Advertisement

Infinite-Dimensional Hyperbolic Sets and Spatio-Temporal Chaos in Reaction Diffusion Systems in \({\mathbb{R}^{n}}\)

  • Alexander Mielke
  • Sergey V. ZelikEmail author
Article

The paper is devoted to a rigorous construction of a parabolic system of partial differential equations which displays space–time chaotic behavior in its global attractor. The construction starts from a periodic array of identical copies of a temporally chaotic reaction-diffusion system (RDS) on a bounded domain with Dirichlet boundary conditions. We start with the case without coupling where space–time chaos, defined via embedding of multi- dimensional Bernoulli schemes, is easily obtained. We introduce small coupling by replacing the Dirichlet boundary conditions by strong absorption between the active islands. Using hyperbolicity and delicate PDE estimates we prove persistence of the embedded Bernoulli scheme. Furthermore we smoothen the nonlinearity and obtain a RDS which has polynomial interaction terms with space and time-periodic coefficients and which has a hyperbolic invariant set on which the dynamics displays spatio-temporal chaos. Finally we show that such a system can be embedded in a bigger system which is autonomous and homogeneous and still contains space–time chaos. Obviously, hyperbolicity is lost in this step.

Keywords

Hyperbolicity Bernoulli shifts space–time chaos global attractor weighted Sobolev spaces 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Afraimovich V., Babin A.V., Chow S.N. (1996). Spatial chaotic structure of attractors of reaction-diffusion systems. Trans. Am. Math. Soc. 348(12): 5031–5063zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Afendikov A., Mielke A. (2001). Multi-pulse solutions to the Navier–Stokes problem between parallel plates. Z. Angew. Math. Phys. 52(1): 79–100zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Afraimovich V., Fernandez B. (2000). Topological properties of linearly coupled map lattices. Nonlinearity 13: 973–993zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Agmon S., Nirenberg L. (1967). Lower bounds and uniqueness theorems for solutions of differential equations in a Hilbert space. Comm. Pure Appl. Math. 20: 207–229zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Amann, H. (1995). Linear and Quasilinear Parabolic Problems, Vol. I: Abstract Linear Theory. Monographs in Mathematics, Vol. 89, Birkhauser Boston.Google Scholar
  6. 6.
    Angenent, S. (1987). The shadowing lemma for elliptic PDE. In Dynamics of Infinite-Dimensional Systems (Lisbon, 1986), Vol. 37 of NATO Adv. Sci. Inst. Ser. F Comput. Sci., Springer, Berlin, pp. 7–22.Google Scholar
  7. 7.
    Babin, A. V., and Vishik, M. I. (1989). Attraktory Evolyutsionnykh uravnenii (English transl. Stud. Math. Appl., 25, North Holland, Amsterdam, 1992), Nauka.Google Scholar
  8. 8.
    Babin A.V., Vishik M.I. (1990). Attractors of partial differential evolution equations in an unbounded domain. Proc. Roy. Soc. Edinburgh Sect. A 116(3–4): 221–243MathSciNetzbMATHGoogle Scholar
  9. 9.
    Babin A.V. (2000). Topological invariants and solutions with a high complexity for scalar semilinear PDE. J. Dynam. Differ. Equ. 12(3): 599–646zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Browder F. (1959). Estimates and existence theorems for elliptic boundary value problems. Proc. Nat. Acad. Sci. U.S.A. 45: 365–372zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bunimovich, L. A. (1999). space–time chaos in spatially continuous systems. Phys. D 131(1–4), 31–37. Classical chaos and its quantum manifestations (Toulouse, 1998).Google Scholar
  12. 12.
