Index of content:
Volume 11, Issue 4, December 2001

Scroll waves in spherical shell geometries
View Description Hide DescriptionThe evolution of scroll waves in excitable media with spherical shell geometries is studied as a function of shell thickness and outer radius. The motion of scroll wave filaments that are the locii of phaseless points in the medium and organize the wave pattern is investigated. When the inner radius is sufficiently large the filaments remain attached to both the inner and outer surfaces. The minimum size of the sphere that supports spiral waves and the maximum number of spiral waves that can be sustained on a sphere of given size are determined for both regular and random initial distributions. When the inner radius is too small to support spiral waves the filaments detach from the inner surface and form a curved filament connecting the two spiral tips in the surface. In certain parameter domains the filament is an arc of a circle that shrinks with constant shape. For parameter values close to the meandering border, the filament grows and collisions with the sphere walls lead to turbulent filament dynamics.

Dynamics of Parkinsonian tremor during deep brain stimulation
View Description Hide DescriptionThe mechanism by which chronic, high frequency, electrical deep brain stimulation (HFDBS) suppresses tremor in Parkinson’s disease is unknown. Rest tremor in subjects with Parkinson’s disease receiving HFDBS was recorded continuously throughout switching the deep brain stimulator on (at an effective frequency) and off. These data suggest that the stimulation induces a qualitative change in the dynamics, called a Hopf bifurcation, so that the stable oscillations are destabilized. We hypothesize that the periodic stimulation modifies a parameter affecting the oscillation in a time dependent way and thereby induces a Hopf bifurcation. We explore this hypothesis using a schematic network model of an oscillator interacting with periodic stimulation. The mechanism of timedependent change of a control parameter in the model captures two aspects of the dynamics observed in the data: (1) a gradual increase in tremor amplitude when the stimulation is switched off and a gradual decrease in tremor amplitude when the stimulation is switched on and (2) a time delay in the onset and offset of the oscillations. This mechanism is consistent with these rest tremor transition data and with the idea that HF–DBS acts via the gradual change of a network property.

Advection of passive particles in the Kolmogorov flow
View Description Hide DescriptionA statistical analysis of the advection of passive particles in a flow governed by driven twodimensional Navier–Stokes equations (Kolmogorov flow) is presented. Different regimes are studied, all corresponding to a chaotic behavior of the flow. The diffusion is found to be strongly asymmetric with a very weak transport perpendicular to the forcing direction. The trajectories of the particles are characterized by the presence of traps and flights. The trapping time distributions show algebraic decrease, and strong anomalous diffusion is observed in transient phases. Different regimes lead to different types of diffusion, i.e., no universal behavior of diffusion is observed, and both time and space properties are needed to define anomalous transport.

Branching of 2D tori off an equilibrium of a cosymmetric system (codimension1 bifurcation)
View Description Hide DescriptionThe standard object for vector fields with a nontrivial cosymmetry is a continuous oneparameter family of equilibria. Characteristically, the stability spectrum of equilibrium varies along such a family, though the spectrum always contains a zero point. Consequently, in the general position a family consists of stable and unstable arcs separated by boundary equilibria, which are neutrally stable in the linear approximation. In the present paper the central manifold method and the Lyapunov–Schmidt method are used to investigate the branching bifurcation of invariant twodimensional tori in cosymmetric systems off a boundary equilibrium whose spectrum contains, besides the requisite point 0, two pairs of purely imaginary eigenvalues. A number of new effects, as compared with the classic case of an isolated equilibrium, are found: the bifurcation studied has codimension 1 (2 for an isolated equilibrium); it is accompanied by a branching bifurcation of a normal limit cycle; and, a stable arc can be created on an unstable arc.

Multifractal structure of a riddled basin
View Description Hide DescriptionFor a twodimensional piecewise linear map with a riddled basin, a multifractal spectrum which characterizes the “skeletons” of the riddled basin, is introduced. With the uncertainty exponent is obtained by a variational principle, which enables us to introduce a concept of a “boundary” for the riddled basin. A conjecture on the relation between and the “stable sets” of various ergodic measures, which coexist with the natural invariant measure on the chaotic attractor, is also proposed.

Mushrooms and other billiards with divided phase space
View Description Hide DescriptionWe present the first natural and visible examples of Hamiltonian systems with divided phase space allowing a rigorous mathematical analysis. The simplest such family (mushrooms) demonstrates a continuous transition from a completely chaotic system (stadium) to a completely integrable one (circle). In the course of this transition, an integrable island appears, grows and finally occupies the entire phase space. We also give the first examples of billiards with a “chaotic sea” (one ergodic component) and an arbitrary (finite or infinite) number of KAM islands and the examples with arbitrary (finite or infinite) number of chaotic (ergodic) components with positive measure coexisting with an arbitrary number of islands. Among other results is the first example of completely understood (rigorously studied) billiards in domains with a fractal boundary.

