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.
Gardini, L., Tramontana, F., Avrutin, V., Schanz, M., Border-collision bifurcations in 1D piecewise-linear maps and Leonov's approach, <<INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS IN APPLIED SCIENCES AND ENGINEERING>>, 2010; 20 (10): 3085-3104. [doi:10.1142/S021812741002757X] [http://hdl.handle.net/10807/127513]
Border-collision bifurcations in 1D piecewise-linear maps and Leonov's approach
Tramontana, Fabio;
2010
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.