Bifurcations of a fused triple cycle of a piecewise-smooth continuous dynamical system

V. Sh. Roitenberg

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 16 (2024) no. 1, pp. 39-48

Résumé

The study of bifurcations of dynamic systems defined by piecewise-smooth continuous vector fields is interesting from a theoretical and practical point of view. Nonlocal bifurcations in generic one-parameter families of such systems on a plane have already been described. In this paper, we consider a generic two-parameter family of piecewise-smooth continuous planar vector fields. At zero values of the parameters, it is assumed that the vector field has a smooth stable closed trajectory $\Gamma$, which has a simple tangency with the field switching line. A bifurcation diagram of a family is obtained – a partition of the neighborhood of zero on the parameter plane into sets, for whose elements the corresponding vector fields of the family have the same number and type of closed trajectories in some fixed neighborhood of $\Gamma$. In particular, we show that the maximum number of closed trajectories born from $\Gamma$ when the parameters change is three.

Détail

Keywords: piecewise-smooth continuous vector field, dynamical system, closed trajectory, bifurcation diagram.