BORDER-COLLISION BIFURCATIONS IN 1D PIECEWISE-LINEAR MAPS AND LEONOV'S APPROACH

作者：Gardini Laura^*; Tramontana Fabio; Avrutin Viktor; Schanz Michael

来源：International Journal of Bifurcation and Chaos, 2010, 20(10): 3085-3104.

DOI：10.1142/S021812741002757X

• Discontinuous piecewise-linear 1D maps
• border collision bifurcations
• adding mechanism
• Farey structure

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.

• 出版日期2010-10