ON THE SEPARATRIS LOOP BIFURCATIONS OF TWO-DIMENSIONAL PIECEWISE-SMOOTH DYNAMIC SYSTEM
Background. Bifurcations in generic one- and two-parameter families of smooth dynamical systems on the plane are almost completely studied. For applications, piecewise-smooth dynamical systems in the plane are of considerable interest. There are much more different types of bifurcations for them...
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| Format: | Article |
| Language: | English |
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Penza State University Publishing House
2020-03-01
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| Series: | Известия высших учебных заведений. Поволжский регион: Физико-математические науки |
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| _version_ | 1818502157752598528 |
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| author | V. Sh. Roytenberg |
| author_facet | V. Sh. Roytenberg |
| author_sort | V. Sh. Roytenberg |
| collection | DOAJ |
| description | Background. Bifurcations in generic one- and two-parameter families of smooth
dynamical systems on the plane are almost completely studied. For applications,
piecewise-smooth dynamical systems in the plane are of considerable interest. There
are much more different types of bifurcations for them than for smooth dynamical
systems. Some of them are already described. However, the continuation of the
study of bifurcations in generic two-parameter families of two-dimensional piecewise-
smooth dynamical systems seems to be still relevant.
Materials and methods. We use methods of the qualitative theory of differential
equations.
Results. We consider a two-dimensional piecewise smooth vector field X. Let S
be a point on the line of discontinuity of the field, and in its semi-neighborhoods V1
and V2 the field coincides with smooth vector fields, respectively, Х1 and Х2. For
the field Х1, the point S is a saddle with nonzero saddle value, whose invariant
manifolds are transversal to the line of discontinuity. At the point S vector field Х2
is transversal to the line of discontinuity and directed inwards V1. The outgoing and
incoming separatrixes of the saddle S that start at V1 do not contain singular points
and form a loop together with S. For generic two-parameter deformations of the
considered vector fields in the neighborhood of the loop, bifurcation diagrams are
obtained.
Conclusions. Bifurcations of the separatrix loop of singular point on the line of
discontinuity of the vector field are described. |
| first_indexed | 2024-12-10T21:06:02Z |
| format | Article |
| id | doaj.art-653e09f466bb41169dfce5de7fec381e |
| institution | Directory Open Access Journal |
| issn | 2072-3040 |
| language | English |
| last_indexed | 2024-12-10T21:06:02Z |
| publishDate | 2020-03-01 |
| publisher | Penza State University Publishing House |
| record_format | Article |
| series | Известия высших учебных заведений. Поволжский регион: Физико-математические науки |
| spelling | doaj.art-653e09f466bb41169dfce5de7fec381e2022-12-22T01:33:38ZengPenza State University Publishing HouseИзвестия высших учебных заведений. Поволжский регион: Физико-математические науки2072-30402020-03-01110.21685/2072-3040-2020-1-3ON THE SEPARATRIS LOOP BIFURCATIONS OF TWO-DIMENSIONAL PIECEWISE-SMOOTH DYNAMIC SYSTEMV. Sh. Roytenberg0Yaroslavl State Technical UniversityBackground. Bifurcations in generic one- and two-parameter families of smooth dynamical systems on the plane are almost completely studied. For applications, piecewise-smooth dynamical systems in the plane are of considerable interest. There are much more different types of bifurcations for them than for smooth dynamical systems. Some of them are already described. However, the continuation of the study of bifurcations in generic two-parameter families of two-dimensional piecewise- smooth dynamical systems seems to be still relevant. Materials and methods. We use methods of the qualitative theory of differential equations. Results. We consider a two-dimensional piecewise smooth vector field X. Let S be a point on the line of discontinuity of the field, and in its semi-neighborhoods V1 and V2 the field coincides with smooth vector fields, respectively, Х1 and Х2. For the field Х1, the point S is a saddle with nonzero saddle value, whose invariant manifolds are transversal to the line of discontinuity. At the point S vector field Х2 is transversal to the line of discontinuity and directed inwards V1. The outgoing and incoming separatrixes of the saddle S that start at V1 do not contain singular points and form a loop together with S. For generic two-parameter deformations of the considered vector fields in the neighborhood of the loop, bifurcation diagrams are obtained. Conclusions. Bifurcations of the separatrix loop of singular point on the line of discontinuity of the vector field are described.dynamical systempiecewise-smooth vector fieldseparatrix loopbifurcationsbifurcation diagramperiodic trajectory |
| spellingShingle | V. Sh. Roytenberg ON THE SEPARATRIS LOOP BIFURCATIONS OF TWO-DIMENSIONAL PIECEWISE-SMOOTH DYNAMIC SYSTEM Известия высших учебных заведений. Поволжский регион: Физико-математические науки dynamical system piecewise-smooth vector field separatrix loop bifurcations bifurcation diagram periodic trajectory |
| title | ON THE SEPARATRIS LOOP BIFURCATIONS OF TWO-DIMENSIONAL PIECEWISE-SMOOTH DYNAMIC SYSTEM |
| title_full | ON THE SEPARATRIS LOOP BIFURCATIONS OF TWO-DIMENSIONAL PIECEWISE-SMOOTH DYNAMIC SYSTEM |
| title_fullStr | ON THE SEPARATRIS LOOP BIFURCATIONS OF TWO-DIMENSIONAL PIECEWISE-SMOOTH DYNAMIC SYSTEM |
| title_full_unstemmed | ON THE SEPARATRIS LOOP BIFURCATIONS OF TWO-DIMENSIONAL PIECEWISE-SMOOTH DYNAMIC SYSTEM |
| title_short | ON THE SEPARATRIS LOOP BIFURCATIONS OF TWO-DIMENSIONAL PIECEWISE-SMOOTH DYNAMIC SYSTEM |
| title_sort | on the separatris loop bifurcations of two dimensional piecewise smooth dynamic system |
| topic | dynamical system piecewise-smooth vector field separatrix loop bifurcations bifurcation diagram periodic trajectory |
| work_keys_str_mv | AT vshroytenberg ontheseparatrisloopbifurcationsoftwodimensionalpiecewisesmoothdynamicsystem |