scholarly journals Flows with collective dynamics on a sphere

2021 ◽  
Vol 14 (1) ◽  
pp. 60-79
Author(s):  
Андрій Прус ◽  
Олександр Пришляк ◽  
Софія Гурака
Keyword(s):  
Topological Invariant ◽  
Collective Dynamics ◽  
Compact Surfaces

In this article different properties of flow codes are studied and a diagram is constructed as a whole topological invariant of them. In particular, flows with no more than 6 saddles are described. Two types of simple bifurcations: positive and negative – are considered as well. Summarizing the results on compact surfaces with boundary remains an interesting question for future works.  

scholarly journals Topology of optimal flows with collective dynamics on closed orientable surfaces

2020 ◽  
Vol 13 (2) ◽  
pp. 50-67
Author(s):  
Alexandr Olegovich Prishlyak ◽  
Mariya Viktorovna Loseva
Keyword(s):  
Morse Function ◽  
Hamiltonian Dynamics ◽  
Closed Surface ◽  
Topological Invariant ◽  
The Other ◽  
Heteroclinic Cycles ◽  
Collective Dynamics ◽  
Gradient Dynamics ◽  
Orientable Surfaces

We consider flows on a closed surface with one or more heteroclinic cycles that divide the surface into two regions. One of the region has gradient dynamics, like Morse fields. The other region has Hamiltonian dynamics generated by the field of the skew gradient of the simple Morse function. We construct the complete topological invariant of the flow using the Reeb and Oshemkov-Shark graphs and study its properties. We describe all possible structures of optimal flows with collective dynamics on oriented surfaces of genus no more than 2, both for flows containing a center and for flows without it.

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