Skip to main content
SpringerLink
Log in

Computational Aspects in Chaotic and Stochastic Dynamics

  • Chapter
Dynamic Motion: Chaotic and Stochastic Behaviour

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 340))

  • 116 Accesses

Abstract

The computational aspects of the study of chaotic and stochastic motion can be framed in that is presently know as numerical dynamics. With these two words, one defines the modelling, the simulation and the understanding of an evolutionary physical system by means of the computer. It is to stress that, by means of the simulation of the physical system through a computer experiment, one hopes to achieve a better understanding of the unknown and forbidden structures of the system [1, 2, 3, 4, 5, 6]. In this way the study of a dynamical system through the computer is sometimes similar to the classical experimental work.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight 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. Abelson H., Disessa A., La geometria della tarta.ruga. Esplorare la matematica con il computer., (in Italian ), Franco Muzzio, 1986.

    Google Scholar 

  2. Gould H., Tobochnik J., An Introduction to Computer Simulation Methods., Addison-Wesley Publishing Company, Inc., Reading Massachusetts, USA, 1987.

    Google Scholar 

  3. Kocak H., Differential and Difference Equations through Computer Experiments., Springer-Verlag, 1989.

    Google Scholar 

  4. Kreutzer E., Numerische Untersuchung nichtlinearer dynamischer Systeme (in German)., Springer Verlag, (1987).

    Google Scholar 

  5. Morrison F., The art of modeling dynamic systems., John Wiley & Sons, 1991.

    Google Scholar 

  6. Whitney C.A., Random Processes in Physical Systems., John Wiley & Sons, 1990.

    Google Scholar 

  7. Casciati F., Faravelli L., Fragility Analysis of Complex Structural Systems., Research Studies Press, Tauton, 1990.

    Google Scholar 

  8. Bender C.M., Orszag S.A., Advanced Mathematical Methods for Scientist and Engineers., McGraw-Hill, 1978.

    Google Scholar 

  9. Peitgen H.O., Jurgens H., Saupe D., Chaos and Fractals. New Frontiers of Science. Springer-Verlag, 1992.

    Google Scholar 

  10. Baker G.L., Gollub J.P., Chaotic Dynamics. An Introduction., Cambridge University Press, 1990.

    MATH  Google Scholar 

  11. Arnold V.I., Equazioni differenziali ordinarie., (in Italian), Editori Riuniti, 1979.

    Google Scholar 

  12. Arnold V.I., Metodi geometrici della teoria delle equazioni differenziali ordinarie., (in Italian), Editori Riuniti, 1989.

    Google Scholar 

  13. Birkhoff G., Rota G.C., Ordinary differential equations. John Wiley & Sons, 1959.

    Google Scholar 

  14. Pagani C.D., Salsa S., Analisi Matematica., (in Italian), Masson, 1991.

    Google Scholar 

  15. Schuster H.G., Deterministic Chaos. An introduction., VCH Weinheim, (1989).

    MATH  Google Scholar 

  16. Seydel R., From Equilibrium to Chaos., Elsevier, (1988).

    Google Scholar 

  17. Bontempi F., Casciati F., Analisi stocastica di un oscillatore caotico., (in Italian), Atti dell’AIMETA 92, XI Congresso Nazionale dell’associazione Italiana di Meccanica Teorica ed Applicata, Trento 28 settembre - 2 ottobre 1992.

    Google Scholar 

  18. Bontempi F., Casciati F., Chaotic Motion and Stochastic Excitation., accepted for publication in Nonlinear Dynamics.

    Google Scholar 

  19. Bontempi F., Casciati F., Stochastic and Chaotic Dynamics., in Computational Stochastic Mechanics, Computational Pubblications Ashuerst Lodge, 1993.

    Google Scholar 

  20. Bontempi F., Transizione al caos in problemi di dinamica strutturale. (in Italian), Ph.D. Thesis, Polytechnic of Milan, Milan 1993.

    Google Scholar 

  21. Abraham R.H., Shaw C.D., Dynamics - The geometry of behaviour. Aerial Press, Inc. Santa Cruz, 1983.

    Google Scholar 

  22. Falconer K., Fractal Geometry., John Wiley & Sons, 1990.

    Google Scholar 

  23. Rasband S.N., Chaotic Dynamics of Nonlinear Systems., John Wiley & Sons, 1990.

    Google Scholar 

  24. Bolotin V.V., Statistical Methods in Structural Mechanics., Holden-Day Inc., San Francisco, 1969.

    MATH  Google Scholar 

  25. Bontempi F., Casciati F., Faravelli L., Chaotic Motion versus Stochastic Excitation., Chaos, Solitons and Fractals, Vol. 1,•m. 6, pp. 491–500, 1991.

    Google Scholar 

  26. Bontempi F., Casciati F., Comport.amento caotico e stocastico di un oscillatore non lineare., (in Italian), Atti del Convegno Nazionale del Gruppo AIMETA di Meccanica dei Mat.eriali e delle Structure, Amalfi, 3–5 giugno 1991.

