The limit case response of the archetypal oscillator for smooth and discontinuous dynamics

International Journal of Non-Linear Mechanics

Volumn 43, Issue 6, 2008, Pages 462-473

  Cao, Qingjie ^a,b   Wiercigroch, Marian ^a   Pavlovskaia, Ekaterina E ^a   Grebogi, Celso ^a   Thompson, J Michael T ^a,c  

^a UNIVERSITY OF ABERDEEN   (United Kingdom)

^b SHIJIAZHUANG TIEDAO UNIVERSITY   (China)

^c UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

Chaotic sea; Discontinuity; KAM structure; SD oscillator

Indexed keywords

BIFURCATION (MATHEMATICS); CHAOS THEORY; PARAMETER ESTIMATION;

CHAOTIC SEA; DISCONTINUITY; KAM STRUCTURE; SD OSCILLATOR;

OSCILLATORS (ELECTRONIC);

EID: 44649186996     PISSN: 00207462     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ijnonlinmec.2008.01.003     Document Type: Article

References (34)

1. Filippov A.F. Differential Equations with Discontinuous Right-hand Sides, Mathematics and its Applications (1988), Kluwer Academic Publishers, Dordrecht
2. Feigin M.I. Forced Vibrations of Nonlinear Systems with Discontinuities (1994), Nauka, Moscow (in Russian)
3. Leine R.I., and Van Campen D.H. Discontinuous bifurcations of periodic solutions. Math. Comput. Modelling 36 (2002) 259-273
4. Leine R.I., Van Campen D.H., and Van de Vrande B.L. Bifurcations in nonlinear discontinuous systems. Nonlinear Dyn. 23 (2000) 105-164
5. Leine R.I., Van Campen D.H., De Kraker A., and Van den Steen L. Stick-slip vibrations induced by alternate friction models. Nonlinear Dyn. 16 1 (1998) 41-54
6. Filippov A.F. Differential equations with discontinuous right-hand side. Am. Math. Soc. Transl. Ser. 2 42 (1964) 199-231
7. Foong C.H., Pavlovskaia E., Wiercigroch M., and Deans W.F. Chaos caused by fatigue crack growth. Chaos Solitons Fractals 16 (2003) 651-659
8. Ouyang H., Mottershead J.E., Cartmell M.P., and Brookfiled D.J. Friction-induced vibration of an elastic slider on a vibrating disk. Int. J. Mech. Sci. 41 3 (1999) 325-336
9. Wiercigroch M. Modelling of dynamical systems with motion dependent discontinuities. Chaos Solitons Fractals 11 15 (2000) 2429-2442
10. Pavlovskaia E., and Wiercigroch M. Periodic solution finder for an impact oscillator with a drift. J. Sound Vib. 267 4 (2003) 893-911
11. Xu L., Lu M.W., and Cao Q. Nonlinear vibrations of dynamical systems with a general form of piecewise-linear viscous damping by incremental harmonic balance method. Phys. Lett. A 301 1-2 (2002) 65-73
12. Cao Q., Xu L., Djidjeli K., Price W.G., and Twizell E.H. Analysis of period-doubling and chaos of a non-symmetric oscillator with piecewise-linearity. Chaos Solitons Fractals 12 10 (2001) 1917-1927
13. Di Bernardo M., Kowalczyk P., and Nordmark A. Sliding bifurcations: a novel mechanism for the sudden onset of chaos in dry friction oscillators. Int. J. Bifurcation Chaos 13 10 (2003) 2935-2948
14. Di Bernardo M., Garofalo F., Iannelli L., and Vasca F. Bifurcations in piecewise-smooth feedback systems. Int. J. Control 75 16-17 (2002) 1243-1259
15. Di Bernardo M., Kowalczyk P., and Nordmark A. Bifurcations of dynamical systems with sliding: derivation of normal-form mappings. Physica D 170 3-4 (2002) 175-205
16. Banerjee S., and Grebogi C. Border collision bifurcations in two-dimensional piecewise smooth maps. Phys. Rev. E 59 4 (1999) 4052-4061
17. Banerjee S., Karthik M.S., Yuan G.H., and Yorke J.A. Bifurcations in one-dimensional piecewise smooth maps-theory and applications in switching circuits. IEEE Trans. Circuits Syst. I 47 3 (2000) 389-394
18. Whiston G.S. Singularities in vibro-impact dynamics. J. Sound Vib. 152 33 (1992) 427-460
19. Cao Q., Wiercigroch M., Pavlovskaia E.E., Grebogi C., and Thompson J.M.T. An archetypal oscillator for smooth and discontinuous dynamics. Phys. Rev. E 74 (2006) 046218
20. Cao Q., Wiercigroch M., Pavlovskaia E.E., Grebogi C., and Thompson J.M.T. Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics. Philos. Trans. R. Soc. A 366 (1865) (2008) 635-652
21. Thompson J.M.T., and Hunt G.W. A General Theory of Elastic Stability (1973), Wiley, London
22. Pippard A.B. The elastic arch and its modes of instability. Eur. J. Phys. 11 6 (1990) 359-365
23. Savi M.A., and Pacheco P.M.C.L. Transient chaos in an elasto-plastic beam with hardening. J. Brazilian Soc. Mech. Sci. Eng. 25 2 (2003) 189-193
24. Lichtenberg A.J., and Lieberman M.A. Regular and Chaotic Dynamics (1992), Springer, New York
25. Chernikov A.A., Sagdeev R.Z., Usikov D.A., Zakharov M.Yu., and Zaslavsky G.M. Minimal chaos and stochastic webs. Nature 326 (1987) 559-563
26. Pekarsky S., and Rom-Kedar V. Uniform stochastic web in low-dimensional Hamiltonian systems. Phys. Lett. A 225 (1997) 274-286
27. Luo A.C.J. Chaos and quasi-periodic motions on the homoclinic surface of nonlinear hamiltonian systems with two degrees of freedom. J. Comput. Nonlinear Dyn. 1 2 (2006) 135-142
28. Feudel U., Grebogi C., Hunt B.R., and Yorke J.A. Map with more than 100 coexisting low-period periodic attractors. Phys. Rev. E 54 (1996) 71-81
29. Feudel U., and Grebogi C. Multistability and the control of complexity. Chaos: Interdiscip. J. Nonlinear Sci. 7 4 (1997) 597-604
30. Lai Y.C., and Winslow R.L. Geometric properties of the chaotic saddle responsible for supertransients in spatiotemporal chaotic systems. Phys. Rev. Lett. 74 26 (1995) 5208-5211
31. Tel T. Characteristic exponents of chaotic repellers as eigenvalues. Phys. Lett. A 119 2 (1986) 65-68
32. Grebogi C., Ott E., and Yorke J.A. Crises, sudden changes in chaotic attractors, and transient chaos. Physica D 7 1-3 (1983) 181-200
33. Thompson J.M.T., and Ghaffari R. Chaotic dynamics of an impact oscillator. Phys. Rev. A 27 3 (1983) 1741-1743
34. Thompson J.M.T., and Stewart H.B. Nonlinear Dynamics and Chaos. second ed. (2002), Wiley, London