Regularization and minimal sets for non-smooth dynamical systems

2015 ◽  
Author(s):  
Douglas Duarte Novaes
Keyword(s):  
Dynamical Systems ◽  
Minimal Sets

Method for constructing minimal sets of dynamical systems

2015 ◽  
Vol 51 (7) ◽  
pp. 831-837 ◽  
Author(s):  
A. P. Afanas’ev ◽  
S. M. Dzyuba
Keyword(s):  
Dynamical Systems ◽  
Minimal Sets

On Recurrent Trajectories, Minimal Sets, and Quasiperiodic Motions of Dynamical Systems

2005 ◽  
Vol 41 (11) ◽  
pp. 1544-1549 ◽  
Author(s):  
A. P. Afanas'ev ◽  
S. M. Dzyuba
Keyword(s):  
Dynamical Systems ◽  
Minimal Sets

ON MINIMAL SETS IN DYNAMICAL SYSTEMS

1956 ◽  
Vol 7 (1) ◽  
pp. 5-16 ◽  
Author(s):  
YAEL NAIM DOWKER
Keyword(s):  
Dynamical Systems ◽  
Minimal Sets

About new properties of recurrent motions and minimal sets of dynamical systems

2021 ◽  
pp. 5-14
Author(s):  
Aleksandr P. Afanas’ev ◽  
Sergei M. Dzyuba
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Metric Space ◽  
Short Proof ◽  
Compact Metric Space ◽  
Limit Sets ◽  
Minimal Sets ◽  
Transmission Problems ◽  
Academy Of Sciences ◽  
First Time

The article presents a new property of recurrent motions of dynamical systems. Within a compact metric space, this property establishes the relation between motions of general type and recurrent motions. Besides, this property establishes rather simple behaviour of recurrent motions, thus naturally corroborating the classical definition given in the monograph [V.V. Nemytskii, V.V. Stepanov. Qualitative Theory of Differential Equations. URSS Publ., Moscow, 2004 (In Russian)]. Actually, the above-stated new property of recurrent motions was announced, for the first time, in the earlier article by the same authors [A.P. Afanas’ev, S. M. Dzyuba. On recurrent trajectories, minimal sets, and quasiperoidic motions of dynamical systems // Differential Equations. 2005, v. 41, № 11, p. 1544–1549]. The very same article provides a short proof for the corresponding theorem. The proof in question turned out to be too schematic. Moreover, it (the proof) includes a range of obvious gaps. Some time ago it was found that, on the basis of this new property, it is possible to show that within a compact metric space α- and ω-limit sets of each and every motion are minimal. Therefore, within a compact metric space each and every motion, which is positively (negatively) stable in the sense of Poisson, is recurrent. Those results are of obvious significance. They clearly show the reason why, at present, there are no criteria for existence of non-recurrent motions stable in the sense of Poisson. Moreover, those results show the reason why the existing attempts of creating non-recurrent motions, stable in the sense of Poisson, on compact closed manifolds turned out to be futile. At least, there are no examples of such motions. The key point of the new property of minimal sets is the stated new property of recurrent motions. That is why here, in our present article, we provide a full and detailed proof for that latter property. For the first time, the results of the present study were reported on the 28th of January, 2020 at a seminar of Dobrushin Mathematic Laboratory at the Institute for Information Transmission Problems named after A. A. Kharkevich of the Russian Academy of Sciences.

Properties of Dynamical Systems on Dendrites and Graphs

2021 ◽  
Vol 31 (07) ◽  
pp. 2150100
Author(s):  
Zdeněk Kočan ◽  
Veronika Kurková ◽  
Michal Málek
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Metric Spaces ◽  
Limit Set ◽  
Open Problems ◽  
Homoclinic Trajectory ◽  
Compact Metric Spaces ◽  
Minimal Sets ◽  
Continuous Maps ◽  
Omega Limit Set

Dynamical systems generated by continuous maps on compact metric spaces can have various properties, e.g. the existence of an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory, the existence of an omega-limit set containing two minimal sets and other. In [Kočan et al., 2014] we consider six such properties and survey the relations among them for the cases of graph maps, dendrite maps and maps on compact metric spaces. In this paper, we consider fourteen such properties, provide new results and survey all the relations among the properties for the case of graph maps and all known relations for the case of dendrite maps. We formulate some open problems at the end of the paper.

scholarly journals Ultrafilters and non-Cantor minimal sets in linearly ordered dynamical systems

2008 ◽  
Vol 47 (3) ◽  
pp. 193-203
Author(s):  
M. Hrušák ◽  
M. Sanchis ◽  
Á. Tamariz-Mascarúa
Keyword(s):  
Dynamical Systems ◽  
Minimal Sets
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