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.

[1]  Fabrizio Vestroni,et al.  Experimental evidence of non-standard bifurcations in non-smooth oscillator dynamics , 2006 .

[2]  M. Oestreich,et al.  Bifurcation and stability analysis for a non-smooth friction oscillator , 1996 .

[3]  K. K.,et al.  Stick-slip vibrations and chaos , 1990, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[4]  Erik Mosekilde,et al.  Bifurcations and chaos in piecewise-smooth dynamical systems , 2003 .

[5]  Stephen John Hogan,et al.  Dynamics of a piecewise linear map with a gap , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[6]  A. J. Homburg.  Piecewise smooth interval maps with non-vanishing derivative , 2000, Ergodic Theory and Dynamical Systems.

[7]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[8]  Yuri A. Kuznetsov,et al.  An event-driven method to simulate Filippov systems with accurate computing of sliding motions , 2008, TOMS.

[9]  Arne Nordmark,et al.  Non-periodic motion caused by grazing incidence in an impact oscillator , 1991 .

[10]  D. Saari,et al.  Stable and Random Motions in Dynamical Systems , 1975 .

[11]  F. Moon,et al.  Chaos in a Forced Dry-Friction Oscillator: Experiments and Numerical Modelling , 1994 .

[12]  Vadim I. Utkin,et al.  Sliding Modes in Control and Optimization , 1992, Communications and Control Engineering Series.

[13]  S. Narayanan,et al.  Chaotic vibration in a non-linear oscillator with Coulomb damping , 1991 .

[14]  G. Stépán,et al.  On the periodic response of a harmonically excited dry friction oscillator , 2006 .

[15]  Carlos Canudas de Wit,et al.  A survey of models, analysis tools and compensation methods for the control of machines with friction , 1994, Autom..

[16]  Erik Mosekilde,et al.  Torus birth bifurcations in a DC/DC converter , 2006, IEEE Transactions on Circuits and Systems I: Regular Papers.

[17]  David K. Campbell,et al.  Piecewise linear models for the quasiperiodic transition to chaos. , 1995, Chaos.

[18]  Andy Ruina,et al.  A two degree-of-freedom earthquake model with static/dynamic friction , 1987 .

[19]  Laura Gardini,et al.  The Hicksian floor-roof model for two regions linked by interregional trade , 2003 .

[20]  Mario di Bernardo,et al.  Sliding bifurcations: a Novel Mechanism for the Sudden Onset of Chaos in dry Friction oscillators , 2003, Int. J. Bifurc. Chaos.

[21]  Harry Dankowicz,et al.  Continuous and discontinuous grazing bifurcations in impacting oscillators , 2006 .

[22]  Somnath Maity,et al.  Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation. , 2006, Chaos.

[23]  Harry Dankowicz,et al.  Low-velocity impacts of quasiperiodic oscillations , 2002 .

[24]  Brandon C. Gegg,et al.  Stick and non-stick periodic motions in periodically forced oscillators with dry friction , 2006 .

[25]  Stephen Wiggins,et al.  Global Bifurcations and Chaos , 1988 .

[26]  J. Tyson,et al.  Mathematical model of the cell division cycle of fission yeast. , 2001, Chaos.