Progress in Nonlinear Dynamics and Chaos

  • J. M. T. Thompson
Part of the International Centre for Mechanical Sciences book series (CISM, volume 342)


Like any new scientific discipline, the modern geometrical theory of nonlinear dynamics and chaos has required a galaxy of specialised concepts and terminologies. These can be a major obstacle to applied scientists and engineers wishing to apply the powerful new methods in their own fields. To help overcome this, we provide here an overview of the subject that aims to highlight the central concepts and ideas that will be of particular importance in practical applications. Recent books which the reader may find helpful are those of Guckenheimer & Holmes (1983), Thompson & Stewart (1986), Moon (1987), Arrowsmith & Place (1990) and Abraham & Shaw (1992). Collections of modern applications are edited by Schiehlen (1990), Thompson & Gray (1990), Kim & Stringer (1992), Thompson & Schiehlen (1992) and Mullin (1993).


Phase Portrait Chaotic Attractor Homoclinic Orbit Basin Boundary Saddle Connection 
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.


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  1. Abraham, R.H. & Shaw, C.D. (1992) Dynamics: The Geometry of Behaviour, Addison-Wesley, Redwood City.Google Scholar
  2. Arrowsmith, D.K. & Place, C.M. (1990) An Introduction to Dynamical Systems, Cambridge University Press, Cambridge.MATHGoogle Scholar
  3. Falconer, K. (1990) Fractal Geometry, Wiley, Chichester.MATHGoogle Scholar
  4. Foale, S. & Bishop, S.R. (1992) Dynamical complexities of forced impacting systems, Phil. Trans. R. Soc. Lond. A338, 547–556.MATHMathSciNetCrossRefGoogle Scholar
  5. Foale, S. & Thompson, J.M.T. (1991) Geometrical concepts and computational techniques of nonlinear dynamics, Computer Methods in Applied Mechanics & Engng, 89, 381–394.MathSciNetCrossRefGoogle Scholar
  6. Ghrist, R. & Holmes, P. (1993) Knots and orbit genealogies in three dimensional flows. In Bifurcations and Periodic Orbits of Vector Fields, NATO ASI Series, Kluwer Academic Press.Google Scholar
  7. Grebogi, C., Ott, E. & Yorke, J.A. (1987) Basin boundary metamorphoses: changes in accessible boundary orbits, Physica D24, 243–262.MATHMathSciNetGoogle Scholar
  8. Guckenheimer, J. & Holmes, P. (1983) Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York.MATHCrossRefGoogle Scholar
  9. Kim, J.H. & Stringer, J. (eds) (1992) Applied Chaos, Wiley, New York.MATHGoogle Scholar
  10. Lansbury, A.N., Thompson, J.M.T. & Stewart, H.B. (1992) Basin erosion in the twin-well Duffing oscillator: two distinct bifurcation scenarios, Int. J. Bifn & Chaos, 2, 505–532.MATHMathSciNetCrossRefGoogle Scholar
  11. McDonald, S.W., Grebogi, C., Ott, E. & Yorke, J.A. (1985) Fractal basin boundaries, Physica D17, 125–153.MATHMathSciNetGoogle Scholar
  12. McRobie, F.A. (1992a) Bifurcational precedences in the braids of periodic orbits of spiral 3-shoes in driven oscillators, Proc. R. Soc. Lond. A438, 545–569.MATHMathSciNetCrossRefGoogle Scholar
  13. McRobie, F.A. (1992b) Birkhoff signature change: a criterion for the instability of chaotic resonance, Phil. Trans. R. Soc. Lond. A338, 557–568.MATHMathSciNetCrossRefGoogle Scholar
  14. McRobie, F.A. & Thompson, J.M.T. (1991) Lobe dynamics and the escape from a potential well, Proc. R. Soc. Lond. A435, 659–672.MATHMathSciNetCrossRefGoogle Scholar
  15. McRobie, F.A. & Thompson, J.M.T. (1992) Invariant sets of planar diffeomorphisms in nonlinear vibrations, Proc. R. Soc. Lond. A436, 427–448.MATHMathSciNetCrossRefGoogle Scholar
  16. McRobie, F.A. & Thompson, J.M.T. (1993) Braids and knots in driven oscillators, Int. J. Bifn & Chaos, in press.Google Scholar
  17. McRobie, F.A. & Thompson, J.M.T. (1994) Criteria for escape phenomena in driven oscillators using Melnikov-like energy estimates, Nonlinear Dynamics, in press.Google Scholar
