Invariant polygons in systems with grazing-sliding.
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 Poincare 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.
Szalai
R
R
Bristol Centre for Applied Nonlinear Mathematics, University of Bristol, University Walk, Bristol, BS8 1TR, United Kingdom. r.szalai@bristol.ac.uk
Osinga
H M
HM
D000465:Algorithms; D001703:Biophysics / Q000379:methods*; D003198:Computer Simulation; D008954:Models, Biological; D015233:Models, Statistical; D008962:Models, Theoretical; D017711:Nonlinear Dynamics; D009991:Oscillometry / Q000295:instrumentation / Q000379:methods*; D012815:Signal Processing, Computer-Assisted; D013598:Systems Theory