C-bifurcations and period-adding in one-dimensional piecewise smooth maps

CK Halse, ME Homer, M Di Bernardo

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

49 Citations (Scopus)


This paper examines the behaviour of piecewise-smooth, continuous, one-dimensional maps that have been derived in the literature as normal forms for grazing and sliding bifurcations. These maps are linear for negative values of the parameter and non-linear for positive values of the parameter. Both C-1 and C-2 maps of this form are considered. These maps display both period-adding and period-doubling behaviour. For maps with a squared or 3/2 term the stability and existence conditions of fixed points and period-2 orbits in the vicinity of the border-collision are found analytically. These agree with the Feigin classification proposed by di Bernardo et al. [Chaos Solitons and Fractals 10 (1999) 188 1]. The period-adding behaviour is examined in these maps, where analytical solutions for the boundaries of periodic solutions are found. Implicit equations for the boundaries of periodic windows for varying power term are also found and plotted. Thus, it is proved that period-adding scenarios are generic in maps of this form.
Translated title of the contributionC-bifurcations and period-adding in one-dimensional piecewise smooth maps
Original languageEnglish
Pages (from-to)953 - 976
Number of pages24
JournalChaos, Solitons and Fractals
Volume18 (5)
Publication statusPublished - Dec 2003

Bibliographical note

Publisher: Pergamon-Elsevier Science BV

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