From topology to dynamics in biochemical networks
View Description Hide DescriptionAbstract formulations of the regulation of gene expression as random Boolean switching networks have been studied extensively over the past three decades. These models have been developed to make statistical predictions of the types of dynamics observed in biological networks based on network topology and interaction bias, p. For values of mean connectivity chosen to correspond to real biological networks, these models predict disordered dynamics. However, chaotic dynamics seems to be absent from the functioning of a normal cell. While these models use a fixed number of inputs for each element in the network, recent experimental evidence suggests that several biological networks have distributions in connectivity. We therefore study randomly constructed Boolean networks with distributions in the number of inputs, K, to each element. We study three distributions: delta function, Poisson, and power law (scale free). We analytically show that the critical value of the interaction bias parameter, p, above which steady state behavior is observed, is independent of the distribution in the limit of the number of elements We also study these networks numerically. Using three different measures (types of attractors, fraction of elements that are active, and length of period), we show that finite, scalefree networks are more ordered than either the Poisson or delta function networks below the critical point. Thus the topology of scalefree biochemical networks, characterized by a wide distribution in the number of inputs per element, may provide a source of order in living cells.

Lagrangian chaos and Eulerian chaos in shear flow dynamics
View Description Hide DescriptionShear flowdynamics described by the twodimensional incompressible Navier–Stokes equations is studied for a onedimensional equilibrium vorticity profile having two minima. These lead to two linear Kelvin–Helmholtz instabilities; the resulting nonlinear waves corresponding to the two minima have different phase velocities. The nonlinear behavior is studied as a function of two parameters, the Reynolds number and a parameter λ specifying the width of the minima in the vorticity profile. For parameters such that the instabilities grow to a sufficient level, there is Lagrangianchaos, leading to mixing of vorticity, i.e., momentum transport, between the chains of vortices or cat’s eyes. Lagrangianchaos is quantified by plotting the finite time Lyapunov exponents on a grid of initial points, and by the probability distribution of these exponents. For moderate values of λ, there is Lagrangianchaos everywhere except near the centers of the vortices and near the boundaries, and there are competing effects of homogenization of vorticity and formation of structures associated with secondary resonances. For smaller values of λ Lagrangianchaos occurs in the regions in the centers of the vortices, and the Eulerian behavior of the flow undergoes bifurcations leading to Eulerian chaos, as measured by the time series of several Galilean invariant quantities. A discussion of Lagrangianchaos and its relation to Eulerian chaos is given.

Dynamics of one and twodimensional kinks in bistable reaction–diffusion equations with quasidiscrete sources of reaction
View Description Hide DescriptionWe study the evolution of fronts in a bistable reaction–diffusion system when the nonlinear reaction term is spatially inhomogeneous. This equation has been used to model wave propagation in various biological systems. Extending previous works on homogeneous reaction terms, we derive asymptotically an equation governing the front motion, which is strongly nonlinear and, for the twodimensional case, generalizes the classical mean curvature flow equation. We study the motion of one and twodimensional fronts, finding that the inhomogeneity acts as a “potential function” for the motion of the front; i.e., there is wave propagation failure and the steady state solution depends on the structure of the function describing the inhomogeneity.

A new approximate analytical approach for dispersion relation of the nonlinear Klein–Gordon equation
View Description Hide DescriptionA novel approach is presented for obtaining approximate analytical expressions for the dispersion relation of periodic wavetrains in the nonlinear Klein–Gordon equation with even potential function. By coupling linearization of the governing equation with the method of harmonic balance, we establish two general analytical approximate formulas for the dispersion relation, which depends on the amplitude of the periodic wavetrain. These formulas are valid for small as well as large amplitude of the wavetrain. They are also applicable to the large amplitude regime, which the conventional perturbation method fails to provide any solution, of the nonlinear system under study. Three examples are demonstrated to illustrate the excellent approximate solutions of the proposed formulas with respect to the exact solutions of the dispersion relation.

Complexity of regular invertible padic motions
View Description Hide DescriptionWe consider issues of computational complexity that arise in the study of quasiperiodic motions (Siegel discs) over the adic integers, where is a prime number. These systems generate regular invertible dynamics over the integers modulo for all and the main questions concern the computation of periods and orbit structure. For a specific family of polynomial maps, we identify conditions under which the cycle structure is determined solely by the number of Siegel discs and two integer parameters for each disc. We conjecture the minimal parametrization needed to achieve—for every odd prime —a twodisc tessellation with maximal cycle length. We discuss the relevance of Cebotarev’s density theorem to the probabilistic description of these dynamical systems.

On the wavelet formalism for multifractal analysis
View Description Hide DescriptionIt is proved that the multifractal characterizations of diametrically regular measures that are provided by the wavelet and by the Hentschel–Procaccia formalisms are identical.