  18. Moon, F.C. (1987) Chaotic Vibrations, Wiley, New York.MATHGoogle Scholar
  19. Mullin, T. (ed) (1993) The Nature of Chaos, Oxford University Press, Oxford.MATHGoogle Scholar
  20. Parker, T.S. & Chua, L.O. (1989) Practical Numerical Algorithms for Chaotic Systems, Springer, New York.MATHCrossRefGoogle Scholar
  21. Schiehlen, W. (ed) (1990) Nonlinear Dynamics in Engineering Systems, Proc. IUTAM Symposium, Stuttgart, Aug 1989. Springer, Berlin.Google Scholar
  22. Soliman, M.S. & Thompson, J.M.T. (1992a) Indeterminate sub-critical bifurcations in parametric resonance, Proc. R. Soc. Lond. A438, 511–518.MATHMathSciNetCrossRefGoogle Scholar
  23. Soliman, M.S. & Thompson, J.M.T. (1992b) Indeterminate trans-critical bifurcations in parametrically excited systems, Proc. R. Soc. Lond. A439, 601–610.MATHMathSciNetCrossRefGoogle Scholar
  24. Soliman, M.S. & Thompson, J.M.T. (1992c) Global dynamics underlying sharp basin erosion in nonlinear driven oscillators, Physical Review A45, 3425–3431.CrossRefGoogle Scholar
  25. Stewart, H.B. (1987) A chaotic saddle catastrophe in forced oscillators. In Dynamical Systems Approaches to Nonlinear Problems in Systems and Circuits. Salam, F. & Levi, M. (eds.), SIAM, Philadelphia.Google Scholar
  26. Stewart, H.B. & Ueda, Y. (1991) Catastrophes with indeterminate outcome, Proc. R. Soc. Lond. A432, 113–123.MATHMathSciNetCrossRefGoogle Scholar
  27. Thompson, J.M.T. (1989) Chaotic phenomena triggering the escape from a potential well, Proc. R. Soc. Lond. A421, 195–225.MATHCrossRefGoogle Scholar
  28. Thompson, J.M.T. (1992) Global unpredictability in nonlinear dynamics: capture, dispersal and the indeterminate bifurcations, Physica D58, 260–272.MATHMathSciNetGoogle Scholar
  29. Thompson, J.M.T. & Gray, P. (eds) (1990) Chaos and Dynamical Complexity in the Physical Sciences, First Theme Issue, Phil. Trans. R. Soc. Lond. A332, 49–186.Google Scholar
  30. Thompson, J.M.T. & McRobie, F.A. (1993) Indeterminate bifurcations and the global dynamics of driven oscillators, Plenary lecture, First European Nonlinear Oscillations Conference, Hamburg, Aug 1993. Edited by E. Kreuzer & G. Schmidt, Akademie Verlag, Berlin, 1993. pp 107–128.Google Scholar
  31. Thompson, J.M.T., Rainey, R.C.T. & Soliman, M.S. (1990) Ship stability criteria based on chaotic transients from incursive fractals, Phil. Trans. R. Soc. Lond. A332, 149–167.MATHMathSciNetCrossRefGoogle Scholar
  32. Thompson, J.M.T., Rainey, R.C.T. & Soliman, M.S. (1992) Mechanics of ship capsize under direct and parametric wave excitation, Phil. Trans. R. Soc. Lond. A338, 471–490.MATHCrossRefGoogle Scholar
  33. Thompson, J.M.T. & Schiehlen, W. (eds) (1992) Nonlinear Dynamics of Engineering Systems, Theme Issue, Phil. Trans. R. Soc. Lond. A338, 451–568.Google Scholar
  34. Thompson, J.M.T. & Soliman, M.S. (1990) Fractal control boundaries of driven oscillators and their relevance to safe engineering design, Proc. R. Soc. Lond. A428, 1–13.MATHCrossRefGoogle Scholar
  35. Thompson, J.M.T. & Soliman, M.S. (1991) Indeterminate jumps to resonance from a tangled saddle-node bifurcation, Proc. R. Soc. Lond. A432, 101–111.MATHMathSciNetCrossRefGoogle Scholar
  36. Thompson, J.M.T. & Stewart, H.B. (1986) Nonlinear Dynamics and Chaos, Wiley, Chichester.MATHGoogle Scholar
  37. Thompson, J.M.T. & Stewart, H.B. (1993), A tutorial glossary of geometrical dynamics, Int. J. Bifn & Chaos, 3, 223–239.MATHMathSciNetCrossRefGoogle Scholar
  38. Thompson, J.M.T., Stewart, H.B. & Ueda, Y. (1993) Safe, explosive and dangerous bifurcations in dissipative dynamical systems, Phys. Rev. (E), in press.Google Scholar

Copyright information

© Springer-Verlag Wien 1995

Authors and Affiliations

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

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