    Bunimovich L.A., Sina#xi8012c;, Ya. G. (1988). Spacetime chaos in coupled map lattices. Nonlinearity 1(4): 491–516zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Collet P., Eckmann J.-P. (1999). The definition and measurement of the topological entropy per unit volume in parabolic PDEs. Nonlinearity 12(3): 451–473zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Collet P., Eckmann J.-P. (1999). Extensive properties of the complex Ginzburg- Landau equation. Comm. Math. Phys. 200(3): 699–722zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Dangelmayr, G., Fiedler, B., Kirchgässner, K., and Mielke, A. (1996). Dynamics of Nonlinear Waves in Dissipative Systems: Reduction, Bifurcation and Stability, Vol. 352 of Pitman Research Notes in Mathematics Series, Longman, Harlow With a contribution by G. Raugel.Google Scholar
  16. 16.
    Efendiev M., Zelik S.V. (2001). The attractor for a nonlinear reaction-diffusion system in an unbounded domain. Comm. Pure Appl. Math. 54: 625–688zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Efendiev M., Zelik S.V. (2002). Upper and lower bounds for the Kolmogorov entropy of the attractor for the RDE in an unbounded domain. J. Dynam. Differ. Equ. 14(2): 369–403zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Fiedler B., Polačcik, P. (1990). Complicated dynamics of scalar reaction diffusion equations with a nonlocal term. Proc. Roy. Soc. Edinburgh Sect. A 115(1–2): 167–192MathSciNetzbMATHGoogle Scholar
  19. 19.
    Goren G., Eckmann J.-P., Procaccia I. (1998). Scenario for the onset of space–time chaos. Phys. Rev. E (3). 57(4): 4106–4134CrossRefMathSciNetGoogle Scholar
  20. 20.
    Katok, A., Hasselblatt, B. (1995). Introduction to The Modern Theory of Dynamical Systems, Cambridge University Press.Google Scholar
  21. 21.
    Kirchgässner K. (1982). Wave-solutions of reversible systems and applications. J. Differ. Equ. 45(1): 113–127zbMATHCrossRefGoogle Scholar
  22. 22.
    Kirchgässner, K. (1985). Nonlinear wave motion and homoclinic bifurcation. In Theoretical and Applied Mechanics (Lyngby, 1984), North-Holland, Amsterdam, pp. 219–231.Google Scholar
  23. 23.
    Kolmogorov A.N., Tihomirov V.M. (1961). \({\varepsilon}\) -entropy and \({\varepsilon}\) -capacity of sets in functional space. Am. Math. Soc. Transl. 17(2): 277–364MathSciNetGoogle Scholar
  24. 24.
    Ladyzhenskaya, O. A., Solonnikov, V. A., and Uraltseva, N. N. (1967). Linear and Quasilinear Equations of Parabolic Type, Nauka.Google Scholar
  25. 25.
    Manneville, P. (1990). Dissipative Structures and Weak Turbulence, Perspectives in Physics, Academic Press Inc., Boston, MA.Google Scholar
  26. 26.
    Manneville, P. (1995). Dynamical systems, temporal vs. spatio-temporal chaos, and climate. In Nonlinear Dynamics and Pattern Formation in the Natural Environment (Noordwijkerhout, 1994), Vol. 335 of Pitman Res. Notes Math. Ser., Longman, Harlow, pp. 168–187.Google Scholar
  27. 27.
    Mielke A. Holmes P.J. (1988). Spatially complex equilibria of buckled rods. Arch. Rational Mech. Anal. 101(4): 319–348zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Mielke A. (1986). A reduction principle for nonautonomous systems in infinite- dimensional spaces. J. Differ. Equ. 65(1): 68–88zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Mielke A. (1987). Über maximale L p-Regularität für Differentialgleichungen in Banach- und Hilbert-Räumen. Math. Ann. 288: 121–133CrossRefMathSciNetGoogle Scholar
  30. 30.
    Mielke A. (1997). Instability and stability of rolls in the Swift–Hohenberg equation. Comm. Math. Phys. 189:829–853zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Mielke, A. (1997). The complex Ginzburg–Landau equation on large and unbounded domains: sharper bounds and attractors. Nonlinearity 10, 199–222.Google Scholar
  32. 32.
