Mogens H. Jensen

Fractal measures and their singularities: The characterization of strange sets☆

Abstract We propose a description of normalized distributions (measures) lying upon possibly fractal sets; for example those arising in dynamical systems theory. We focus upon the scaling properties of such measures, by considering their singularities, which are characterized by two indices: α, which determines the strength of their singularities; and f , which describes how densely they are distributed. The spectrum of singularities is described by giving the possible range of α values and the function f ( α ). We apply this formation to the 2 ∞ cycle of period doubling, to the devils staircase of mode locking, and to trajectories on 2-tori with golden-mean winding numbers. In all cases the new formalism allows an introduction of smooth functions to characterize the measures. We believe that this formalism is readily applicable to experiments and should result in new tests of global universality.

Spectra of scaling indices for fractal measures: theory and experiment

Abstract We propose a new formalism to characterize fractal measures, such as strange attractors of dynamical systems, which uses a function f(α). Here α is the scaling index of the measure about a point on the fractal and f(α) is the dimension of the set of points on the fractal with the same value of α. The spectrum describes the global scaling structure of the measure. The spectrum includes as special points the Hausdorff dimension of the fractal and the scaling indices of the most rarefied and the most concentrated regions of the measure. We apply this formalism to two well-known transitions to chaos: the route via period-doubling and the route via quasiperiodicity. In both cases we find a universal spectrum which includes previously discovered scaling numbers. Both spectra have been measured experimentally on a forced Rayleigh-Benard system.

Universal strange attractors on wrinkled tori

Strange attractors in dynamical systems that go to chaos via quasiperiodicity are considered. It is shown that there exists an infinite number of points in parameter space where the topology of the strange attractors is universal. At such points the periodic points belonging to unstable periodic orbits can be organised on ternary trees which are pruned by local rules. The grammar is universal, and thus the topological entropy is universal at each of these points in parameter space. The complete understanding of the topology is used to calculate systematically the metric properties of the attractors. The spectrum of scaling indices f( alpha ) is computed. It is found that there is no metric universality, although some aspects of the metric properties are universal. Experiments to test some of the predictions of this theory are suggested.

Intermittency and predictability in a shell model for three-dimensional turbulence

Abstract We review some recent work on scaling laws in turbulence. The Navier-Stokes equations and the equation for the transport of passive scalar fields are formulated in terms of shell models in Fourier space, which incorporate the appropriate conservation laws. We find that the Kolmogorov scaling for the energy cascade and the Obukhov-Corrsin scaling for the passive scalar cascade are dynamically unstable. This fact gives rise to corrections to classical values of the scaling exponents. We relate these corrections to the dynamical behavior of the intermittency bursts present in the shell models. Our numerical results are in good quantitative agreement with the experimental results in fluids. Finally we discuss the problem of predictability.

Turbulence and Linear Stability in a Discrete Ginzburg-Landau Model

The Complex Ginzburg-Landau partial differential equation appears in many interesting none-quilibrium dynamical systems. It describes an extended system close to a global Hopf bifurcation [1] such as occurs e.g. in oscillatory chemical reactions like the Belousov-Zhabotinsky reaction [2]. In two recent papers [3–4] we have discussed a discrete, “map lattice” version of this equation, analysed the dynamics of vortices and the onset of turbulence. The main results were that vortices can get bound together in “entangled” states where their cores do not move and that the system has a well-defined transition to turbulence below the linear instability threshold for the uniform state. It remains to be seen which of our results will be valid for the continuum Ginzburg-Landau equation; but recently an analytic treatment of the motion of a pair of vortices leads to bound states analogously to our entangled states [5].

Global University at the onset of chaos: Results of a forced Rayleigh-Bénard experiment☆

Abstract We study an experimental orbit on a two-torus with a golden-mean winding number obtained from a forced Rayleigh-Benard system at the onset of chaos. This experimental orbit is compared with the orbit generated by a simple theoretical model, the circle map, at its golden-mean winding number at the onset of chaos. The “spectrum of singularities” of the two orbits are compared. Within error, these are identical. Since the spectrum characterizes the metric properties of the entire orbit, this result confirms theoretical speculations that these orbits, taken as a whole, enjoy a kind of universality. PACS numbers: 47.20. + m, 05.45. + b, 47.25. −c

CIRCLE MAPS IN THE COMPLEX PLANE

Circle maps of polynomial, exponential, and rational polynomial types are studied numerically in the complex plane. The golden mean universality for real circle maps does not extend into the complex plane.

Reply to some comments on a proposed universal strange attractor on wrinkled tori

The conditions under which one expects universal symbolic dynamics for strange attractor on wrinkled tori are restated and clarified.

Time ordering and the thermodynamics of strange sets: Theory and experimental tests

Abstract From the spectrum of dimensions of a fractal invariant measure of a dynamical system one can extract information about the dynamical process that gave rise to the measure. This is equivalent to finding the class of Hamiltonians of an Ising model with a given thermodynamics.