The European Physical Journal Special Topics

, Volume 226, Issue 10, pp 2247–2262 | Cite as

Singular eigenvalue limit of advection-diffusion operators and properties of the strange eigenfunctions in globally chaotic flows

Regular Article
Part of the following topical collections:
  1. Aspects of Statistical Mechanics and Dynamical Complexity


Enforcing the results developed by Gorodetskyi et al. [O. Gorodetskyi, M. Giona, P. Anderson, Phys. Fluids 24, 073603 (2012)] on the application of the mapping matrix formalism to simulate advective-diffusive transport, we investigate the structure and the properties of strange eigenfunctions and of the associated eigenvalues up to values of the Péclet number Pe ~ 𝒪(108). Attention is focused on the possible occurrence of a singular limit for the second eigenvalue, ν2, of the advection-diffusion propagator as the Péclet number, Pe, tends to infinity, and on the structure of the corresponding eigenfunction. Prototypical time-periodic flows on the two-torus are considered, which give rise to toral twist maps with different hyperbolic character, encompassing Anosov, pseudo-Anosov, and smooth nonuniformly hyperbolic systems possessing a hyperbolic set of full measure. We show that for uniformly hyperbolic systems, a singular limit of the dominant decay exponent occurs, log|ν2| → constant≠0 for Pe → ∞, whereas log |ν2| → 0 according to a power-law in smooth non-uniformly hyperbolic systems that are not uniformly hyperbolic. The mere presence of a nonempty set of nonhyperbolic points (even if of zero Lebesgue measure) is thus found to mark the watershed between regular vs. singular behavior of ν2 with Pe as Pe → ∞.


