Skip to main content
SpringerLink
Log in

Complexity for Extended Dynamical Systems

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider dynamical systems for which the spatial extension plays an important role. For these systems, the notions of attractor, ϵ-entropy and topological entropy per unit time and volume have been introduced previously. In this paper we use the notion of Kolmogorov complexity to introduce, for extended dynamical systems, a notion of complexity per unit time and volume which plays the same role as the metric entropy for classical dynamical systems. We introduce this notion as an almost sure limit on orbits of the system. Moreover we prove a kind of variational principle for this complexity.

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. Benci V., Bonanno C., Galatolo S., Menconi G. and Virgilio M. (2004). Dynamical systems and computable information. Discrete Contin. Dyn. Syst. Ser. B 4: 935–960

    MATH  MathSciNet  Google Scholar 

  2. Billingsley P. (1968). Convergence of Probability Measures. Wiley, New York

    MATH  Google Scholar 

  3. Brudno A.A. (1983). Entropy and the complexity of the trajectories of a dynamical system. Trans. Moscow Math. Soc. 2: 127–151

    Google Scholar 

  4. Collet P. (1994). Thermodynamic limit of the Ginzburg-Landau equations. Nonlinearity 7(4): 1175–1190

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Collet, P.: Non-linear parabolic evolutions in unbounded domains. In: Dynamics, Bifurcations and Symmetries, P. Chossat editor, Nato ASI 437, New York-London: Plenum 1994, pp 97–104

  6. Collet P. (2002). Extensive quantities for extended systems. Fields Institute Communications 21: 65–74

    ADS  MathSciNet  Google Scholar 

  7. Collet P. and Eckmann J.-P. (1990). Instabilities and Fronts in Extended Sytems. Princeton University Press, Princeton, NJ

    Google Scholar 

  8. Collet P. and Eckmann J.-P. (1999). Extensive properties of the Ginzburg-Landau equation. Commun. Math. Phys. 200: 699–722

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Collet, P., Eckmann, J.-P.: The definition and measurement of the topological entropy per unit volume in parabolic pde’s. Nonlinearity 12, 451–475 (1999). Erratum: Nonlinearity 14, 907, (2001)

    Google Scholar 

  10. Collet P. and Eckmann J.-P. (2000). Topological entropy and ϵ-entropy for damped hyperbolic equations. Ann. Henri Poincaré 1: 715–752

    Article  MATH  MathSciNet  Google Scholar 

  11. Denker, M., Grillenberger, C., Sigmund, K.: Ergodic Theory on Compact Spaces, LNM 527, Berlin- Heidelberg: Springer-Verlag, 1976

  12. Derrienic Y. (1983). Un theoreme ergodique presque sous-additif. Ann. Probab. 11: 669–677

    MathSciNet  Google Scholar 

  13. Efendiev M., Miranville A. and Zelik S. (2004). Infinite-dimensional exponential attractors for nonlinear reaction-diffusion systems in unbounded domains and their approximation. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460(2044): 1107–1129

    MATH  ADS  MathSciNet  Google Scholar 

  14. Feireisl E. (1996). Bounded locally compact global attractors for semilinear damped wave equations on R N. Differ. Integral Eq. 9: 1147–1156

    MATH  MathSciNet  Google Scholar 

  15. Ginibre J. and Velo G. (1997). The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. II. Contraction methods. Commun. Math. Phys. 187(1): 45–79

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. del Junco A. and Rosenblatt J. (1979). Counterexamples in ergodic theory and number theory. Math. Ann. 245: 185–197

    Article  MATH  MathSciNet  Google Scholar 

  17. del Junco A. and Steele J.M. (1977). Moving averages of ergodic processes. Metrika 24: 35–43

    Article  MATH  MathSciNet  Google Scholar 

  18. Kolmogorov, A.N., Tihomirov, V.T.: ϵ-entropy and ϵ-capacity of sets in functions spaces. In: Selected works of A.N.Kolmogorov, Vol. III, A.N. Shiryayev. ed., Dordrecht: Kluwer (1993)

  19. Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications. Second edition, GTCS, Berlin-Heielberg Newyork: Springer-Verlag, 1997

  20. Mielke A. (1997). The complex Ginzburg-Landau equation on large and unbounded domains: sharper bounds and attractors. Nonlinearity 10(1): 199–222

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. Mielke A. and Schneider G. (1995). Attractors for modulation equations on unbounded domains—existence and comparison. Nonlinearity 8(5): 743–768

    Article  MATH  ADS  MathSciNet  Google Scholar 

  22. Mielke, A., Zelik, S.: Infinite-dimensional hyperbolic sets and spatio-temporal chaos in reaction-diffusion systems in R n. J. Dynam. Differ. Eqs., to appear

  23. Milnor J. (1988). On the entropy geometry of cellular automata. Complex Systems 2: 357–385

    MATH  ADS  MathSciNet  Google Scholar 

  24. Misiurewicz, M.: A short proof of variational principle for \(\mathbb {Z}^n\) action on a compact space, In: International Conference on Dynamical Systems in Mathematical Physics, (Rennes, 1975), Asterisque, no. 40, Paris: Soc. Math. France, 1976, pp. 147–157

  25. Schürger K. (1991). Almost subadditive extensions of Kingman’s ergodic theorem. Ann. Probab. 19: 1575–1586

    MATH  MathSciNet  Google Scholar 

  26. Shannon, C.E.: A mathematical theory of communication. Bell System Tech. J. 27, 379–423, 623–656 (1948)

    Google Scholar 

  27. Takač P., Bollerman P., Doelman A., van Harten A. and Titi E.S. (1996). Analyticity of essentially bounded solutions to semilinear parabolic systems and validity of the Ginzburg-Landau equation. SIAM J. Math. Anal. 27: 424–448

    Article  MATH  MathSciNet  Google Scholar 

  28. Tagi-Zade, A.T.: A variational characterization of the topological entropy of continuous groups of transformations. The case of \(\mathbb {R}^n\) actions. Mat. Zametki 49, 114–123 (1991); English transl., Mat. Notes49, 305–311 (1991)

    Google Scholar 

  29. White H. (1993). Algorithmic complexity of points in dynamical systems. Ergodic Theory Dynam. Systems 13(4): 807–830

    Article  MATH  MathSciNet  Google Scholar 

  30. Zelik S. (2003). Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity. Comm. Pure Appl. Math. 56(5): 584–637

    Article  MATH  MathSciNet  Google Scholar 

  31. Zelik S. (2004). Multiparameter semigroups and attractors of reaction-diffusion equations in \(\mathbb {R}^n\) Trans. Moscow Math. Soc. 65: 105–160

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica Applicata, Università di Pisa, via F. Buonarroti 1/c, 56127, Pisa, Italy

    Claudio Bonanno

  2. Centre de Physique Théorique, École Polytechnique, CNRS UMR 7644, F-91128, Palaiseau Cedex, France

    Pierre Collet

Authors
  1. Claudio Bonanno
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pierre Collet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Claudio Bonanno.

Additional information

Communicated by A. Kupiainen

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bonanno, C., Collet, P. Complexity for Extended Dynamical Systems. Commun. Math. Phys. 275, 721–748 (2007). https://doi.org/10.1007/s00220-007-0313-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-007-0313-4

Keywords

Search

Navigation