The paper investigates generic three-dimensional nonsmooth systems with a periodic orbit near grazing-sliding. We assume that the periodic orbit is unstable with complex multipliers so that two dominant frequencies are present in the system. Because grazing-sliding induces a dimension loss and the instability drives every trajectory into sliding, the system has an attractor that consists of forward sliding orbits. We analyze this attractor in a suitably chosen Poincaré section using a three-parameter generalized map that can be viewed as a normal form. We show that in this normal form the attractor must be contained in a finite number of lines that intersect in the vertices of a polygon. However the attractor is typically larger than the associated polygon. We classify the number of lines involved in forming the attractor as a function of the parameters. Furthermore, for fixed values of parameters we investigate the one-dimensional dynamics on the attractor.
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June 2008
Research Article|
June 05 2008
Invariant polygons in systems with grazing-sliding
R. Szalai;
R. Szalai
a)
Bristol Centre for Applied Nonlinear Mathematics,
University of Bristol
, University Walk, Bristol, BS8 1TR, United Kingdom
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H. M. Osinga
H. M. Osinga
b)
Bristol Centre for Applied Nonlinear Mathematics,
University of Bristol
, University Walk, Bristol, BS8 1TR, United Kingdom
Search for other works by this author on:
Chaos 18, 023121 (2008)
Article history
Received:
August 07 2007
Accepted:
March 06 2008
Citation
R. Szalai, H. M. Osinga; Invariant polygons in systems with grazing-sliding. Chaos 1 June 2008; 18 (2): 023121. https://doi.org/10.1063/1.2904774
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