Two-parameter sliding bifurcations of periodic solutions in a dry-friction oscillator

Physica D: Nonlinear Phenomena

Volumn 237, Issue 8, 2008, Pages 1053-1073

  Kowalczyk, P ^a   Piiroinen, P T ^b  

^a UNIVERSITY OF EXETER   (United Kingdom)

^b NATIONAL UNIVERSITY OF IRELAND   (Ireland)

Author keywords

Codimension two points; Numerical continuation; Sliding bifurcations

Indexed keywords

FRICTION; NUMERICAL METHODS; OSCILLATORS (ELECTRONIC); PARAMETER ESTIMATION;

CODIMENSION-TWO POINTS; NUMERICAL CONTINUATION; PERIODIC SOLUTIONS; SLIDING BIFURCATIONS;

BIFURCATION (MATHEMATICS);

EID: 43049133262     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2007.12.007     Document Type: Article

References (46)

1. M.I. Feigin Forced Oscillations in Systems With Discontinuous Nonlinearities 1994 Nauka Moscow (in Russian)
2. M.I. Feigin The increasingly complex structure of the bifurcation tree of a piecewise-smooth system J. Appl. Maths. Mech. 59 1995 853 863
3. L.E. Nusse J.A. Yorke Border-collision bifurcations including ‘period two to period three’ for piecewise smooth systems Physica D 57 1992 39 57
4. L.E. Nusse J.A. Yorke Border collision bifurcation: An explanation for observed bifurcation phenomena Phys. Rev. E 49 1994 1073 1076
5. L.E. Nusse J.A. Yorke Border-collision bifurcations for piece-wise smooth one-dimensional maps Internat. J. Bifur. Chaos 5 1995 189 207
6. H. Nusse E. Ott J. Yorke Border collision bifurcations: An explanation for observed bifurcation phenomena Phys. Rev. E 49 1994 1073 1076
7. S. Banerjee J.A. Yorke C. Grebogi Robust chaos Phys. Rev. Lett 80 1998 3049 3052
8. S. Banerjee G. Verghese Nonlinear Phenomena in Power Electronics 2001 IEEE Press New York
9. S. Banerjee C. Grebogi Border collision bifurcations at the change of state-space dimension Chaos 12 2002 1054 1069
10. S. Banerjee C. Grebogi Border collision bifurcations in two-dimensional piecewise smooth maps Phys. Rev. E 59 1999 4052 4061
11. M. di Bernardo C.J. Budd A.R. Champneys Corner–collision implies border-collision bifurcation Physica D 154 2001 171 194
12. A.B. Nordmark Non-periodic motion caused by grazing incidence in impact oscillators J. Sound Vibration 2 1991 279 297
13. C.J. Budd P.T. Piiroinen Corner bifurcations in non-smoothly forced impact oscillators Physica D 220 2006 127 145
14. Part 1: Theoretical analysis of n-multiple (1/n)-impact solutions
15. C.J. Budd F. Dux Chattering and related behaviour in impact oscillators Phil. Trans. Roy. Soc. Lond. A 347 1994 365 389
16. C.J. Budd F. Dux Intermittency in impact oscillators close to resonance Nonlinearity 7 1994 1191 1224
17. W.P.M.H. Heemels B.B. Brogliato The complementarity class of hybrid dynamical systems Eur. J. Control 9 2003 311 319
18. P. Kowalczyk M. di Bernardo On a novel class of bifurcations in hybrid dynamical systems — the case of relay feedback systems Proceedings of Hybrid Systems Computation and Control 2001 Springer–Verlag 361 374
19. G. Rega A. Salvatori Bifurcation structure at 1/3-sub-harmonic resonance in an asymmetric nonlinear elastic oscillator Internat. J. Bifur. Chaos 6 1996 1529 1546
20. L.N. Virgin C.J. Begely Grazing bifurcations and basins of attraction in an impact-friction oscillator Physica D 130 1999 43 57
21. K. Popp N. Hinrichs M. Oestreich Dynamical behaviour of friction oscillators with simultaneous self and external excitation Sadhana (Indian Academy of Sciences) 20 1995 627 654
22. U. Galvanetto S.R. Bishop Dynamics of a simple damped oscillator undergoing stick-slip vibrations Meccanica 34 2000 337 347
23. A.F. Filippov Differential Equations with Discontinuous Righthand Sides 1988 Kluwer Academic Publishers Dortrecht
