Skip to main content
SpringerLink
Log in

Application of Cell Mapping methods to a discontinuous dynamic system

  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The Cell Mapping method is a robust tool for investigating nonlinear dynamic systems. It is capable of finding the attractors and corresponding basins of attraction of a system under investigation. To investigate the applicability of the Cell Mapping method to discontinuous systems, a “forced zero-stiffness impact oscillator” is chosen as an application. The numerical integration algorithm, the basic element in the Cell Mapping method, is adjusted to overcome the discontinuity. Four types of Cell Mapping techniques are applied: Simple Cell Mapping, Generalized Cell Mapping, Interpolated Cell Mapping, and Mixed Cell Mapping. The last type is a new modification to existing types. Each type of Cell Mapping is briefly explained. The results are compared to the exact solutions. The Interpolated Cell Mapping and Mixed Cell Mapping methods are found to produce the most accurate results for this case.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Hénon, M., ‘On the numerical computation of Poincaré maps’, Physica D 5, 1982, 412–414.

    Google Scholar 

  2. Hsu, C. S., ‘A theory of cell-to-cell mapping dynamical systems’, Journal of Applied Mechanics 47, 1980, 931–939.

    Google Scholar 

  3. Hsu, C. S., Cell-to-Cell Mapping; A Method of Global Analysis for Nonlinear Systems, Springer-Verlag, New York, Heidelberg, Berlin, Tokyo, 1987.

    Google Scholar 

  4. Isaacson, D. L. and Madsen, R. W., Markov Chains: Theory and Applications, John Wiley & Sons, New York, London, Sidney, Toronto, 1976.

    Google Scholar 

  5. Li, G. X., Rand, R. H., and Moon, F. C., ‘Bifurcation and chaos in a forced zero-stiffness impact oscillator’, International Journal of Non-Linear Mechanics 25, 1990, 417–432.

    Google Scholar 

  6. Tongue, B. H., ‘On obtaining global nonlinear system characteristics through interpolated cell mapping’, Physica D 28, 1987, 401–408.

    Google Scholar 

  7. Tongue, B. H., ‘Interpolated cell mapping of dynamical systems’, Journal of Applied Mechanics 55, 1988, 461–466.

    Google Scholar 

  8. Tongue, B. H., ‘A multiple-map strategy for interpolated mapping’, International Journal of Non-Linear Mechanics 25, 1990, 177–186.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    J. A. W. Van Der Spek, C. A. L. De Hoon, A. De Kraker & D. H. Van Campen

Authors
  1. J. A. W. Van Der Spek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. A. L. De Hoon
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. De Kraker
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. D. H. Van Campen
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Van Der Spek, J.A.W., De Hoon, C.A.L., De Kraker, A. et al. Application of Cell Mapping methods to a discontinuous dynamic system. Nonlinear Dyn 6, 87–99 (1994). https://doi.org/10.1007/BF00045434

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00045434

Key words

Search

Navigation