Fractal Control Boundaries of Driven Oscillators and Their Relevance to Safe Engineering Design

  • J. M. T. Thompson
  • M. S. Soliman
Part of the International Centre for Mechanical Sciences book series (CISM, volume 319)


When a metastable, damped oscillator is driven by strong periodic forcing, the catchment basin of constrained motions in the space of the starting conditions {x(0), (0)} develops a fractal boundary associated with a homoclinic tangling of the governing invariant manifolds. The four-dimensional basin in the phase-control space spanned by {x(0), (0), F,ω}, where F is the magnitude and ω the frequency of the excitation, will likewise acquire a fractal boundary, and we here explore the engineering significance of the control cross section corresponding, for example, to x(0) = (0) = 0. The fractal boundary in this section is a failure locus for a mechanical or electrical system subjected, while resting in its ambient equilibrium state, to a sudden pulse of excitation. We assess here the relative magnitude of the uncertainties implied by this fractal structure for the optimal escape from a universal cubic potential well. Absolute and transient basins are examined, giving control-space maps analogous to familiar pictures of the Mandelbrot set. Generalizing from this prototype study, it is argued that in engineering design, against boat capsize or earthquake damage, for example, a study of safe basins should augment, and perhaps entirely replace, conventional analysis of the steady-state attracting solutions.


Fractal Dimension Fractal Boundary Control Space Basin Boundary Homoclinic Tangency 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bird, H. & Morali, A. 1986 Research towards realistic stability criteria. Proc. Int. Cont on the safeship project: ship stability and safety. London: Royal Institute of Naval Architects.Google Scholar
  2. Eschenazi, E., Solari, H. G. & Gilmore, R. 1989 Basins of attraction in driven dynamical systems. Phys. Rev. A 39, 2609–2627.MathSciNetCrossRefGoogle Scholar
  3. Fleischmann, M., Tildesley, D. J. & Ball, R. C. (eds) 1989 Fractals in the Natural Sciences, R.yal Society Discussion Meeting. Proc. R. Soc. Lond. A 423, 1–200.Google Scholar
  4. Grebogi, C., McDonald, S. W., Ott, E. & Yorke, J. A. 1983 Final state sensitivity: an obstruction to predictability. Phys. Lett. A 99, 415–419.MathSciNetCrossRefGoogle Scholar
  5. Mandelbrot, B. B. 1977 The fractal geometry of nature. New York: Freeman.Google Scholar
  6. Miller, D. R., Tam, G., Rainey, R. C. T. & Ritch, R. 1986 Investigation of the use of modern ship motion prediction models in identifying ships with a larger than acceptable risk of dynamic capsize. Report prepared by Arctec Canada Ltd for the Transportation Development Centre of the Canadian Government. Report no. TP7407E.Google Scholar
  7. Moon, F. C. 1987 Chaotic vibrations: an introduction far applied scientists and engineers. New York: Wiley.Google Scholar
  8. McDonald, S. W., Grebogi, C., Ott, E. & Yorke, J. A. 1985 Fractal basin boundaries. Physica D 17, 125–153.MathSciNetMATHGoogle Scholar
  9. Peitgen, H. O. & Richter, P. H. 1986 The beauty of fractals. Berlin: Springer-Verlag.CrossRefMATHGoogle Scholar
  10. Pezeshki, C. & Dowell, E. H. 1987 An examiantion of initial condition maps for the sinusoidally excited buckled beam modeled by the Duffing’s equation. J. Sound Vibration 117, 219–232.MathSciNetCrossRefMATHGoogle Scholar
  11. Rainey, R. C. T. & Thompson, J. M. T. 1990 The transient capsize diagram — a new method of quantifying stability in waves. J. Ship Research. (In the press.)Google Scholar
  12. Soliman, M. S. & Thompson, J. M. T. 1989 Integrity measures quantifying the erosion of smooth and fractal basins of attraction. J. Sound Vibration. (In the press.)Google Scholar
  13. Thompson, J. M. T. 1989 Chaotic phenomena triggering the escape from a potential well. Proc. R. Soc. Lond. A 421, 195–225.CrossRefMATHGoogle Scholar
  14. Thompson, J. M. T. & Stewart, H. B. 1986 Nonlinear dynamics and chaos. Chichester: Wiley.MATHGoogle Scholar
  15. Thompson, J. M. T. & Ueda, Y. 1989 Basin boundary metamorphoses in the canonical escape equation. Dynamics Stability Syst. 4, 285–294.MathSciNetCrossRefGoogle Scholar
  16. Virgin, L. N. 1987 The nonlinear rolling response of a vessel including chaotic motions leading to capsize in regular seas. Appl. Ocean Res. 9, 89–95.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 1991

Authors and Affiliations

  • J. M. T. Thompson
    • 1
  • M. S. Soliman
    • 1
  1. 1.University College LondonLondonUK

Personalised recommendations