24. M. di Bernardo C.J. Budd A.R. Champneys Grazing, skipping and sliding: Analysis of the nonsmooth dynamics of the DC/DC buck converter Nonlinearity 11 1998 858 890
25. M. di Bernardo K.H. Johansson F. Vasca Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations Internat. J. Bifur. Chaos 11 4 2001 1121 1140
26. M. di Bernardo P. Kowalczyk A. Nordmark Sliding bifurcations: A novel mechanism for the sudden onset of chaos in dry-friction oscillators Internat. J. Bifur. Chaos 13 10 2003 2935 2948
27. Yu.A. Kuznetsov S. Rinaldi A. Gragnani One-parameter bifurcations in planar Filippov systems Internat. J. Bifur. Chaos 13 2003 2157 2188
28. J. Adolfsson H. Dankowicz A. Nordmark 3D passive walkers: Finding periodic gaits in the presence of discontinuities Nonlinear Dynam. 24 2001 205 229
29. H. Dankowicz P.T. Piiroinen Exploiting discontinuities for stabilization of recurrent motions Dynam. Syst. 17 2002 317 342
30. V.I. Utkin Sliding Modes in Control Optimization 1992 Springer-Verlag New York
31. M. di Bernardo P. Nordmark Bifurcations of dynamical systems with sliding: Derivation of normal-form mappings Physica D 170 2002 175 205
32. P.T. Piiroinen Yu.A. Kuznetsov An event-driven method to simulate Filippov systems with accurate computing of the vector field ACM Trans. Math. Softw. 34 3 2008
33. U.M. Ascher L.R. Petzold Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations 1998 SIAM Philadelphia, USA
34. Yu.A. Kuznetsov Elements of Applied Bifurcation Theory 2nd ed. Applied Mathematical Sciences, vol. 112 1998 Springer–Verlag New York
35. R.I. Leine H. Nijmeijer Dynamics and Bifurcations of Non-Smooth Mechanical Systems Lecture Notes in Applied and Computational Mechanics vol. 18 2004 Springer–Verlag Berlin, Heidelberg
36. M.H. Frederiksson A.B. Nordmark Bifuractions caused by grazing incidence in many degrees of freedom impact oscillators Proc. Royal Soc. Lond. A 453 1997 1261 1276
37. R.I. Leine, Bifurcations in discontinuous mechanical systems of Filippov-type. Ph.D. Thesis, Teknische Universiteit Eindhoven, The Netherlands, 2000
38. P.C. Müller Calculations of Lyapunov exponents for dynamical systems with discontinuities Chaos Solitons Fractals 5 1995 1671 1681
39. H.B. Keller Lectures on Numerical Methods In Bifurcation Problems 1987 Springer-Verlag
40. P. Kowalczyk M. di Bernardo Two-parameter degenerate sliding bifurcations in Filippov systems Physica D 204 2005 204 229
41. P. Kowalczyk M. di Bernardo A.R. Champneys S.J. Hogan M. Homer Yu.A. Kuznetsov A.B. Nordmark P.T. Piiroinen Two-parameter nonsmooth bifurcations of limit cycles: Classification and open problems Internat. J. Bifur. Chaos 16 3 2006
42. A.B. Nordmark P. Kowalczyk A codimension-two scenario of sliding solutions in grazing-sliding bifurcations Nonlinearity 19 2006 1 26
43. G. Csernák G. Stépán On the periodic response of a harmonically excited dry friction oscillator J. Sound Vibration 295 2006 649 658
44. F. Dercole Yu.A. Kuznetsov SlideCont: An AUTO97 driver for sliding bifurcation analysis ACM Trans. Math. Softw. 31 2005 95 119
45. P. Thota, H. Dankowicz, On a boundary-value formulation for the continuation of solution trajectories in hybrid dynamical systems and its implementation in the software toolbox TC-HAT, SIADS (submitted for publication)
46. E.J. Doedel, A.R. Champneys, T.F. Fairgrieve, T.F. Kuznetsov, Yu.A. Sandstede, B. Wang, Auto97: Continuation and bifurcation software for ordinary differential equations (with HomCont). Computer Science, Concordia University, Montreal, Canada, 1997 ftp.cs.concordia.ca/doedel/doc/auto