Graduate School of Informatics,

Kyoto University, Japan

One of eminent statistical characteristics of homogeneous, isotropic developed turbulence in the three dimensional system is the self-similar energy cascade in the wavenumber space. This results in the power law behavior for the velocity structure function , the -th moment of the velocity difference at two positions separated by . is a universal function of . Several years ago, it was found that even when the turbulence is not fully developed, i.e., the Reynolds number is not extremely high and the the scaling range where the above scaling law holds is not wide enough, scaling behaviors of which has an extended form of that in developed turbulence holds. They are called the extended self-similarity (ESS) and the generalized extended self-similarity (GESS). I will talk about a phenomenological derivation of ESS and GESS, proposing a new scaling hypothesis on the basis of the large deviation theory of probability theory. Furthermore, using numerical and experimental data, I will examine the validity of the present approach.

**References**

Kolmogorov A N 1941, *Dokl. Akad. Nauk SSSR* **30** 9

Kolmogorov A N 1962, *J. Fluid Mech.* **13** 82

Obukhov A M 1962, *J. Fluid Mech.* **13** 77.

Parisi G and Frisch U 1985, in *Turbulence and Predictability in
Geophysical Fluid Dynamics*, Proceed. Intern. School of Physics `Enrico
Fermi', 1983, Varenna, Italy, p.84, eds. M. Ghil, R. Benzi and G. Parisi,
(Amsterdam: North-Holland)

Frisch U 1995, *Turbulence: The legacy of A. N.Kolmogorov* (Cambridge:
Cambridge Univ. Press)

Benzi R, Ciliberto S, Tripiccione R, Baudet C and Succi S 1993, *Phys.
Rev. E* **48** R29

Benzi R, Ciliberto S, Baudet C and Chavarria G R 1995, *Physica D* **80** 385

Benzi R, Biferale L, Ciliberto S, Struglia M V and Tripiccione R 1996, *Phys. Rev. E* **53** R3025;

Benzi R, Biferale L, Ciliberto S, Struglia M V, Tripiccione R 1996, *Physica D* **96** 162.

Fujisaka H and Inoue M 1987, *Prog. Theor. Phys.* **77** 1334

Watanabe T and Fujisaka H 2000, *J. of Phys. Soc. Japan* **69** 1672

Fujisaka H and Grossmann S 2001, *Phys. Rev. E* **63** 026305

Fujisaka H, Nakayama Y, Watanabe T and Grossmann S 2002, *Phys. Rev. E*
**65** 046307.

Graduate School of Informatics,

Kyoto University, Japan

Intermittency is a quite ubiquitous phenomenon in nonlinear dynamics. The
intermittency observed when a particular dynamical state undergoes the
instability is called the modulational (often called the on-off)
intermittency. Recently, an experimental confirmation of the on-off
intermittency in the electrohydrodynamic convection in nematics under
dichotomous noise was reported by John *et al.*. An eminent statistics
of the observation is the intermittent generation of convective pattern.

In my talk, in order to elucidate the experiment I will first propose a
phenomenological nonlinear stochastic model which has the structure of the
Swift-Hohenberg equation for local convection variable with fluctuating
threshold. Then, I will discuss results of numerical integration of the
model equation associated with the intermittent emergence of convective
pattern. Detailed analysis on the statistics of the intermittent pattern
dynamics will be addressed.

**References**

Behn U, Lange A and John Th 1998, *Phys. Rev. E* **58** 2047

John Th, Stannarius R and Behn U 1999, *Phys. Rev. Lett.* **83** 749

Fujisaka H, Ouchi K and Ohara H 2001, *Phys. Rev. E* **64** 036201

John Th, Behn U and R Stannarius 2002, *Phys. Rev. E* **65** 046229

For the modulational intermittency (on-off intermittency), see the
following references:

Fujisaka H and Yamada T 1985, *Prog. Theor. Phys.* **74** 918

Fujisaka H and Yamada T 1986, *Prog. Theor. Phys.* **75** 1087

Yamada T and Fujisaka H 1986, *Prog. Theor. Phys.* **76** 582

Fujisaka H and Yamada T 1987, Prog. Theor. Phys. **77** 1045

Platt N, Spiegel E A and Tresser C 1993, *Phys. Rev. Lett.* **70** 279

Heagy J F, Platt N and Hammel S M 1994, *Phys. Rev. E* **49** 1140

Yamada T, Fukushima K and Yazaki T 1989, *Prog. Theor. Phys. Suppl.*
No.99, 120

Ott E and Sommerer J C 1994, *Phys. Lett. A* **188** 39

Lai Y C and Grebogi C 1995, *Phys. Rev. E* **52** R3313

Cenys A, Namajunas A, Tamserius A and Schneider T 1996, *Phys.
Lett. A* **213** 259

Venkataramani S C, Antonsen Jr. T M, Ott E and Sommerer J C 1996, *Physica D* **96** 66

Lai Y C 1996, *Phys. Rev. E* **54** 321

Rödelsperger F, Cenys A and Benner H 1995, *Phys. Rev. Lett.*
**75** 2594

Becker J, Rödelsperger F, Weyrauch Th, Benner H, Just W and Cenys A
1999, *Phys. Rev. E* **59** 1622

Fujisaka H, Ouchi K, Hata H, Masaoka B and Miyazaki S 1998, *Physica D*
**114** 237

Pikovsky A, Rosenblum M and Kurths J 2001, *Synchronization: A
universal concept in nonlinear sciences* (Cambridge: Cambridge Univ. Press)