Title
Discontinuity Induced Bifurcations of Nonhyperbolic Cycles in Nonsmooth Systems
Abstract
We analyze three codimension-two bifurcations occurring in nonsmooth systems, when a nonhyperbolic cycle (fold, flip, and Neimark-Sacker cases, in both continuous and discrete time) interacts with one of the discontinuity boundaries characterizing the system's dynamics. Rather than aiming at a complete unfolding of the three cases, which would require specific assumptions on both the class of nonsmooth system and the geometry of the involved boundary, we concentrate on the geometric features that are common to all scenarios. We show that, at a generic intersection between the smooth and discontinuity induced bifurcation curves, a third curve generically emanates tangentially to the former. This is the discontinuity induced bifurcation curve of the secondary invariant set (the other cycle, the double-period cycle, or the torus, respectively) involved in the smooth bifurcation. The result can be explained intuitively, but its validity is proved here rigorously under very general conditions. Three examples from different fields of science and engineering are also reported.
Year
DOI
Venue
2010
10.1137/080732377
SIAM J. Applied Dynamical Systems
Keywords
DocType
Volume
smooth bifurcation,non-hyperbolic,nonsmooth system,border collision,nonhyperbolic cycle,bifurcation,nonhyperbolic cycles,discontinuity boundary,different field,involved boundary,nonsmooth systems,discontinuity induced bifurcation curve,nonsmooth,codimension-two,discrete time,discontinuity induced bifurcations,double-period cycle,neimark-sacker case
Journal
3
Issue
ISSN
Citations 
1
SIAM J. Applied Dynamical Sysytems Vol. 9, No. 1, pp. 62-83, 2010
6
PageRank 
References 
Authors
0.89
7
2
Name
Order
Citations
PageRank
A. Colombo115111.06
Fabio Dercole24714.32