properties of decimation-invariant sequences, we focus our interest on their periodicity. Necessary and sufficient conditions for the existence of periodic decimation invariant sequences are presented.

On the threshold strategies in optimal stopping problems for diffusion processes

2017 ◽  
Vol 54 (3) ◽  
pp. 963-969 ◽  
Author(s):  
Vadim Arkin ◽  
Alexander Slastnikov
Keyword(s):  
Diffusion Process ◽  
Optimal Stopping ◽  
Diffusion Processes ◽  
Sufficient Conditions ◽  
Payoff Function ◽  
Threshold Strategy ◽  
One Dimensional ◽  
Stopping Set ◽  
Optimal Stopping Problems ◽  
Necessary And Sufficient

Abstract We study a problem when the optimal stopping for a one-dimensional diffusion process is generated by a threshold strategy. Namely, we give necessary and sufficient conditions (on the diffusion process and the payoff function) under which a stopping set has a threshold structure.

System Properties of One-Dimensional Distributed Systems

1977 ◽  
Vol 99 (2) ◽  
pp. 85-90 ◽  
Author(s):  
L. S. Bonderson
Keyword(s):  
Distributed Systems ◽  
Sufficient Conditions ◽  
State Variables ◽  
One Dimensional ◽  
First Order ◽  
Lumped Element ◽  
Order Form ◽  
Power Product ◽  
System Properties ◽  
Necessary And Sufficient

The system properties of passivity, losslessness, and reciprocity are defined and their necessary and sufficient conditions are derived for a class of linear one-dimensional multipower distributed systems. The utilization of power product pairs as state variables and the representation of the dynamics in first-order form allows results completely analogous to those for lumped-element systems.

scholarly journals The Structure of Locally Finite Two-Connected Graphs

10.37236/1211 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Carl Droms ◽  
Brigitte Servatius ◽  
Herman Servatius
Keyword(s):  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Locally Finite ◽  
Connected Graphs ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Necessary And Sufficient ◽  
Block Tree ◽  
Labeled Tree

We expand on Tutte's theory of $3$-blocks for $2$-connected graphs, generalizing it to apply to infinite, locally finite graphs, and giving necessary and sufficient conditions for a labeled tree to be the $3$-block tree of a $2$-connected graph.

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