Local analysis of \(C\)-bifurcations in \(n\)-dimensional piecewise-smooth dynamical systems
Many models in engineering and applications involve piecewise-smooth, continuous or discontinuous, maps or differential equations, e.g. impact oscillators. The authors reformulate and extend a theory of local codimension one bifurcations in such \(n\)-dimensional systems, which was originated by one of the authors, \textit{M. I. Feigin}, in the 70s in Russia [Prikl. Mat. Mekh. 34, 861-869 (1970; Zbl 0224.34021), ibid. 42, 820-829 (1978; Zbl 0419.58014)], in a work on \(C\)-bifurcations (border-collision bifurcation). In the ODE formulation, for a suitable section of a periodic orbit, the authors derive a local Poincaré map, whose linearization yields conditions for typical \(C\)-bifurcations (related to period doubling). These conditions can be given in terms of the spectrum of the linearized local map which completely characterize this bifurcation scenario. For a class of two dimensional piecewise continuous maps, a classification of all codimension one bifurcations is given. More complexity is pictured for a three parameter 1-D piecewise smooth map. Finally the methods are applied to a three parameter ODE. Numerical data and figures illustrate these examples.
scientific article; zbMATH DE number 1371680