A direct transition to chaos in hysteretic systems with focus dynamics

A specific transition to chaos is detected in the study of periodic orbits of hysteretic systems with symmetry and dynamics of focus type. The corresponding bifurcation is rigorously justified by resorting to the analysis of transition maps, whose mathematical expressions are adequately derived. It is shown that, depending on a parameter related to the location of equilibria, such transition maps can pass from being a smooth function to a discontinuous, piecewise-smooth function. We deal with the intermediate situation for which the transition map is continuous but nonsmooth. Using a second parameter, and previous known results on chaotic maps, we show in a rigorous way the existence of a chaos boundary crisis bifurcation, where the transition from a configuration without periodic orbits to another with bounded chaotic solutions occurs.