The structure of mode-locking regions of piecewise-linear continuous maps: I. Nearby mode-locking regions and shrinking points

The mode-locking regions of a dynamical system are subsets of parameter space within which there exists an attracting periodic solution. For piecewise-linear continuous maps, these regions have a distinctive chain structure with points of zero width called shrinking points. In this paper a local ana...

Full description

Saved in:
Bibliographic Details
Published inNonlinearity Vol. 30; no. 1; pp. 382 - 444
Main Author Simpson, D J W
Format Journal Article
LanguageEnglish
Published IOP Publishing 01.01.2017
Subjects
Arnold tongue
border-collision bifurcation
nonsmooth
periodicity region
piecewise-smooth
resonance
Online AccessGet full text

Cover

More Information
Summary:The mode-locking regions of a dynamical system are subsets of parameter space within which there exists an attracting periodic solution. For piecewise-linear continuous maps, these regions have a distinctive chain structure with points of zero width called shrinking points. In this paper a local analysis about an arbitrary shrinking point is performed. This is achieved by studying the symbolic itineraries of periodic solutions in nearby mode-locking regions and performing an asymptotic analysis on one-dimensional centre manifolds in order to build a comprehensive theoretical framework for the local dynamics. The main results are universal quantitative descriptions for the shape of nearby mode-locking regions, the location of nearby shrinking points, and the key properties of these shrinking points. The results are applied to the three-dimensional border-collision normal form, a model of an oscillator subject to dry friction, and a model of a DC/DC power converter.
Bibliography:NON-101579.R1
London Mathematical Society
ISSN:0951-7715
1361-6544
DOI:10.1088/1361-6544/aa4f49