    Google Scholar 

  27. Abramowitz M., Stegun I.A., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables., Dover, 1959

    Google Scholar 

  28. Belytschko T., Hughes T.J.R., Computational Methods for Transient Analysis., North-Holland, 1983.

    Google Scholar 

  29. Bathe K.J., Finite Element Procedures in Engineering Analysis., Prentice-Hall, 1982.

    Google Scholar 

  30. Argyris J., Mlejnek Ii.P., Dynamics of Structures., Text on Computational Mechanics Volume V, North-Holland, 1991.

    MATH  Google Scholar 

  31. Griffiths D.V., Smith 1.M., Numerical Methods for Engineers., Blackwell Scientific Publications, 1991.

    Google Scholar 

  32. Hughes T.J.R., The Finite Element Method., Prentice-Hall, 1987.

    Google Scholar 

  33. Press W.H., Flannery B.P., Teukolsky S.A., Vetterling W.T., Numerical Recipes. The Art of Scientific Computating., Cambridge University Press, 1986.

    Google Scholar 

  34. St.oer J., Burlish R., Introduction to Numerical Analysis. Springer Verlag, 1980.

    Google Scholar 

  35. Augusti G., Baratta A., Casciati F., Probabilistic Methods in Structural Engineering., Chapman & Hall, 1984.

    Google Scholar 

  36. Gardiner C.W., Handbook of Stochastic Methods., Springer Series in Synergetics, Springer Verlag, 1983.

    Google Scholar 

  37. Wedig W., Vom Chaos zur Ordnung., GAMM–Mitteilungen, Heft 2, pp. 3–31, 1988.

    Google Scholar 

  38. Wedig W., Analysis and Simulation of Nonlinear Stochastic Systems., Proceedings of the IUTAM Symposium on Nonlinear Dynamics in Engineering Systems Stuttgart/Germany 1989.

    Google Scholar 

  39. Tabor M., Chaos and Integrability in Nonlinear Dynamics., John Wiley & Sons, (1989).

    Google Scholar 

  40. Jackson A., Perspectives in Nonlinear Dynamics. Vol. II., Cambridge University Press, (1991).

    Google Scholar 

  41. Wedig W., Lyapunov Exponents of Stochastic Systems and related Bifurcation Problem., Euromech 229, Nonlinear Applied Dynamics, Stuttgart October 5–9, 1987.

    Google Scholar 

  42. Wedig W., Lyapunov Exponents and Invariant Measures of Dynamics Systems., International Series of Mathematics, Vol. 97, 1991 Birkhauser Verlag Basel.

    Google Scholar 

  43. Lichtenberg A.J., Lieberman, M.A., Regular and Stochastic Motion., Springer Verlag, New York, 1983.

    Book  MATH  Google Scholar 

  44. Wolf A., Swift J.B., Swinney H.L., Vastano J.A., Determining Lyapunov exponents from a time series., Physical Review A, Vol. 29, N. 3, March 1984.

    Google Scholar 

  45. Mayer-Kress G., (Ed.), Dimensions and Entropies in Chaotic Systems., Springer Verlag, Berlin, 1986.

    MATH  Google Scholar 

  46. Hamilton I., Brumer P., Relaxation rates for two-dimensional deterministic mappings., Physical Review A, Vol. 25, N. 6, June 1982.

    Google Scholar 

  47. Termonia Y., Kolmogorov entropy from a time series., Physical Review A, Vol. 29, N. 3, March 1984.

    Google Scholar 

  48. Abraham N.B., Albano A.M., Passamante A., Rapp P.E., Measures of Complexity and Chaos., NATO ASI Series, Plenum Press, New York 1989.

    Google Scholar 

  49. Risken H., The Fokker-Planck Equation., Springer Series in Synergetics, Springer Verlag, 1984.

    Google Scholar 

  50. Fletcher C.A.J., Coinputational Techniques for Fluid Dynamics., Springer Series in Computational Physics, Springer Verlag, 1990.

    Google Scholar 

  51. Johnson C., Numerical solution of partial differential equations by the finite element method., Cambridge University Press, 1990.

    Google Scholar 

  52. Kunert A., Pfeiffer F., Description of Chaotic Motion by an Invariant Distribution at the Example of the Driven Duffing Oscillator., International Series of Numerical Mathematics, Vol. 97, (1991) Birkhauser Verlag Basel.

    Google Scholar 

  53. Kunert A., Naeherung fuer Invariante Distributionen chaotischer Systeme (in German)., ZAMM - Z.angew.Math.Mech. 71 (1991), T131 - T134.

    Google Scholar 

  54. Kunert A., Pfeiffer F., Description of Chaotic Motion by an Invariant Probability Density. Third Conference on Nonlinear Vibrations, Stability and Dynamics of Structures and Mechanism, Blacksburg VA June 25–27, 1990.

    Google Scholar 

  55. Langtangen H.P., A general numerical solution method for Fokker-Planck equations with applications to structural reliability., Probabilistic Engineering Mechanics, (1991), Vol. 6, No. 1, pp. 33–48.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Polytechnic of Milan, Milan, Italy

    F. Bontempi

Authors
  1. F. Bontempi
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Pavia, Italy

    F. Casciati

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer-Verlag Wien

About this chapter

Cite this chapter

Bontempi, F. (1993). Computational Aspects in Chaotic and Stochastic Dynamics. In: Casciati, F. (eds) Dynamic Motion: Chaotic and Stochastic Behaviour. International Centre for Mechanical Sciences, vol 340. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2682-0_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-7091-2682-0_4

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-82517-4

  • Online ISBN: 978-3-7091-2682-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation