Communications in Mathematical Physics

, Volume 275, Issue 3, pp 721–748 | Cite as

Complexity for Extended Dynamical Systems

Article

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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 MATHMathSciNetGoogle Scholar
  2. 2.
    Billingsley P. (1968). Convergence of Probability Measures. Wiley, New York MATHGoogle Scholar
  3. 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. 4.
    Collet P. (1994). Thermodynamic limit of the Ginzburg-Landau equations. Nonlinearity 7(4): 1175–1190 MATHCrossRefADSMathSciNetGoogle Scholar
  5. 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 97–104Google Scholar
  6. 6.
    Collet P. (2002). Extensive quantities for extended systems. Fields Institute Communications 21: 65–74 ADSMathSciNetGoogle Scholar
  7. 7.
    Collet P. and Eckmann J.-P. (1990). Instabilities and Fronts in Extended Sytems. Princeton University Press, Princeton, NJ Google Scholar
  8. 8.
    Collet P. and Eckmann J.-P. (1999). Extensive properties of the Ginzburg-Landau equation. Commun. Math. Phys. 200: 699–722 MATHCrossRefADSMathSciNetGoogle Scholar
  9. 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. 10.
    Collet P. and Eckmann J.-P. (2000). Topological entropy and ϵ-entropy for damped hyperbolic equations. Ann. Henri Poincaré 1: 715–752 MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Denker, M., Grillenberger, C., Sigmund, K.: Ergodic Theory on Compact Spaces, LNM 527, Berlin- Heidelberg: Springer-Verlag, 1976Google Scholar
  12. 12.
    Derrienic Y. (1983). Un theoreme ergodique presque sous-additif. Ann. Probab. 11: 669–677 MathSciNetGoogle Scholar
  13. 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 MATHADSMathSciNetGoogle Scholar
  14. 14.
    Feireisl E. (1996). Bounded locally compact global attractors for semilinear damped wave equations on R N. Differ. Integral Eq. 9: 1147–1156 MATHMathSciNetGoogle Scholar
  15. 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 MATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    del Junco A. and Rosenblatt J. (1979). Counterexamples in ergodic theory and number theory. Math. Ann. 245: 185–197 MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    del Junco A. and Steele J.M. (1977). Moving averages of ergodic processes. Metrika 24: 35–43 MATHCrossRefMathSciNetGoogle Scholar
  18. 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)Google Scholar
  19. 19.
    Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications. Second edition, GTCS, Berlin-Heielberg Newyork: Springer-Verlag, 1997Google Scholar
  20. 20.
    Mielke A. (1997). The complex Ginzburg-Landau equation on large and unbounded domains: sharper bounds and attractors. Nonlinearity 10(1): 199–222 MATHCrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Mielke A. and Schneider G. (1995). Attractors for modulation equations on unbounded domains—existence and comparison. Nonlinearity 8(5): 743–768 MATHCrossRefADSMathSciNetGoogle Scholar
  22. 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 appearGoogle Scholar
  23. 23.
    Milnor J. (1988). On the entropy geometry of cellular automata. Complex Systems 2: 357–385 MATHADSMathSciNetGoogle Scholar
  24. 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. 147–157Google Scholar
  25. 25.
    Schürger K. (1991). Almost subadditive extensions of Kingman’s ergodic theorem. Ann. Probab. 19: 1575–1586 MATHMathSciNetGoogle Scholar
  26. 26.
    Shannon, C.E.: A mathematical theory of communication. Bell System Tech. J. 27, 379–423, 623–656 (1948)Google Scholar
  27. 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 CrossRefMATHMathSciNetGoogle Scholar
  28. 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. 29.
    White H. (1993). Algorithmic complexity of points in dynamical systems. Ergodic Theory Dynam. Systems 13(4): 807–830 MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Zelik S. (2003). Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity. Comm. Pure Appl. Math. 56(5): 584–637 MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Zelik S. (2004). Multiparameter semigroups and attractors of reaction-diffusion equations in \(\mathbb {R}^n\) Trans. Moscow Math. Soc. 65: 105–160 CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Dipartimento di Matematica ApplicataUniversità di PisaPisaItaly
  2. 2.Centre de Physique Théorique, École PolytechniqueCNRS UMR 7644Palaiseau CedexFrance

Personalised recommendations