A periodically forced, piecewise linear system. Part I: Local singularity and grazing bifurcation

Abstract A methodology for the local singularity of non-smooth dynamical systems is systematically presented in this paper, and a periodically forced, piecewise linear system is investigated as a sample problem to demonstrate the methodology. The sliding dynamics along the separation boundary are investigated through the differential inclusion theory. For this sample problem, a perturbation method is introduced to determine the singularity of the sliding dynamics on the separation boundary. The criteria for grazing bifurcation are presented mathematically and numerically. The grazing flows are illustrated numerically. This methodology can be very easily applied to predict grazing motions in other non-smooth dynamical systems. The fragmentation of the strange attractors of chaotic motion will be presented in the second part of this work.

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

[2]  piecewise linear systems , 2005 .

[3]  James A. Yorke,et al.  Border-collision bifurcations in the buck converter , 1998 .

[4]  Ray P. S. Han,et al.  Chaotic motion of a horizontal impact pair , 1995 .

[5]  Werner Schiehlen,et al.  Local and global stability of a piecewise linear oscillator , 1992, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[6]  Stephen P. Timoshenko,et al.  Vibration problems in engineering , 1928 .

[7]  M. di Bernardo,et al.  Bifurcations of dynamical systems with sliding: derivation of normal-form mappings , 2002 .

[8]  Ott,et al.  Border-collision bifurcations: An explanation for observed bifurcation phenomena. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[9]  Albert C. J. Luo,et al.  A theory for non-smooth dynamic systems on the connectable domains , 2005 .

[10]  P. Holmes,et al.  A periodically forced piecewise linear oscillator , 1983 .

[11]  J. Yorke,et al.  Bifurcations in one-dimensional piecewise smooth maps-theory and applications in switching circuits , 2000 .

[12]  S. Foale.  Analytical determination of bifurcations in an impact oscillator , 1994, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[13]  Grebogi,et al.  Grazing bifurcations in impact oscillators. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

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

[15]  H. Dankowicz,et al.  On the origin and bifurcations of stick-slip oscillations , 2000 .

[16]  A. P. Ivanov,et al.  Stabilization Of An Impact Oscillator Near Grazing Incidence Owing To Resonance , 1993 .

[17]  Albert C. J. Luo,et al.  An Unsymmetrical Motion in a Horizontal Impact Oscillator , 2002 .

[18]  Graph theory and piecewise smooth dynamical systems of arbitrary dimension , 1999 .

[19]  Alan R. Champneys,et al.  Normal form maps for grazing bifurcations in n -dimensional piecewise-smooth dynamical systems , 2001 .

[20]  Albert C. J. Luo,et al.  Global chaos in a periodically forced, linear system with a dead-zone restoring force , 2004 .

[21]  Albert C. J. Luo.  On the symmetry of solutions in non-smooth dynamical systems with two constraints , 2004 .

[22]  Albert C. J. Luo,et al.  The mapping dynamics of periodic motions for a three-piecewise linear system under a periodic excitation , 2005 .

[23]  S. Theodossiades,et al.  NON-LINEAR DYNAMICS OF GEAR-PAIR SYSTEMS WITH PERIODIC STIFFNESS AND BACKLASH , 2000 .

[24]  S. Natsiavas,et al.  Periodic response and stability of oscillators with symmetric trilinear restoring force , 1989 .

[25]  Richard H. Rand,et al.  Bifurcations and chaos in a forced zero-stiffness impact oscillator , 1990 .