# Breaking the continuity of a piecewise linear map

Viktor Avrutin; Michael Schanz; Björn Schenke

ESAIM: Proceedings (2012)

- Volume: 36, page 73-105
- ISSN: 1270-900X

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topAvrutin, Viktor, Schanz, Michael, and Schenke, Björn. Fournier-Prunaret, D., Gardini, L., and Reich, L., eds. " Breaking the continuity of a piecewise linear map ." ESAIM: Proceedings 36 (2012): 73-105. <http://eudml.org/doc/251253>.

@article{Avrutin2012,

abstract = {Knowledge about the behavior of discontinuous piecewise-linear maps is important for a
wide range of applications. An efficient way to investigate the bifurcation structure in
2D parameter spaces of such maps is to detect specific codimension-2 bifurcation points,
called organizing centers, and to describe the bifurcation structure in their
neighborhood. In this work, we present the organizing centers in the 1D discontinuous
piecewise-linear map in the generic form, which can be used as a normal form for these
bifurcations in other 1D discontinuous maps with one discontinuity. These organizing
centers appear when the continuity of the system function is broken in a fixed point. The
type of an organizing center depends on the slopes of the piecewise-linear map. The
organizing centers that occur if the slopes have an absolute value smaller than one were
already described in previous works, so we concentrate on presenting the organizing
centers that occur if one or both slopes have absolute values larger than one. By doing
this, we also show that the behavior for each organizing center can be explained using
four basic bifurcation scenarios: the period incrementing and the period adding scenarios
in the periodic domain, as well as the bandcount incrementing and the bandcount adding
scenarios in the chaotic domain.},

author = {Avrutin, Viktor, Schanz, Michael, Schenke, Björn},

editor = {Fournier-Prunaret, D., Gardini, L., Reich, L.},

journal = {ESAIM: Proceedings},

keywords = {Discontinuous piecewise-linear map; codimension-2 bifurcation; organizing center; discontinuous piecewise-linear map},

language = {eng},

month = {8},

pages = {73-105},

publisher = {EDP Sciences},

title = { Breaking the continuity of a piecewise linear map },

url = {http://eudml.org/doc/251253},

volume = {36},

year = {2012},

}

TY - JOUR

AU - Avrutin, Viktor

AU - Schanz, Michael

AU - Schenke, Björn

AU - Fournier-Prunaret, D.

AU - Gardini, L.

AU - Reich, L.

TI - Breaking the continuity of a piecewise linear map

JO - ESAIM: Proceedings

DA - 2012/8//

PB - EDP Sciences

VL - 36

SP - 73

EP - 105

AB - Knowledge about the behavior of discontinuous piecewise-linear maps is important for a
wide range of applications. An efficient way to investigate the bifurcation structure in
2D parameter spaces of such maps is to detect specific codimension-2 bifurcation points,
called organizing centers, and to describe the bifurcation structure in their
neighborhood. In this work, we present the organizing centers in the 1D discontinuous
piecewise-linear map in the generic form, which can be used as a normal form for these
bifurcations in other 1D discontinuous maps with one discontinuity. These organizing
centers appear when the continuity of the system function is broken in a fixed point. The
type of an organizing center depends on the slopes of the piecewise-linear map. The
organizing centers that occur if the slopes have an absolute value smaller than one were
already described in previous works, so we concentrate on presenting the organizing
centers that occur if one or both slopes have absolute values larger than one. By doing
this, we also show that the behavior for each organizing center can be explained using
four basic bifurcation scenarios: the period incrementing and the period adding scenarios
in the periodic domain, as well as the bandcount incrementing and the bandcount adding
scenarios in the chaotic domain.

LA - eng

KW - Discontinuous piecewise-linear map; codimension-2 bifurcation; organizing center; discontinuous piecewise-linear map

UR - http://eudml.org/doc/251253

ER -

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