Chaotic Dynamics

In this work we present a theoretical and numerical study of the behaviour of the maximum Lyapunov exponent for a generic coupled-map-lattice in the weak-coupling regime. We explain the observed results by introducing a suitable continuous-time formulation of the tangent dynamics. The first general result is that the deviation of the Lyapunov exponent from the uncoupled-limit limit is function of a single scaling parameter which, in the case of strictly positive multipliers, is the ratio of the coupling strength with the variance of local multipliers. Moreover, we find an approximate analytic expression for the Lyapunov exponent by mapping the problem onto the evolution of a chain of nonlinear Langevin equations, which are eventually reduced to a single stochastic equation. The probability distribution of this dynamical equation provides an excellent description for the behaviour of the Lyapunov exponent. Furthermore, multipliers with random signs are considered as well, finding that the Lyapunov exponent still depends on a single scaling parameter, which however has a different expression.

The field distributions and eigenfrequencies of a microwave resonator which is composed of 20 identical cells have been measured. With external screws the periodicity of the cavity can be perturbed arbitrarily. If the perturbation is increased a transition from extended to localized field distributions is observed. For very large perturbations the field distributions show signatures of Anderson localization, while for smaller perturbations the field distribution is extended or weakly localized. The localization length of a strongly localized field distribution can be varied by adjusting the penetration depth of the screws. Shifts in the frequency spectrum of the resonator provide further evidence for Anderson localization.

We found universal anizopropic spectra of acoustic turbulence with the linear dispersion law \bbox{ ω(k)=ck } within the framework of generalized kinetic equation which takes into account the finite time of three-wave interactions. This anisotropic spectra can assume both scale-invariant and non scale-invariant form. The implications for the evolution of the acoustic turbulence with non-isotropic pumping are discussed. The main result of the article is that the spectra of acoustic turbulence tend to become more isotropic.

A first analytic assessment of the role of anisotropic corrections to the isotropic anomalous scaling exponents is given for the d -dimensional kinematic magneto-hydrodynamics problem in the presence of a mean magnetic field. The velocity advecting the magnetic field changes very rapidly in time and scales with a positive exponent ξ . Inertial-range anisotropic contributions to the scaling exponents, ζ j , of second-order magnetic correlations are associated to zero modes and have been calculated non-perturbatively. For d=3 , the limit ξ↦0 yields $\protect{\zeta_j=j-2+ \xi (2j^3 +j^2 -5 j - 4)/[2(4 j^2 - 1)]}$ where j ( j≥2 ) is the order in the Legendre polynomial decomposition applied to correlation functions. Conjectures on the fact that anisotropic components cannot change the isotropic threshold to the dynamo effect are also made.

The major difficulty in developing theories for anomalous scaling in hydrodynamic turbulence is the lack of a small parameter. In this Letter we introduce a shell model of turbulence that exhibits anomalous scaling with a tunable small parameter. The small parameter ϵ represents the ratio between deterministic and random components in the coupling between N identical copies of the turbulent field. We show that in the limit N→∞ anomalous scaling sets in proportional to ϵ 4 . Moreover we give strong evidences that the birth of anomalous scaling appears at a finite critical ϵ , being ϵ c ≈0.6 .

Kraichnan's model of passive scalar advection is studied as a case model for understanding the anomalous scaling in the anisotropic sectors. We show here that the solutions of the Kraichnan equation for the n order correlations foliate into sectors that are classified by the irreducible representations of the SO(d) group. We find a discrete spectrum of universal anomalous exponents in every sector. Generically the correlations and structure functions appear as sums over all the contributions, with non-universal amplitudes which are determined by the anisotropic boundary conditions. The isotropic sector is always characterized by the smallest exponent, and therefore for sufficiently small scales local isotropy is always restored. The calculation of the anomalous exponents is done in two complementary ways. In the first they are obtained from the analysis of correlations of gradient fields. The corresponding theory involves the control of logarithmic divergences which translate into anomalous scaling with the ratio of the inner and the outer scales appearing in the final result. In the second way we compute the exponents from the zero modes of the Kraichnan equation for the correlations of the scalar field itself. In this case the renormalization scale is the outer scale. The two approaches lead to the same scaling exponents for the same statistical objects, illuminating the relative role of the outer and inner scales as renormalization scales. We derive fusion rules which govern the small scale asymptotics of the correlation functions in the sectors of the symmetry group, in all dimensions.

