METHOD OF TOPOLOGICAL ROUGHNESS OF DYNAMIC SYSTEMS: APPLICATIONS TO SYNERGETIC SYSTEMS
The paper presents a method of dynamic system roughness research, based on Andronov-Pontryagin concept of roughness (method of topological roughness). Andronov-Pontryagin concept of roughness has been formulated. Reachability conditions of dynamic system required roughness are defined. Concept defin...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Saint Petersburg National Research University of Information Technologies, Mechanics and Optics (ITMO University)
2020-04-01
|
| Series: | Naučno-tehničeskij Vestnik Informacionnyh Tehnologij, Mehaniki i Optiki |
| Subjects: | |
| Online Access: | https://ntv.ifmo.ru/file/article/19526.pdf |
| _version_ | 1818126368167165952 |
|---|---|
| author | Roman O. Omorov |
| author_facet | Roman O. Omorov |
| author_sort | Roman O. Omorov |
| collection | DOAJ |
| description | The paper presents a method of dynamic system roughness research, based on Andronov-Pontryagin concept of roughness (method of topological roughness). Andronov-Pontryagin concept of roughness has been formulated. Reachability conditions of dynamic system required roughness are defined. Concept definition for maximum roughness and minimum non-roughness of dynamic systems is given. Theorems on necessary and sufficient conditions of reachability of maximum roughness and minimum non-roughness and occurrence of bifurcations of dynamic system topological structures are formulated. It is claimed that the sets of rough and non-rough systems are continuous in terms
of the set roughness. The condition number of the matrix of bringing to the diagonal (quasi-diagonal) view of the Jacobi matrix at special points of the system phase space is used as an indicator of roughness. The method gives the possibility to control the roughness of control systems based on a theorem formulated using Sylvester’s matrix equation. The basic concepts on synergetics and synergetic systems are presented. The method can be used for studies of roughness and
bifurcations of dynamic systems, as well as synergetic systems and chaos of various physical nature. The method is tested on the examples of many synergetic systems: Lorenz and Rössler, Belousov-Zhabotinsky, Chua, “predator-prey”, Henon, and Hopf bifurcation. The main provisions of the topological roughness method are given. The possibilities of the method are illustrated by examples of Belousov-Zhabotinsky and Chua synergetic systems. |
| first_indexed | 2024-12-11T07:00:21Z |
| format | Article |
| id | doaj.art-9bfc49834e014eb4a6b0d6d67784d5d2 |
| institution | Directory Open Access Journal |
| issn | 2226-1494 2500-0373 |
| language | English |
| last_indexed | 2024-12-11T07:00:21Z |
| publishDate | 2020-04-01 |
| publisher | Saint Petersburg National Research University of Information Technologies, Mechanics and Optics (ITMO University) |
| record_format | Article |
| series | Naučno-tehničeskij Vestnik Informacionnyh Tehnologij, Mehaniki i Optiki |
| spelling | doaj.art-9bfc49834e014eb4a6b0d6d67784d5d22022-12-22T01:16:38ZengSaint Petersburg National Research University of Information Technologies, Mechanics and Optics (ITMO University)Naučno-tehničeskij Vestnik Informacionnyh Tehnologij, Mehaniki i Optiki2226-14942500-03732020-04-0120225726210.17586/2226-1494-2020-20-2-257-262METHOD OF TOPOLOGICAL ROUGHNESS OF DYNAMIC SYSTEMS: APPLICATIONS TO SYNERGETIC SYSTEMSRoman O. Omorov0https://orcid.org/0000-0003-3555-1323D.Sc., Professor, Corresponding member of National Academy of Sciences of the Kyrgyz Republic (KR), Chief Researcher, Institute of Physics of National Academy of Sciences KR, Bishkek, 720071, Kyrgyz RepublicThe paper presents a method of dynamic system roughness research, based on Andronov-Pontryagin concept of roughness (method of topological roughness). Andronov-Pontryagin concept of roughness has been formulated. Reachability conditions of dynamic system required roughness are defined. Concept definition for maximum roughness and minimum non-roughness of dynamic systems is given. Theorems on necessary and sufficient conditions of reachability of maximum roughness and minimum non-roughness and occurrence of bifurcations of dynamic system topological structures are formulated. It is claimed that the sets of rough and non-rough systems are continuous in terms of the set roughness. The condition number of the matrix of bringing to the diagonal (quasi-diagonal) view of the Jacobi matrix at special points of the system phase space is used as an indicator of roughness. The method gives the possibility to control the roughness of control systems based on a theorem formulated using Sylvester’s matrix equation. The basic concepts on synergetics and synergetic systems are presented. The method can be used for studies of roughness and bifurcations of dynamic systems, as well as synergetic systems and chaos of various physical nature. The method is tested on the examples of many synergetic systems: Lorenz and Rössler, Belousov-Zhabotinsky, Chua, “predator-prey”, Henon, and Hopf bifurcation. The main provisions of the topological roughness method are given. The possibilities of the method are illustrated by examples of Belousov-Zhabotinsky and Chua synergetic systems.https://ntv.ifmo.ru/file/article/19526.pdfdynamic systemtopological roughnesssynergetic systemandronov-pontryagin roughnessbifurcationmaximum roughness and minimum non-roughness of systems |
| spellingShingle | Roman O. Omorov METHOD OF TOPOLOGICAL ROUGHNESS OF DYNAMIC SYSTEMS: APPLICATIONS TO SYNERGETIC SYSTEMS Naučno-tehničeskij Vestnik Informacionnyh Tehnologij, Mehaniki i Optiki dynamic system topological roughness synergetic system andronov-pontryagin roughness bifurcation maximum roughness and minimum non-roughness of systems |
| title | METHOD OF TOPOLOGICAL ROUGHNESS OF DYNAMIC SYSTEMS: APPLICATIONS TO SYNERGETIC SYSTEMS |
| title_full | METHOD OF TOPOLOGICAL ROUGHNESS OF DYNAMIC SYSTEMS: APPLICATIONS TO SYNERGETIC SYSTEMS |
| title_fullStr | METHOD OF TOPOLOGICAL ROUGHNESS OF DYNAMIC SYSTEMS: APPLICATIONS TO SYNERGETIC SYSTEMS |
| title_full_unstemmed | METHOD OF TOPOLOGICAL ROUGHNESS OF DYNAMIC SYSTEMS: APPLICATIONS TO SYNERGETIC SYSTEMS |
| title_short | METHOD OF TOPOLOGICAL ROUGHNESS OF DYNAMIC SYSTEMS: APPLICATIONS TO SYNERGETIC SYSTEMS |
| title_sort | method of topological roughness of dynamic systems applications to synergetic systems |
| topic | dynamic system topological roughness synergetic system andronov-pontryagin roughness bifurcation maximum roughness and minimum non-roughness of systems |
| url | https://ntv.ifmo.ru/file/article/19526.pdf |
| work_keys_str_mv | AT romanoomorov methodoftopologicalroughnessofdynamicsystemsapplicationstosynergeticsystems |