On local bifurcations in nonlinear continuous-discrete dynamical systems

The dynamics of nonlinear continuous-discrete (hybrid) systems and its dependence on the sampling step $h$ are studied. Such systems contain phase variables and equations with both continuous and discrete time. The main focus of the work is the issue of local bifurcations during loss of stability of equilibrium points of hybrid systems. Sufficient signs of bifurcations are given, the properties of bifurcations are studied, and possible bifurcation scenarios are determined. The concept of transversal bifurcation is introduced, meaning that the corresponding eigenvalue of the matrix of the linearized problem passes through the unit circle when the parameter $h$ passes through the bifurcation point $h_0$. It is shown that in a one-parameter formulation, two main scenarios are typical: transversal bifurcation of period doubling and transversal Andronov–Hopf bifurcation, while the scenario of transversal bifurcation of multiple equilibrium, as a rule, is not realized. Examples are given to illustrate the effectiveness of the proposed approaches in the problem of studying bifurcations in hybrid systems.