On the bifurcations of two periodic trajectories of a piecewise-smooth dynamical system with central symmetry
Background. A large number of scientific works are devoted to the description of bifurcations in generic families of piecewise-smooth dynamical systems on the plane. Although dynamical systems with symmetry are often encountered in applied problems, the bifurcations of piecewise-smooth systems wi...
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| Format: | Article |
| Language: | English |
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Penza State University Publishing House
2022-01-01
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| Series: | Известия высших учебных заведений. Поволжский регион: Физико-математические науки |
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| _version_ | 1811303512273846272 |
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| author | V.Sh. Roytenberg |
| author_facet | V.Sh. Roytenberg |
| author_sort | V.Sh. Roytenberg |
| collection | DOAJ |
| description | Background. A large number of scientific works are devoted to the description of
bifurcations in generic families of piecewise-smooth dynamical systems on the plane. Although
dynamical systems with symmetry are often encountered in applied problems, the bifurcations
of piecewise-smooth systems with symmetry have been studied not enough.
Therefore, the consideration of bifurcations in generic families of such dynamical systems
is of undoubted interest. Materials and methods. The methods of the qualitative theory of
differential equations are applied. The behavior of the Poincare mappings and the corresponding
divergence functions for different values of the parameters is investigated. We
used estimates for the derivatives of local correspondence functions along trajectories at the
points of tangency of the trajectories with the line of discontinuity of the vector field. Results.
We consider a piecewise-smooth vector field X on a plane, “sewn” from smooth vector
fields specified in the upper and lower half-planes, respectively, and having periodic trajectories
tangent to the x-axis, which is invariant under symmetry transformation about the
origin. The bouquet Г, composed of the indicated periodic trajectories, is a periodic trajectory
of the field X. For a two-parameter family in general position, which is a deformation
of the field X in the space of piecewise-smooth vector fields with central symmetry, bifurcations
are described in a neighborhood U of the contour Г. The bifurcation diagram is obtained
– a partition of a neighborhood of zero in the parameter planes into topological
equivalence classes in U of vector fields of the family. Conclusions. Generic two-parameter
bifurcations in a neighborhood of the considered bouquet of periodic trajectories are described. |
| first_indexed | 2024-04-13T07:48:11Z |
| format | Article |
| id | doaj.art-808ea294f5c24f219e5252150eb42d2c |
| institution | Directory Open Access Journal |
| issn | 2072-3040 |
| language | English |
| last_indexed | 2024-04-13T07:48:11Z |
| publishDate | 2022-01-01 |
| publisher | Penza State University Publishing House |
| record_format | Article |
| series | Известия высших учебных заведений. Поволжский регион: Физико-математические науки |
| spelling | doaj.art-808ea294f5c24f219e5252150eb42d2c2022-12-22T02:55:37ZengPenza State University Publishing HouseИзвестия высших учебных заведений. Поволжский регион: Физико-математические науки2072-30402022-01-01410.21685/2072-3040-2021-4-1On the bifurcations of two periodic trajectories of a piecewise-smooth dynamical system with central symmetryV.Sh. Roytenberg0Yaroslavl State Technical UniversityBackground. A large number of scientific works are devoted to the description of bifurcations in generic families of piecewise-smooth dynamical systems on the plane. Although dynamical systems with symmetry are often encountered in applied problems, the bifurcations of piecewise-smooth systems with symmetry have been studied not enough. Therefore, the consideration of bifurcations in generic families of such dynamical systems is of undoubted interest. Materials and methods. The methods of the qualitative theory of differential equations are applied. The behavior of the Poincare mappings and the corresponding divergence functions for different values of the parameters is investigated. We used estimates for the derivatives of local correspondence functions along trajectories at the points of tangency of the trajectories with the line of discontinuity of the vector field. Results. We consider a piecewise-smooth vector field X on a plane, “sewn” from smooth vector fields specified in the upper and lower half-planes, respectively, and having periodic trajectories tangent to the x-axis, which is invariant under symmetry transformation about the origin. The bouquet Г, composed of the indicated periodic trajectories, is a periodic trajectory of the field X. For a two-parameter family in general position, which is a deformation of the field X in the space of piecewise-smooth vector fields with central symmetry, bifurcations are described in a neighborhood U of the contour Г. The bifurcation diagram is obtained – a partition of a neighborhood of zero in the parameter planes into topological equivalence classes in U of vector fields of the family. Conclusions. Generic two-parameter bifurcations in a neighborhood of the considered bouquet of periodic trajectories are described.piecewise smooth vector fieldsymmetryperiodic trajectorybifurcationbifurcation diagram |
| spellingShingle | V.Sh. Roytenberg On the bifurcations of two periodic trajectories of a piecewise-smooth dynamical system with central symmetry Известия высших учебных заведений. Поволжский регион: Физико-математические науки piecewise smooth vector field symmetry periodic trajectory bifurcation bifurcation diagram |
| title | On the bifurcations of two periodic trajectories of a piecewise-smooth dynamical system with central symmetry |
| title_full | On the bifurcations of two periodic trajectories of a piecewise-smooth dynamical system with central symmetry |
| title_fullStr | On the bifurcations of two periodic trajectories of a piecewise-smooth dynamical system with central symmetry |
| title_full_unstemmed | On the bifurcations of two periodic trajectories of a piecewise-smooth dynamical system with central symmetry |
| title_short | On the bifurcations of two periodic trajectories of a piecewise-smooth dynamical system with central symmetry |
| title_sort | on the bifurcations of two periodic trajectories of a piecewise smooth dynamical system with central symmetry |
| topic | piecewise smooth vector field symmetry periodic trajectory bifurcation bifurcation diagram |
| work_keys_str_mv | AT vshroytenberg onthebifurcationsoftwoperiodictrajectoriesofapiecewisesmoothdynamicalsystemwithcentralsymmetry |