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On generation of a limit cycle from a separatrix loop of a sewn saddle-node

Authors :
Roitenberg, Vladimir Shleymovich
Source :
Известия Саратовского университета. Новая серия. Серия Математика. Механика. Информатика, Vol 22, Iss 2, Pp 159-168 (2022)
Publication Year :
2022
Publisher :
Saratov State University, 2022.

Abstract

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.

Details

Language :
English, Russian
ISSN :
18169791 and 25419005
Volume :
22
Issue :
2
Database :
Directory of Open Access Journals
Journal :
Известия Саратовского университета. Новая серия. Серия Математика. Механика. Информатика
Publication Type :
Academic Journal
Accession number :
edsdoj.f28ae4631d4f4d2d89622f72eac6ec79
Document Type :
article
Full Text :
https://doi.org/10.18500/1816-9791-2022-22-2-159-168