Border-collision bifurcations in 1D piecewise-linear maps and Leonov's approach

Laura Gardini, Fabio Tramontana, Viktor Avrutin, Michael Schanz

Research output: Contribution to journalArticle

56 Citations (Scopus)


50 years ago (1959) in a series of publications by Leonov, a detailed analytical study of the nested period adding bifurcation structure occurring in piecewise-linear discontinuous 1D maps was presented. The results obtained by Leonov are barely known, although they allow the analytical calculation of border-collision bifurcation subspaces in an elegant and much more efficient way than it is usually done. In this work we recall Leonov's approach and explain why it works. Furthermore, we slightly improve the approach by avoiding an unnecessary coordinate transformation, and also demonstrate that the approach can be used not only for the calculation of border-collision bifurcation curves. © 2010 World Scientific Publishing Company.
Original languageEnglish
Pages (from-to)3085-3104
Number of pages20
JournalInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering
Publication statusPublished - 2010


  • Applied Mathematics
  • Discontinuous piecewise-linear 1D maps
  • Farey structure
  • Modeling and Simulation
  • adding mechanism
  • border collision bifurcations


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