On bifurcations of a limit cycle passing through a junction point of lines of discontinuity of a vector field and tangent to one of them

Author(s):  V.Sh. Roitenberg, candidate of Sciences, associate Professor, Yaroslavl State Technical University, Yaroslavl, Russia, vroitenberg@mail.ru

Issue:  Volume 50, №1

Rubric:  Mathematics

Annotation:  We consider a piecewise-smooth vector field X0 on the plane with partition into submanifolds Mi ( i 1,2,... ) with angles. Let the point O be 1-angular for M1 , 2-angular for the remaining elements of the partition adjacent to the point O, and through this point there passes a periodic trajectory Г tangent to M1 . It is assumed that on the transversal to Г is defined Poincare map f () , f (0)  0, (0)   1 f  . Two cases, ( 1)( 1)  0 and ( 1)( 1)  0 , are considered. In each case, for a family of vector fields that is a generic two-parameter deformation of a vector field X0 considered in a sufficiently small ring neighborhood of Г, we obtain the bifurcation diagram. In the first case, Г bifurcates similarly to the triple cycle of a smooth vector field; in particular, three coarse periodic trajectories can be born from it. In the second case, bifurcations of Г are similar to bifurcations of the double cycle of a smooth vector field.

Keywords:  piecewise smooth vector field, periodic trajectory, bifurcation diagram, bifurcations

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