Bifurcations of hidden orbits in discontinuous maps

Viktor Avrutin*, Mike R Jeffrey*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

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One-dimensional maps with discontinuities are known to exhibit bifurcations
somewhat different to those of continuous maps. Freed from the constraints of continuity, and hence from the balance of stability that is maintained through fold, flip, and other standard bifurcations, the attractors of discontinuous maps can appear as if from nowhere, and change period or stability almost arbitrarily. But in fact this is misleading, and if one includes states inside the discontinuity in the map, highly unstable “hidden orbits” are created that have iterates on the discontinuity. These populate the bifurcation diagrams of discontinuous maps with just the necessary unstable branches to make them resemble those of continuous maps, namely fold, flip, and other familiar bifurcations. Here we analyse such bifurcations in detail, focussing first on folds and flips, then on bifurcations characterized by creating infinities of orbits, chaotic repellers, and infinite accumulations of sub-bifurcations. We show the role that hidden orbits play, and how they capture the topological structures of continuous maps with steep branches. This suggests both that a more universal dynamical systems theory marrying continuous and discontinuous systems is possible, and shows how discontinuities can be used to approximate steep jumps in continuous systems without losing any of their topological structure.
Original languageEnglish
Pages (from-to)6140–6172
Number of pages34
Volume34 (2021)
Early online date26 Jul 2021
Publication statusE-pub ahead of print - 26 Jul 2021


  • discontinuous maps
  • steep maps
  • stiff maps
  • border collision bifurcations
  • hidden orbits
  • unstable orbits


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