    Mielke, A. (2002). The Ginzburg–Landau equation in its role as a modulation equation. In Fiedler, B. (ed.), Handbook of Dybnamical Systems, Vol. 2, Elsevier, pp. 759–834.Google Scholar
  33. 33.
    Mielke A., Schneider G. (1995). Attractors for modulation equations on unbounded domains—existence and comparison. Nonlinearity 8: 743–768zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Mielke, A., and Zelik, S. V. (2002). Infinite-dimensional trajectory attractors for elliptic boundary-value problems on cylindrical domains. Russian Math. Surveys 57, 753–784 (Uspekhi Mat. Nauk. 57(4(345)), 2002, 119–150).Google Scholar
  35. 35.
    Pesin Y.B., SinaȜ Y.G. (1988). space–time chaos in the system of weakly interacting hyperbolic systems. J. Geom. Phys. 5(3): 483–492zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Pesin, Y. B., and SinaȜ, Y. G. (1991). space–time chaos in chains of weakly interacting hyperbolic mappings, In Adv. Soviet Math., Vol. 3, Dynamical Systems and Statistical Mechanics (Moscow, 1991), Am. Math. Soc., Providence, RI, pp. 165–198.Google Scholar
  37. 37.
    Polačcik, P. (2002). Parabolic equations: asymptotic behavior and dynamics on invariant manifolds. In Handbook of Dynamical Systems, Vol. 2, North-Holland, Amsterdam, pp. 835–883.Google Scholar
  38. 38.
    Rabinowitz, P. H. (1993). Multibump solutions of a semilinear elliptic PDE on \({\mathbb{R}^n}\) . In Degenerate Diffusions (Minneapolis, MN, 1991), IMA Vol. Math. Appl., Vol 47, Springer, New York, pp. 175–185.Google Scholar
  39. 39.
    Shilnikov, L. P., Shilnikov, A. L., Turaev, D., and Chua, L. (2001). Methods of Qualitative Theory in Nonlinear Dynamics, Part II, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 5, World Scientific Publishing Co., Inc., River Edge, NJ.Google Scholar
  40. 40.
    Temam, R. (1988). Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag.Google Scholar
  41. 41.
    Triebel, H. (1978). Interpolation Theory, Function Spaces, Differential Operators, North-Holland.Google Scholar
  42. 42.
    Vishik M.I., Chepyzhov V.V. (1998). Kolmogorov \({\epsilon}\) -entropy of attractors of reaction-diffusion systems. Sb. Math. 189(1–2): 235–263CrossRefMathSciNetGoogle Scholar
  43. 43.
    Zelik, S. V. (1999). An attractor of a nonlinear system of reaction-diffusion equations in R n and estimates for its \({\epsilon}\) -entropy. Math. Notes. 65(5/6), 790–792 (English transl.).Google Scholar
  44. 44.
    Zelik S.V. (2001). The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete Contin. Dynam. Systems 7(3): 593–641zbMATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    Zelik, S. V. (to appear). Spatial and dynamical chaos generated by reaction diffusion systems in unbounded domains. J. Dynam. Differ. Eqs.Google Scholar
  46. 46.
    Zelik S.V. (2003). The attractor for an nonlinear reaction-diffusion system in an unbounded domain and Kolmogorov’s \({\epsilon }\) –entropy. Mathem. Nachr. 248-249(1): 72–96CrossRefMathSciNetGoogle Scholar
  47. 47.
    Zelik S.V. (2003). The attractors of reaction-diffusion systems in unbounded domains and their spatial complexity. Comm. Pure Appl. Math. 56(5): 584–637zbMATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Zelik S.V. (2004). Multiparametrical semigroups and attractors of reaction-diffusion systems in \({\mathbb{R}^n}\) . Proc. Moscow Math. Soc. 65: 69–130Google Scholar
  49. 49.
    Zelik, S. V., and Mielke, A. (2006). Multi-pulse evolution and space–time chaos in systems. To appear in Memoirs of AMS.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  1. 1.Weierstraß–Institut für Angewandte Analysis und StochastikBerlinGermany
  2. 2.Department of MathematicsUniversity of SurreyGuildfordUK

Personalised recommendations