In a helical flow there is a subrange of the inertial range in which there is a cascade of both energy and helicity. In this range the scaling exponents associated with the cascade of helicity can be defined. These scaling exponents are calculated from a simulation of the GOY shell model. The scaling exponents for even moments are associated with the scaling of the symmetric part of the probability density functions while the odd moments are associated with the anti-symmetric part of the probability density functions.

The field theoretic renormalization group and operator product expansion are applied to the problem of a passive scalar advected by the Gaussian nonsolenoidal velocity field with finite correlation time, in the presence of large-scale anisotropy. The energy spectrum of the velocity in the inertial range has the form E(k)∝ k 1−ϵ , and the correlation time at the wavenumber k scales as k −2+η . It is shown that, depending on the values of the exponents ϵ and η , the model exhibits various types of inertial-range scaling regimes with nontrivial anomalous exponents. Explicit asymptotic expressions for the structure functions and other correlation functions are obtained; they are represented by superpositions of power laws with nonuniversal amplitudes and universal (independent of the anisotropy) anomalous exponents, calculated to the first order in ϵ and η in any space dimension. These anomalous exponents are determined by the critical dimensions of tensor composite operators built of the scalar gradients, and exhibit a kind of hierarchy related to the degree of anisotropy: the less is the rank, the less is the dimension and, consequently, the more important is the contribution to the inertial-range behavior. The leading terms of the even (odd) structure functions are given by the scalar (vector) operators. The anomalous exponents depend explicitly on the degree of compressibility.

The field theoretic renormalization group (RG) is applied to the problem of a passive scalar advected by the Gaussian self-similar velocity field with finite correlation time and in the presence of an imposed linear mean gradient. The energy spectrum in the inertial range has the form E(k)∝ k 1−ϵ , and the correlation time at the wavenumber k scales as k −2+η . It is shown that, depending on the values of the exponents ϵ and η , the model in the inertial range reveals various types of scaling regimes associated with the infrared stable fixed points of the RG equations: diffusive-type regimes for which the advection can be treated within ordinary perturbation theory, and three nontrivial convection-type regimes for which the correlation functions exhibit anomalous scaling behavior. Explicit asymptotic expressions for the structure functions and other correlation functions are obtained; they are represented by superpositions of power laws with nonuniversal amplitudes and universal (independent of the anisotropy) anomalous exponents, calculated to the first order in ϵ and η in any space dimension. For the first nontrivial regime the anomalous exponents are the same as in the rapid-change version of the model; for the second they are the same as in the model with time-independent (frozen) velocity field. In these regimes, the anomalous exponents are universal in the sense that they depend only on the exponents entering into the velocity correlator. For the last regime the exponents are nonuniversal (they can depend also on the amplitudes); however, the nonuniversality can reveal itself only in the second order of the RG expansion. Comments: Extended version accepted to Phys. Rev. E. 35 pages; REVTeX source with LATeX figures inside.

A two-dimensional system of non-locally coupled complex Ginzburg-Landau oscillators is investigated numerically for the first time. As already known for the one-dimensional case, the system exhibits anomalous spatio-temporal chaos characterized by power-law spatial correlations. In this chaotic regime, the amplitude difference between neighboring elements shows temporal noisy on-off intermittency. The system is also spatially intermittent in this regime, which is revealed by multi-scaling analysis; the amplitude field is multi-affine and the difference field is multi-fractal. Correspondingly, the probability distribution function of the measure defined for each field is strongly non-Gaussian, showing scale-dependent deviations in the tails due to intermittency.