Skip to main content
SpringerLink
Log in

Model of a spatially inhomogeneous one-dimensional active medium

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We investigate the dynamics of one-dimensional discrete models of a one-component active medium analytically. The models represent spatially inhomogeneous diffusively concatenated systems of one-dimensional piecewise-continuous maps. The discontinuities (the defects) are interpreted as the differences in the parameters of the maps constituting the model. Two classes of defects are considered: spatially periodic defects and localized defects. The area of regular dynamics in the space of the parameters is estimated analytically. For the model with a periodic inhomogeneity, an exact analytic partition into domains with regular and with chaotic types of behavior is found. Numerical results are obtained for the model with a single defect. The possibility of the occurrence of each behavior type for the system as a whole is investigated.

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. K. Kaneko,Physica D,34, 1–41 (1989).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. K. Kaneko,Physica D,37, 60–82 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  3. L. A. Bunimovich and Ya. G. Sinai, “Statistical mechanics of coupled map lattices,” in:Theory and Applications of Coupled Map Lattices (K. Kaneko, ed.), Wiley, San Francisco (1993), pp. 169–189.

    Google Scholar 

  4. A. S. Mikhailov and A. Yu. Loskutov,Chaos and Noise, Springer, Berlin (1996).

    MATH  Google Scholar 

  5. M. Bär and M. Eiswirth,Phys. Rev. E,48, 1635–1637 (1993).

    Article  ADS  Google Scholar 

  6. Zhilin Qu, J. N. Weiss, and A. Garfinkel,Phys. Rev. Lett,58, 1378–1390 (1997).

    Google Scholar 

  7. S. Morita,Phys. Lett. A,226, 172–178 (1997).

    Article  ADS  Google Scholar 

  8. A. P. Muñuzuri, V. Perez-Muñuzuri, M. Gomez-Gesteira, L. O. Chua, and V. Perez-Villar,Int. J. Bifurcation Chaos Appl. Sci. Eng.,5, No. 1, 17–50 (1995).

    Article  MATH  Google Scholar 

  9. L. A. Bunimovich,Physica D,86, 248–255 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  10. K. Kaneko, ed.,Theory and Applications of Coupled Map Lattices, Wiley, San Francisco (1993).

    MATH  Google Scholar 

  11. Chaos Focus Issue on Coupled Map Lattices,Chaos,2, No. 3 (1992).

  12. K. Kaneko,Physica D,41, 137–172 (1990).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. L. A. Bunimovich and Ya. G. Sinai,Nonlinearity,1, 491–504 (1988).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. G. Perez, S. Sinha, and H. Cerdeira,Phys. Rev. E,54, 6936–6939 (1996).

    Article  ADS  Google Scholar 

  15. W. Just,J. Stat. Phys.,79, 429–449 (1995).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  16. K. Kaneko,Physica D,54, 5–19 (1991).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. V. S. Afraimovich, V. I. Nekorkin, G. V. Osipov, and V. D. Shalfeev,Stability, Structures and Chaos in Nonlinear Synchronization Nets [in Russian], Inst. Appl. Phys., Acad. Sci., Gor'kiy (1989).

    Google Scholar 

  18. V. A. Vasil'ev, Yu. M. Romanovskii, and V. G. Yakhno,Autowave Processes [in Russian], Nauka, Moscow (1987); English transl.: V. A. Vasil'ev, Yu. M. Romanovskii, D. S. Chernavskii, and V. G. YakhnoAutowave Processes in Kinetic Systems: Spatial and Temporal Self-Organization in Physics, Chemistry, Biology, and Medicine, Kluwer, Dordrecht (1987).

    MATH  Google Scholar 

  19. A. V. Gaponov-Grekhov, ed.,Autowave Processes in Systems with Diffusion [in Russian], (Collected papers), Inst. Appl. Phys., Acad. Sci., Gor'kiy (1981).

    Google Scholar 

  20. Ya. B. Zel'dovich, G. I. Barenblatt, and V. B. Librovich,Mathematical Theory of Combustion and Explosion [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  21. D. G. Aronson and H. F. Weinberger, “Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation,” in:Partial Differential Equations and Related Topics (J. A. Goldstein, ed.) (Lect. Notes Math., Vol. 446), Springer, Berlin (1975), pp. 5–49.

    Chapter  Google Scholar 

  22. P. A. Horn and C. R. Johnson,Matrix Analysis, Cambridge Univ. Press, Cambridge (1986).

    Google Scholar 

  23. A. Yu. Loskutov, S. D. Rybalko, and D. N. Udin, “Dynamics of an inhomogeneous net of nonautonomous diffusively connected piecewise-linear maps,”Regular and Chaotic Dynamics (forthcoming).

  24. A. Loskutov, S. D. Rybalko, D. N. Udin, and N. V. Brillianton, “Space-time chaos and order in nonuniform coupled map lattices,”Physica D (submitted).

Download references

Author information

Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    K. A. Vasil'ev, A. Yu. Loskutov, S. D. Rybalko & D. N. Udin

Authors
  1. K. A. Vasil'ev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Yu. Loskutov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. D. Rybalko
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. D. N. Udin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 506–519, September, 2000.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vasil'ev, K.A., Loskutov, A.Y., Rybalko, S.D. et al. Model of a spatially inhomogeneous one-dimensional active medium. Theor Math Phys 124, 1286–1297 (2000). https://doi.org/10.1007/BF02551005

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02551005

Keywords

Search

Navigation