On generation of a limit cycle from a separatrix loop of a sewn saddle-node
The article considers dynamical systems on the plane, defined by continuous piecewise smooth vector fields. Such systems are used as mathematical models of real processes with switching. An important task is to find the conditions for the generation of periodic trajectories when the parameters chang...
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| Format: | Article |
| Language: | English |
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Saratov State University
2022-05-01
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| Series: | Известия Саратовского университета. Новая серия. Серия Математика. Механика. Информатика |
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| Online Access: | https://mmi.sgu.ru/sites/mmi.sgu.ru/files/text-pdf/2022/05/159-168-roitenberg.pdf |
| _version_ | 1811231832703762432 |
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| author | Roitenberg, Vladimir Shleymovich |
| author_facet | Roitenberg, Vladimir Shleymovich |
| author_sort | Roitenberg, Vladimir Shleymovich |
| collection | DOAJ |
| description | The article considers dynamical systems on the plane, defined by continuous piecewise smooth vector fields. Such systems are used as mathematical models of real processes with switching. An important task is to find the conditions for the generation of periodic trajectories when the parameters change. The paper describes the bifurcation of the birth of a periodic trajectory from the loop of the separatrix of a sewn saddle-node — an analogue of the classical bifurcation of the separatrix loop of a saddle-node of a smooth dynamical system. Consider a one-parameter family $\{ X_\varepsilon \} $ of continuous piecewise-smooth vector fields on the plane. Let $z^0 $ be a point on the switching line. Let's choose the local coordinates $x,y$ in which $z^0 $ has zero coordinates, and the switching line is given by the equation $y = 0$. Let the vector field $X_0 $ in a semi-neighborhood $y \ge 0$ ($y \le 0$) coincide with a smooth vector field $X_0^ + $ ($X_0^ - $), for which the point $z^0 $ is a stable rough node (rough saddle), and the proper subspaces of the matrix of the linear part of the field in $z^0 $ do not coincide with the straight line $y = 0$. The singular point $z^0 $ is called a sewn saddle-node. There is a single trajectory $L_0 $ that is $\alpha $-limit to $z^0 $ — the outgoing separatrix of the point $z^0 $. It is assumed that $L_0 $ is also $\omega $-limit to $z^0$, and enters $z^0 $ in the leading direction of the node of the field $X_0^ + $. For generic family, when the parameter $\varepsilon $ changes, the sewn saddle-node either splits into a rough node and a rough saddle, or disappears. In the paper it is proved that in the latter case the only periodic trajectory of the field $X_\varepsilon $ is generated from the contour $L_0 \cup \{ z^0 \} $ — a stable limit cycle. |
| first_indexed | 2024-04-12T10:51:44Z |
| format | Article |
| id | doaj.art-f28ae4631d4f4d2d89622f72eac6ec79 |
| institution | Directory Open Access Journal |
| issn | 1816-9791 2541-9005 |
| language | English |
| last_indexed | 2024-04-12T10:51:44Z |
| publishDate | 2022-05-01 |
| publisher | Saratov State University |
| record_format | Article |
| series | Известия Саратовского университета. Новая серия. Серия Математика. Механика. Информатика |
| spelling | doaj.art-f28ae4631d4f4d2d89622f72eac6ec792022-12-22T03:36:13ZengSaratov State UniversityИзвестия Саратовского университета. Новая серия. Серия Математика. Механика. Информатика1816-97912541-90052022-05-0122215916810.18500/1816-9791-2022-22-2-159-168On generation of a limit cycle from a separatrix loop of a sewn saddle-nodeRoitenberg, Vladimir Shleymovich0Yaroslavl State Technical University, 88 Moskovskii prospekt, Yaroslavl 150023, RussiaThe article considers dynamical systems on the plane, defined by continuous piecewise smooth vector fields. Such systems are used as mathematical models of real processes with switching. An important task is to find the conditions for the generation of periodic trajectories when the parameters change. The paper describes the bifurcation of the birth of a periodic trajectory from the loop of the separatrix of a sewn saddle-node — an analogue of the classical bifurcation of the separatrix loop of a saddle-node of a smooth dynamical system. Consider a one-parameter family $\{ X_\varepsilon \} $ of continuous piecewise-smooth vector fields on the plane. Let $z^0 $ be a point on the switching line. Let's choose the local coordinates $x,y$ in which $z^0 $ has zero coordinates, and the switching line is given by the equation $y = 0$. Let the vector field $X_0 $ in a semi-neighborhood $y \ge 0$ ($y \le 0$) coincide with a smooth vector field $X_0^ + $ ($X_0^ - $), for which the point $z^0 $ is a stable rough node (rough saddle), and the proper subspaces of the matrix of the linear part of the field in $z^0 $ do not coincide with the straight line $y = 0$. The singular point $z^0 $ is called a sewn saddle-node. There is a single trajectory $L_0 $ that is $\alpha $-limit to $z^0 $ — the outgoing separatrix of the point $z^0 $. It is assumed that $L_0 $ is also $\omega $-limit to $z^0$, and enters $z^0 $ in the leading direction of the node of the field $X_0^ + $. For generic family, when the parameter $\varepsilon $ changes, the sewn saddle-node either splits into a rough node and a rough saddle, or disappears. In the paper it is proved that in the latter case the only periodic trajectory of the field $X_\varepsilon $ is generated from the contour $L_0 \cup \{ z^0 \} $ — a stable limit cycle. https://mmi.sgu.ru/sites/mmi.sgu.ru/files/text-pdf/2022/05/159-168-roitenberg.pdfcontinuous piecewise smooth dynamical systemphase planebifurcationsewn saddle-nodelimit cycle |
| spellingShingle | Roitenberg, Vladimir Shleymovich On generation of a limit cycle from a separatrix loop of a sewn saddle-node Известия Саратовского университета. Новая серия. Серия Математика. Механика. Информатика continuous piecewise smooth dynamical system phase plane bifurcation sewn saddle-node limit cycle |
| title | On generation of a limit cycle from a separatrix loop of a sewn saddle-node |
| title_full | On generation of a limit cycle from a separatrix loop of a sewn saddle-node |
| title_fullStr | On generation of a limit cycle from a separatrix loop of a sewn saddle-node |
| title_full_unstemmed | On generation of a limit cycle from a separatrix loop of a sewn saddle-node |
| title_short | On generation of a limit cycle from a separatrix loop of a sewn saddle-node |
| title_sort | on generation of a limit cycle from a separatrix loop of a sewn saddle node |
| topic | continuous piecewise smooth dynamical system phase plane bifurcation sewn saddle-node limit cycle |
| url | https://mmi.sgu.ru/sites/mmi.sgu.ru/files/text-pdf/2022/05/159-168-roitenberg.pdf |
| work_keys_str_mv | AT roitenbergvladimirshleymovich ongenerationofalimitcyclefromaseparatrixloopofasewnsaddlenode |