Skip to main content
SpringerLink
Log in

Control of Chaos: Impact Oscillators and Targeting

  • Conference paper
IUTAM Symposium on Interaction between Dynamics and Control in Advanced Mechanical Systems

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 52))

  • 297 Accesses

Abstract

We present two applications of chaos control techniques that can be of importance in mechanical systems. First, we apply chaos control to select a desired sequence of impacts in a map that captures the universal properties of impact oscillators near grazing. Next we describe a targeting method that can significantly reduce the chaotic transients that precede stabilization when these control methods are used.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Foale S. and Bishop, S.R. (1992) Dynamical complexities of forced impacting systems, Phil. Trans. R. Soc. Lond., A338, pp. 547–556.

    Article  MathSciNet  Google Scholar 

  2. E. Slivsgaard and H. True (1994) Chaos in railway-vehicle dynamics, in J.M.T. Thompson and S.R. Bishop (eds.), Nonlinearity and Chaos in Engineering Dynamics, John Wiley and Sons Ltd, England, pp. 183–192.

    Google Scholar 

  3. Nordmark, A.B. (1991) Non-periodic motion caused by grazing incidence in an impact oscillator, J. Sound Vib., 145, pp. 279–297.

    Article  Google Scholar 

  4. Budd, C. and Dux, F. (1994) Intermittency in impact oscillators close to resonance, Nonlinearity, 7, pp. 1191–1224.

    Article  MathSciNet  MATH  Google Scholar 

  5. Shaw, S.W. and Holmes, P.J. (1983) A periodically forced piecewise linear oscillator, J. Sound Vib. 90, pp. 129–155.

    Article  MathSciNet  MATH  Google Scholar 

  6. Shaw, S.W. (1985) The dynamics of a harmonically excited system having rigid amplitude constraints, ASME J. Appl. Mech., 52, pp. 453–464.

    Article  Google Scholar 

  7. Nusse, H.E. and Yorke, J.A. (1992) Border-collision bifurcations including ‘period two to period three’ for piecewise smooth systems, Physica D, 57, pp. 39–57.

    Article  MathSciNet  MATH  Google Scholar 

  8. Budd, C., Dux, F. and Cliffe, A. (1995) The effect of frequency and clearance variations on single degree of freedom impact oscillators, J. Sound Vib., 184, pp. 475–502.

    Article  MATH  Google Scholar 

  9. E. Ott, C. Grebogi and J.A. Yorke (1990) Controlling chaos, Phys. Rev. Lett, 64, pp. 1196–1199; F.J. Romeiras, C. Grebogi, E. Ott and W.P. Dayawansa (1992) Controlling chaotic dynamical systems, Physica D 58, pp. 165–192; E. Barreto and C. Grebogi (1995) Multiparameter control of chaos, Phys. Rev. E, 52, pp. 3553–3557. See also E. Barreto, Y.C. Lai and C. Grebogi (1996), Controlling Chaos with applications to Mechanical Systems, in F.C. Moon (ed.), Nonlinear Dynamics of Material Processing and Manufacturing, John Wiley and Sons, Inc., New York, to be published.

    Article  MathSciNet  MATH  Google Scholar 

  10. Chin, W., Ott, E., Nusse, H.E. and Grebogi, C. (1994) Grazing bifurcations in impact oscillators, Phys. Rev. E, 50, pp. 4427–4444.

    Article  MathSciNet  Google Scholar 

  11. Chin, W., Ott, E., Nusse, H.E. and Grebogi, C. (1995) Universal behavior of impact oscillators near grazing incidence, Phys. Lett. A, 201, pp. 197–204.

    Article  MathSciNet  MATH  Google Scholar 

  12. Casas, F., Chin, W., Grebogi, G. and Ott, E. (1995) Universal grazing bifurcations in impact oscillators, submitted for publication.

    Google Scholar 

  13. U. Dressier and G. Nitsche (1992) Controlling chaos using time delay coordinates, Phys. Rev. Lett, 68, pp. 1–4; P. So and E. Ott (1995) Controlling chaos using time delay coordinates via stabilization of periodic orbits, Phys. Rev. E, 51, pp. 2955–2962.

    Article  Google Scholar 

  14. E.J. Kostelich, C. Grebogi, E. Ott and J.A. Yorke (1987) Multi-dimensioned intertwined basin boundaries: basin structure of the kicked double rotor, Physica D, 25, pp. 347–360; C. Grebogi, E.J. Kostelich, E. Ott and J.A. Yorke (1986) Multi-dimensioned intertwined basin boundaries and the kicked double rotor, Phys. Lett. A, 118, pp. 448–452 and errata(1987), 120, pp. 497.

    Article  MathSciNet  Google Scholar 

  15. E. Barreto, E.J. Kostelich, C. Grebogi, E. Ott and J.A. Yorke (1995) Efficient switching between controlled unstable periodic orbits in higher dimensional chaotic systems, Phys. Rev. E, 51, pp 4169–4172.

    Article  Google Scholar 

  16. T. Shinbrot, E. Ott, C. Grebogi and J.A. Yorke (1990) Using chaos to direct trajectory to targets, Phys. Rev. Lett., 65, pp. 3215–3218.

    Article  Google Scholar 

  17. T. Shinbrot, W. Ditto, C. Grebogi, E. Ott, M. Spano and J.A. Yorke (1992) Using the sensitive dependence of chaos (the ‘butterfly effect’) to direct trajectories in an experimental chaotic system, Phys. Rev. Lett., 68, pp. 2863–2866.

    Article  MathSciNet  MATH  Google Scholar 

  18. E.J. Kostelich, C. Grebogi, E. Ott, and J.A. Yorke (1993) Higher-dimensional targeting, Phys. Rev. E, 47, p. 305–310.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, University of Maryland, College Park, MD, 20742, USA

    Ernest Barreto

  2. Institute for Plasma Research, University of Maryland, College Park, MD, 20742, USA

    Ernest Barreto & Celso Grebogi

  3. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Celso Grebogi

  4. Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA

    Celso Grebogi

  5. Departament de Matemàtiques, Universitat Jaume I, 12071, Castellón, Spain

    Fernando Casas

  6. Department of Mathematics, Arizona State University, Tempe, AZ, 85287, USA

    Eric J. Kostelich

Authors
  1. Ernest Barreto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fernando Casas
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Celso Grebogi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Eric J. Kostelich
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

    D. H. Van Campen

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer Science+Business Media Dordrecht

About this paper

Cite this paper

Barreto, E., Casas, F., Grebogi, C., Kostelich, E.J. (1997). Control of Chaos: Impact Oscillators and Targeting. In: Van Campen, D.H. (eds) IUTAM Symposium on Interaction between Dynamics and Control in Advanced Mechanical Systems. Solid Mechanics and Its Applications, vol 52. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5778-0_3

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-5778-0_3

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6439-2

  • Online ISBN: 978-94-011-5778-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation