Index of content:
Volume 7, Issue 1, March 1997

Stable periodic motions in the problem on passage through a separatrix
View Description Hide DescriptionA Hamiltonian system with one degree of freedom depending on a slowly periodically varying in time parameter is considered. For every fixed value of the parameter there are separatrices on the phase portrait of the system. When parameter is changing in time, these separatrices are pulsing slowly periodically, and phase points of the system cross them repeatedly. In numeric experiments region swept by pulsing separatrices looks like a region of chaotic motion. However, it is shown in the present paper that if the system possesses some additional symmetry (like a pendulum in a slowly varying gravitational field), then typically in the region in question there are many periodic solutions surrounded by stability islands; total measure of these islands does not vanish and does not tend to 0 as rate of changing of the parameter tends to 0.

Pesin’s dimension for Poincaré recurrences
View Description Hide DescriptionA new characteristic of Poincaré recurrences is introduced. It describes an average return time in the framework of a general construction for dimensionlike characteristics. Some examples are considered including rotations on the circle and the Denjoy example.

Dynamics of spatial averages
View Description Hide DescriptionWe study the dynamics of spatial averages of spatially extended dynamical systems. We present various examples of lattice dynamical systems to show the possibility of different behaviors, including asymptotically constant, periodic, and nonperiodic, of spatial averages. We explain that the fluctuation in spatial averages is caused by the transitivity and the lack of symmetry of the dynamics of local subsystems.

On a general concept of multifractality: Multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity
View Description Hide DescriptionWe introduce the mathematical concept of multifractality and describe various multifractal spectra for dynamical systems, including spectra for dimensions and spectra for entropies. We support the study by providing some physical motivation and describing several nontrivial examples. Among them are subshifts of finite type and onedimensional Markov maps. An essential part of the article is devoted to the concept of multifractal rigidity. In particular, we use the multifractal spectra to obtain a “physical’’ classification of dynamical systems. For a class of Markov maps, we show that, if the multifractal spectra for dimensions of two maps coincide, then the maps are differentiably equivalent.

Monte Carlo methods for turbulent tracers with long range and fractal random velocity fields
View Description Hide DescriptionMonte Carlo methods for computing various statistical aspects of turbulent diffusion with long range correlated and even fractal random velocity fields are described here. A simple explicit exactly solvable model with complex regimes of scaling behavior including trapping, subdiffusion, and superdiffusion is utilized to compare and contrast the capabilities of conventional Monte Carlo procedures such as the Fourier method and the moving average method; explicit numerical examples are presented which demonstrate the poor convergence of these conventional methods in various regimes with long range velocity correlations. A new method for computing fractal random fields involving wavelets and random plane waves developed recently by two of the authors [J. Comput. Phys. 117, 146 (1995)] is applied to compute pair dispersion over many decades for systematic families of anisotropicfractal velocity fields with the Kolmogorov spectrum. The important associated preconstant for pair dispersion in the Richardson law in these anisotropic settings is compared with the one obtained over many decades recently by two of the authors [Phys. Fluids 8, 1052 (1996)] for an isotropic fractal field with the Kolmogorov spectrum.

Quasiperiodicity, global stability and scaling in a model of Hamiltonian roundoff
View Description Hide DescriptionWe investigate the effects of roundoff errors on the orbits of a linear symplectic map of the plane, with rational rotation number . Uniform discretization transforms this map into a permutation of the integer lattice . We study in detail the case , exploiting the correspondence between and a suitable domain of algebraic integers. We completely classify the orbits, proving that all of them are periodic. Using higherdimensional embedding, we establish the quasiperiodicity of the phase portrait. We show that the model exhibits asymptotic scaling of the periodic orbits and a longrange clustering property similar to that found in repetitive tilings of the plane.

Entropy conservation as in neurobiological dynamical systems
View Description Hide DescriptionThat the topological entropy, of a diffeomorphism, , of a surface,, upon which invariant measure(s) are concentrated, varies as the product of its average leading Lyapunov characteristic exponent, and the Hausdorff dimension of its support, was proven by Pesin [Russ. Math Surveys 32, 55–114 (1977)] for nonuniform partial hyperbolic systems and by Ledreppier and Young [Ergod. Theor. Dyn. Syst. 2, 109–123 (1982)], and Manning [Ergod. Theor. Dyn. Syst. 1, 451–459 (1981)] for uniformly hyperbolic (Axiom A) diffeomorphisms. When considered in conjunction with the postShannon informationencodingtheorems of Adler [Trans. Am. Math. Soc. 114, 309–319 (1965); Mem. Am. Math. Soc., No. 219 (1979)] and others, this suggests a way to differentiate equal entropy behaviors in systems with varying patterns of dynamical behaviors. Here we show this relation to be useful in the quantitative discrimination among the behaviors of abstract neuronal models and two real, finite time, partially and nonuniformly hyperbolic, brainrelated dynamical systems. We observe a tradeoff in finite time between two competing dynamical processes, jittery sticking (tending to increase and convective escaping (more prominently incrementing . In finite time systems, these changes in combination can statistically conserve the dynamical entropy,, while altering the Levy characteristic exponent, (describing the tail of the density distribution of observables, , and the MandelbrotHurst exponent such that implicates sequential correlations and sequential anticorrelation. When the relation fails, the way it does so provides information about the system.

Cosmic lacunarity
View Description Hide DescriptionThe present distribution of galaxies in space is a remnant of their formation and interaction. On a large enough scale, we may represent the galaxies as a set of points and quantify the structures in this set by its generalized dimensions [Beck and Schlögl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, 1986); Paladin and Vulpiani, Phys. Rep. 156, 147 (1987)]. The results of such evaluation are often taken to be evidence of a fractal (or multifractal) distribution of galaxies. However, those results, for some scales, may also reveal the presence of singularities formed in the gravitational processes that produce structure in the galaxy distribution. To try to make some decision about this issue, we look for the more subtle galactic lacunarity. We believe that this quantity is discernible in the currently available data and that it provides important evidence on the galaxyformation process.

The multifractal analysis of Gibbs measures: Motivation, mathematical foundation, and examples
View Description Hide DescriptionWe first motivate the study of multifractals. We then present a rigorous mathematical foundation for the multifractal analysis of Gibbs measures invariant under dynamical systems. Finally we effect a complete multifractal analysis for several classes of hyperbolic dynamical systems.

Plume dynamics in quasi2D turbulent convection
View Description Hide DescriptionWe have studied turbulent convection in a vertical thin (HeleShaw) cell at very high Rayleigh numbers (up to times the value for convective onset) through experiment, simulation, and analysis. Experimentally, convection is driven by an imposed concentration gradient in an isothermal cell. Modelequations treat the fields in two dimensions, with the reduced dimension exerting its influence through a linear wall friction. Linear stability analysis of these equations demonstrates that as the thickness of the cell tends to zero, the critical Rayleigh number and wave number for convective onset do not depend on the velocity conditions at the top and bottom boundaries (i.e., noslip or stressfree). At finite cell thickness , however, solutions with different boundary conditions behave differently. We simulate the modelequations numerically for both types of boundary conditions. Time sequences of the full concentration fields from experiment and simulation display a large number of solutal plumes that are born in thin concentration boundary layers, merge to form vertical channels, and sometimes split at their tips via a RayleighTaylor instability. Power spectra of the concentration field reveal scaling regions with slopes that depend on the Rayleigh number. We examine the scaling of nondimensional heat flux (the Nusselt number, and rms vertical velocity (the Péclet number, with the Rayleigh number for the simulations. Both noslip and stressfree solutions exhibit the scaling that we develop from simple arguments involving dynamics in the interior, away from cell boundaries. In addition, for stressfree solutions a second relation, , is dictated by stagnationpoint flows occurring at the horizontal boundaries; is the number of plumes per unit length. Noslip solutions exhibit no such organization of the boundary flow and the results appear to agree with Priestley’s prediction of .

Indecomposable continua in dynamical systems with noise: Fluid flow past an array of cylinders
View Description Hide DescriptionStandard dynamical systems theory is based on the study of invariant sets. However, when noise is added, there are no bounded invariant sets. Our goal is then to study the fractal structure that exists even with noise. The problem we investigate is fluid flow past an array of cylinders. We study a parameter range for which there is a periodic oscillation of the fluid, represented by vortices being shed past each cylinder. Since the motion is periodic in time, we can study a time1 Poincaré map. Then we add a small amount of noise, so that on each iteration the Poincaré map is perturbed smoothly, but differently for each time cycle. Fix an coordinate and an initial time . We discuss when the set of initial points at a time whose trajectory is semibounded (i.e., for all time) has a fractal structure called an indecomposable continuum. We believe that the indecomposable continuum will become a fundamental object in the study of dynamical systems with noise.

Average exit time for volumepreserving maps
View Description Hide DescriptionFor a volumepreserving map, we show that the exit time averaged over the entry set of a region is given by the ratio of the measure of the accessible subset of the region to that of the entry set. This result is primarily of interest to show two things: First, it gives a simple bound on the algebraic decay exponent of the survival probability. Second, it gives a tool for computing the measure of the accessible set. We use this to compute the measure of the bounded orbits for the Hénon quadratic map.

Parabolic resonances and instabilities
View Description Hide DescriptionA parabolic resonance is formed when an integrable twodegreesoffreedom (d.o.f.) Hamiltonian system possessing a circle of parabolic fixed points is perturbed. It is proved that its occurrence is generic for one parameter families (codimension one phenomenon) of nearintegrable, two d.o.f. Hamiltonian systems. Numerical experiments indicate that the motion near a parabolic resonance exhibits a new type of chaotic behavior which includes instabilities in some directions and long trapping times in others. Moreover, in a degenerate case, near a flat parabolic resonance, large scale instabilities appear. A model arising from an atmospherical study is shown to exhibit flat parabolic resonance. This supplies a simple mechanism for the transport of particles with small (i.e. atmospherically relevant) initial velocities from the vicinity of the equator to high latitudes. A modification of the model which allows the development of atmospherical jets unfolds the degeneracy, yet traces of the flat instabilities are clearly observed.

Selfsimilarity, renormalization, and phase space nonuniformity of Hamiltonian chaotic dynamics
View Description Hide DescriptionA detailed description of fractional kinetics is given in connection to islands’ topology in the phase space of a system. The method of renormalization group is applied to the fractional kinetic equation in order to obtain characteristic exponents of the fractional space and time derivatives, and an analytic expression for the transport exponents. Numerous simulations for the webmap and standard map demonstrate different results of the theory. Special attention is applied to study the singular zone, a domain near the island boundary with a selfsimilar hierarchy of subislands. The birth and collapse of islands of different types are considered.

Chaotic transmission of waves and “cooling” of signals
View Description Hide DescriptionRay dynamics in waveguide media exhibits chaotic motion. For a finite length of propagation, the large distance asymptotics is not uniform and represents a complicated combination of bunches of rays with different intermediate asymptotics. The origin of the phenomena that we call “chaotic transmission,’’ lies in the nonuniformity of the phase space with sticky domains near the boundary of islands. We demonstrate different fractalproperties of ray propagation using underwater acoustics as an example. The phenomenon of the kind of Lévy flights can occur and it can be used as a mechanism of cooling of signals when the width of spatial spectra dispersion is significantly reduced.

Dynamics of a pair of spherical gravitating shells
View Description Hide DescriptionThe dynamical body problem for a system of mass points interacting solely through gravitational forces is not integrable. The difficulties which arise in constructing accurate numerical codes for simulating the motion over long time scales are legend. Thus, in order to test their theories,astronomers and astrophysicists resort to simpler, onedimensional models which avoid the problems of binary formation, escape, and the singularity of the inverse square force law. To date, the most frequently employed “test’’ model consists of a system of parallel mass sheets moving perpendicular to their surface. While this system avoids all of the above problems, the time scale for reaching equilibrium is extremely long and probably arises from the system’s weak ergodic properties, which become manifest even in the three sheet system. Here we consider a different onedimensional gravitating system consisting of nonrotating concentric mass shells. For the case of two shells we investigate the structure of the phase space by studying the stability of periodic trajectories. By employing an event driven algorithm, we are able to directly investigate the influence of the singularity without having to resort to regularization of the force. Although stable structures are present at every energy, we find that the ergodic properties of this system are more robust than its planar counterpart.

On the chaotic nature of turbulence observed in benchmark analysis of nonlinear plasma simulation
View Description Hide DescriptionSimulational results of two dissipative interchange turbulence (Rayleigh–Taylortype instability with dissipation) models with the same physics are compared. The convective nonlinearity is the nonlinear mechanism in the models. They are shown to have different time evolutions in the nonlinear phase due to the different initial value which is attributed to the initial noise. In the first model (A), a single pressure representing the sum of the ion and electron components is used (onefluid model). In the second model (B) the ion and electron components of the pressure fields are independently solved (twofluid model). Both models become physically identical if we set ion and electron pressure fields to be equal in the model (B). The initial conditions only differ by the infinitesimally small initial noise due to the roundoff errors which comes from the finite difference but not the differentiation. This noise grows in accordance with the nonlinear development of the turbulence mode. Interaction with an intrinsic nonlinearity of the system makes the noise grow, whose contribution becomes almost the same magnitude of the fluctuation itself in the results. The instantaneous deviation shows the chaotic characteristics of the turbulence.