Bifurcations of hidden orbits in discontinuous maps

Abstract One-dimensional maps with discontinuities are known to exhibit bifurcations somewhat different those of continuous maps. Freed from the constraints continuity, and hence balance stability that is maintained through fold, flip, other standard bifurcations, attractors discontinuous can appear as if nowhere, change period or almost arbitrarily. But in fact this misleading, one includes states inside discontinuity map, highly unstable ‘hidden orbits’ created have iterates on discontinuity. These populate bifurcation diagrams just necessary branches make them resemble maps, namely familiar bifurcations. Here we analyse such detail, focussing first folds flips, then characterised by creating infinities orbits, chaotic repellers, infinite accumulations sub-bifurcations. We show role hidden orbits play, how they capture topological structures steep branches. This suggests both a more universal dynamical systems theory marrying possible, shows be used approximate jumps without losing any their structure.