Skip to main content
Log in

Topological methods for transients of driven systems

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

We consider the application of topological methods (such as knot, braid and Nielsen-Thurston theory) to transient, rather than periodic, orbits of periodically-forced nonlinear oscillators. The methods are restricted to systems with a three-dimensional phase space.

Sommario

Si considera l'applicazione di metodi topologici (basati sulle teorie dei nodi, delle trecce e di Nielsen-Thurston) allo studio delle orbite transitorie, piuttosto che stazionarie, di oscillatori nonlineari forzati periodicamente. Tali applicazioni sono ristrette a sistemi aventi spazio delle fasi tridimensionale.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. ThompsonJ.M.T., ‘Chaotic behaviour triggering the escape from a potential well’, Proc. R. Soc. Lond., A421 (1989) 195–225.

    Google Scholar 

  2. KangI.S. and LealI.G., ‘Bubble dynamics in time-periodic straining flows’, J. Fluid Mechanics, 218, (1990) 41–69.

    Google Scholar 

  3. GrimshawR. and TianX., ‘Periodic and chaotic behaviour in a reduction of the perturbed Korteweg-de Vries equation’, Proc. R. Soc. Lond., A445 (1994) 1–21.

    Google Scholar 

  4. CliffordM.J. and BishopS.R., ‘Bifurcational precedences for parametric escape from a symmetrical potential well’, Int. J. Bifurcation and Chaos, 4 (3) (1994) 623–630.

    Google Scholar 

  5. HolmesP.J. and WhitleyD., ‘Bifurcations of one-and two-dimensional maps’, Phil. Trans.R.Soc.Lond., A 311 (1984) 43–102.

    Google Scholar 

  6. Holmes P.J., ‘Knots and orbit genealogies in nonlinear oscillators’. In: Bedford and Swift (eds.), New Directions in Dynamical Systems, LMS 127, Cambridge University Press, 1988.

  7. BoylandP.L., ‘Braid-types of periodic orbits for surface automorphisms’. In: R. S.Mackay (ed.), Notes on Dynamics of Surface Homeomorphisms, Lecture notes, Maths. Institute, Warwick University, 1989.

    Google Scholar 

  8. BoylandP.L., ‘Topological methods in surface dynamics’, Topol. and its Appl. 58 (1994) 223–298.

    Google Scholar 

  9. Hall, T.D. ‘Periodicity in Chaos: the Dynamics of Surface Automorphisms’, Ph.D. thesis, University of Cambridge, UK., 1991.

  10. McRobieF.A., and ThompsonJ.M.T., ‘Braids and knots in driven oscillators’, Int. J. Bifurcation and Chaos, 3, 6 (1993) 1343–1361.

    Google Scholar 

  11. McRobieF.A. and ThompsonJ.M.T., ‘Driven oscillators, knots, braids and Nielsen-Thurston theory’, in J.M.T.Thompson and S.R.Bishop (eds.), Nonlinearity and Chaos in Engineering Dynamics, Wiley, Chichester, 1994.

    Google Scholar 

  12. McRobieF.A. and ThompsonJ.M.T., ‘Knot-types and bifurcation sequences of homoclinic and transient orbits of a single-degree-of-freedom driven oscillator’, Dyn. and Stabil. of Systs. 9, 3 (1994) 223–251.

    Google Scholar 

  13. Tufillaro, N.B., Wyckoff, P., Brown, R., Schreiber, T. and Molteno, T., ‘Topological time series analysis of a string experiment and its synchronized model’ (preprint, Los Alamos), (1994).

  14. White, T., ‘foldtool’ (An implementation of the Bestvina-Handel algorithm) (tadpole@ucrmath.ucr.edu), (1993).

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bishop, S.R., McRobie, F.A. Topological methods for transients of driven systems. Meccanica 31, 225–234 (1996). https://doi.org/10.1007/BF00426989

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00426989

Key words

Navigation