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Supplementary material


  1. 1.
    P.D. Swanson, J.M. Ottino, J. Fluid Mech. 213, 227 (1990)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    A.D. Stroock, S.K. Dertinger, A. Ajdari, I. Mezic, H.A. Stone, G. Whitesides, Science 295, 647 (2002)ADSCrossRefGoogle Scholar
  3. 3.
    H. Aref, J. Fluid Mech. 143, 1 (1984)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    S. Balasuriya, J. Micromech. Microeng. 25, 094005 1 (2015)ADSCrossRefGoogle Scholar
  5. 5.
    V. Toussaint, P. Carriere, F. Raynal, Phys. Fluids 17, 2587 (1995)ADSCrossRefGoogle Scholar
  6. 6.
    T. Tel, G. Karoly, A. Pentek, I. Scheuring, Z. Toroczkai, C. Grebogi, J. Kadtke, Chaos 10, 89 (2000)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    M. Giona, A. Adrover, S. Cerbelli, V. Vitacolonna, Phys. Rev. Lett. 92, 114101 (2004)ADSCrossRefGoogle Scholar
  8. 8.
    J.P. Gleeson, Phys. Fluids 17, 100614 (2005)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Pikovsky, O. Popovych, Europhys. Lett. 61, 625 (2003)ADSCrossRefGoogle Scholar
  10. 10.
    G.M. Zaslavsky, Hamiltonian Chaos & Fractional Dynamics (Oxford University Press, Oxford, 2005)Google Scholar
  11. 11.
    G. Gallavotti, E.G.D. Cohen, J. Stat. Phys. 80, 931 (1995)ADSCrossRefGoogle Scholar
  12. 12.
    G. Gallavotti, E.G.D. Cohen, Phys. Rev. Lett. 74, 2694 (1995)ADSCrossRefGoogle Scholar
  13. 13.
    J. Vanneste, Phys. Fluids 18, 087108 1 (2006)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    O.V. Popovych, A. Pikovsky, B. Eckhardt, Phys. Rev. E 75, 036308 1 (2007)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    L. Liu, G. Haller, Physica D 188, 1 (2004)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    G. Froyland, S. Lloyd, N. Santitissadeekorn, Physica D 239, 1527 (2010)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    M.I. Freidlin, A.D. Wentzell, Random Perturbations of Dynamical Systems (Springer, Berlin, 2012)Google Scholar
  18. 18.
    G. Orsi, M. Roudgar, E. Brunazzi, C. Galletti, R. Mauri, Chem. Eng. Sci. 95, 174 (2013)CrossRefGoogle Scholar
  19. 19.
    C. Galletti, G. Arcolini, E. Brunazzi, R. Mauri, Chem. Eng. Sci. 123, 300 (2015)CrossRefGoogle Scholar
  20. 20.
    C. Liverani, Math. Phys. Electron. J. 10, 1 (2004)MathSciNetGoogle Scholar
  21. 21.
    O. Gorodetskyi, M. Giona, P. Anderson, Europhys. Lett. 97, 14002 (2012)ADSCrossRefGoogle Scholar
  22. 22.
    O. Gorodetskyi, M. Giona, P. Anderson, Phys. Fluids 24, 073603 (2012)ADSCrossRefGoogle Scholar
  23. 23.
    S. Cerbelli, M. Giona, J. Nonlin. Sci. 15, 387 (2005)ADSCrossRefGoogle Scholar
  24. 24.
    R.S. Mackay, J. Nonlin. Sci. 16, 415 (2006)ADSCrossRefGoogle Scholar
  25. 25.
    M.F. Demers, M.P. Wojtkowski, Nonlinearity 22, 1743 (2009)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    A. Lasota, M.C. Mackey, Chaos, Fractals, and Noise (Spinger, New York, 1994)Google Scholar
  27. 27.
    P. Kruijt, O. Galaktionov, P.D. Anderson, G.W. Peters, H. Meijer, AIChE J. 47, 1005 (2001)CrossRefGoogle Scholar
  28. 28.
    S.M. Ulam, A Collection of Mathematical Problems, Vol. 8 (Interscience, New York, 1960)Google Scholar
  29. 29.
    J. Ding, A. Zhou, Statistical Properties of Deterministic Systems (Springer, Berlin, 2009)Google Scholar
  30. 30.
    G. Froyland, Discrete Contin. Byn. Syst, Ser. A 17, 203 (2007)MathSciNetGoogle Scholar
  31. 31.
    O. Gorodetskyi, M.F.M. Speetjens, P.D. Anderson, Chem. Eng. Sci. 107, 30 (2014)CrossRefGoogle Scholar
  32. 32.
    M.K. Singh, M.F.M. Speetjens, P.D. Anderson, Phys. Fluids 21, 093601 (2008)ADSCrossRefGoogle Scholar
  33. 33.
    M.C. Mackey, Time’s Arrow: The Origin of Thermodynamic Behavior (Springer, New York, 1992)Google Scholar
  34. 34.
    V. Artale, G. Boffetta, A. Celani, M. Cencini, A. Vulpiani, Phys. Fluids 9, 3162 (1997)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    G. Boffetta, A. Celani, M. Cencini, G. Lacorata, A. Vulpiani, Chaos 10, 50 (2000)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Y.-K. Tsang, T.M. Antonsen, E. Ott, Phys. Rev. E 71, 066301 (2005)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    M. Giona, A. Adrover, Phys. Rev. Lett. 81, 3864 (1998)ADSCrossRefGoogle Scholar
  38. 38.
    S. Cerbelli, Phys. Rev. E 87, 060102(R) (2013)ADSCrossRefGoogle Scholar
  39. 39.
    R. Devendra, G. Drazer, Anal. Chem. 84, 10621 (2012)CrossRefGoogle Scholar
  40. 40.
    W. Horsthemke, R. Lefever, Noise-Induced Transitions (Springer-Verlag, Berlin, 2006)Google Scholar
  41. 41.
    R. Artuso, Physica D 131, 68 (1999)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    M. Giona, S. Cerbelli, V. Vitacolonna, J. Fluid Mech. 513, 221 (2004)ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    M. Giona, S. Cerbelli, F. Garofalo, J. Fluid Mech. 639, 291 (2009)ADSCrossRefGoogle Scholar

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© EDP Sciences and Springer 2017

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria Chimica DICMA Facoltà di Ingegneria, La Sapienza Università di Roma via Eudossiana 18RomaItaly
  2. 2.Materials Technology, Eindhoven University of TechnologyEindhovenThe Netherlands

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