Border collision bifurcations in two-dimensional piecewise smooth maps

Banerjee, Soumitro ; Grebogi, Celso (1999) Border collision bifurcations in two-dimensional piecewise smooth maps Physical Review E, 59 (4). pp. 4052-4061. ISSN 1063-651X

PDF - Publisher Version

Official URL:

Related URL:


Recent investigations on the bifurcations in switching circuits have shown that many atypical bifurcations can occur in piecewise smooth maps that cannot be classified among the generic cases like saddle-node, pitchfork, or Hopf bifurcations occurring in smooth maps. In this paper we first present experimental results to establish the need for the development of a theoretical framework and classification of the bifurcations resulting from border collision. We then present a systematic analysis of such bifurcations by deriving a normal form - the piecewise linear approximation in the neighborhood of the border. We show that there can be eleven qualitatively different types of border collision bifurcations depending on the parameters of the normal form, and these are classified under six cases. We present a partitioning of the parameter space of the normal form showing the regions where different types of bifurcations occur. This theoretical framework will help in explaining bifurcations in all systems, which can be represented by two-dimensional piecewise smooth maps.

Item Type:Article
Source:Copyright of this article belongs to American Physical Society.
ID Code:21394
Deposited On:20 Nov 2010 12:59
Last Modified:17 May 2016 05:36

Repository Staff Only: item control page