Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 366, Issue 1865, 2008, Pages 635-652

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

^a UNIVERSITY OF ABERDEEN   (United Kingdom)

^b UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

Homoclinic like orbit; Melnikov method; Piecewise linearization; Saddle like singularity

Indexed keywords

HAMILTONIANS; PERTURBATION TECHNIQUES; PIECEWISE LINEAR TECHNIQUES; RESOURCE ALLOCATION; TOPOLOGY;

DISCONTINUOUS DYNAMICS; HOMOCLINIC ORBITS;

ARCH BRIDGES;

EID: 38149133287     PISSN: 1364503X     EISSN: None     Source Type: Journal    
DOI: 10.1098/rsta.2007.2115     Document Type: Conference Paper

References (22)

1. Aguirre, J. & Sanjuán, M. A. F. 2002 Unpredictable behavior in the Duffing oscillator: Wada basins. Physica D 171, 41-51. (doi:10.1016/S0167-2789(02)00565-1)
2. Anna, L. H. & Vered, R. K. 1997 Symmetry-breaking perturbations and strange attractors. Phys. Rev. E 55, 4964-4978. (doi:10.1103/PhysRevE.55.4964)
3. Cao, Q., Wiercigroch, M., Pavlovskaia, E. E., Grebogi, C. & Thompson, J. M. T. 2006 Archetypal oscillator for smooth and discontinuous dynamics. Phys. Rev. E 74, 046218. (doi:10.1103/PhysRevE.74.046218)
4. Chen, Y. & Leung, A. Y. T. 1998 Bifurcation and chaos in engineering. London, UK: Springer.
5. Grassberger, P. & Procaccia, I. 1983 Characterization of strange attractors. Phys. Rev. Lett. 50, 346. (doi:10.1103/ PhysRevLett.50.346)
6. Guckenheimer, J. & Holmes, P. 1983 Nonlinear oscillation, dynamical system and bifurcation of vector fields. New York, NY: Springer.
7. Lai, Y. C. & Lerner, D. 1998 Effective scaling regime for computing the correlation dimension from chaotic time series. Physica D 115, 1-18. (doi:10.1016/S0167-2789(97)00230-3)
8. Melnikov, V. K. 1963 On the stability of the center for time periodic perturbations. Trans. Moscow Math. Soc. 12, 1-57.
9. Nayfeh, A. & Mook, D. 1979 Nonlinear oscillators. New York, NY: Wiley-Interscience.
10. Pavlovskaia, E. E. & Wiercigroch, M. 2004 Analytical drift reconstruction for visco-elastic impact oscillators operating in periodic and chaotic regimes. Chaos Solit. Fract. 19, 151-161. (doi:10.1016/S0960-0779(03)00128-0)
11. Pikovsky, A. S. 1992 Escape exponent for transient chaos and chaotic scattering in non-hyperbolic Hamiltonian systems. J. Phys. A: Math. Gen. 25, 447-481. (doi:10.1088/0305-4470/25/8/016)
12. Pippard, A. B. 1990 The elastic arch and its modes of instability. Eur. J. Phys. 11, 359-365. (doi:10.1088/0143-0807/11/6/008)
13. Savi, M. A. & Pacheco, P. M. C. L. 2003 Transient chaos in an elasto-plastic beam with hardening. J. Braz. Soc. Mech. Sci. Eng. 25, 189-193.
14. Stefanski, A. & Kapitaniak, T. 2000 Using chaos synchronization to estimate the largest Lyapunov exponent of nonsmooth systems. Disc. Dyn. Nat. Soc. 4, 207-215. (doi:10.1155/S1026022600000200)
15. Takens, F. 1981 Detecting strange attractors in turbulence. In Dynamical systems and turbulence. Warwick 1980, vol. 398 (eds D. A. Rand & L.-S. Young). Lecture notes in mathematics, pp. 366-381. Berlin, Germany: Springer.
16. Thompson, J. M. T. & Hunt, G. W. 1973 A general theory of elastic stability. London, UK: Wiley.
17. Thompson, J. M. T. & Ghaffari, R. 1983 Chaotic dynamics of an impact oscillator. Phys. Rev. A 27, 1741-1743. (doi:10.1103/PhysRevA.27. 1741)
18. Thompson, J. M. T. & Stewart, H. B. 2002 Nonlinear dynamics and chaos, 2nd edn. Chichester, UK: Wiley.
19. Tufillaro, N. B. 1989 Nonlinear and chaotic string vibrations. Am. J. Phys. 57, 404-414. (doi:10.1119/1.16011)
20. Tufillaro, N. B. 1990 Torsional parametric oscillations in wires. Eur. J. Phys. 11, 122-124. (doi:10.1088/0143-0807/11/2/011)
21. Tufillaro, N. B., Reilly, J. & Abbott, T. 1992 An experimental approach to nonlinear dynamics and chaos. Boston, MA: Addison-Wesley.
22. Wiggins, S. 1990 Introduction to applied nonlinear dynamical systems and chaos. New York, NY: Springer.