Rotation intervals for chaotic sets

Chaotic invariant sets for planar maps typically contain periodic orbits whose stable and unstable manifolds cross in grid-like fashion. Consider the rotation of orbits around a central fixed point. The intersections of the invariant manifolds of two periodic points with distinct rotation numbers can imply complicated rotational behavior. We show, in particular, that when the unstable manifold of one of these periodic points crosses the stable manifold of the other, and, similarly, the unstable manifold of the second crosses the stable manifold of the first, so that the segments of these invariant manifolds form a topological rectangle, then all rotation numbers between those of the two given orbits are represented. The result follows from a horseshoe-like construction.

     

Quantitation of nine quinolones in chicken tissues by high-performance liquid chromatography with fluorescence detection

A simple reversed-phase high-performance liquid chromatographic method was developed and validated for simultaneous analysis of nine quinolones (ciprofloxacin, danofloxacin, difloxacin, enrofloxacin, flumequine, marbofloxacin, nalidixic acid, oxolinic acid, sarafloxacin) in chicken tissue. The analytes were extracted from homogenized muscle using an acetonitrile basic solution. After centrifugation and partial evaporation, direct injection was possible. Three different HPLC conditions were applied to quantify the residual quinolones. Separation was achieved on a PLRP-S column and detection was performed with a monochromator fluorescence detector. The recovery, the limit of detection, the limit of quantification, the accuracy and the precision of the method were evaluated from spiked tissue samples at concentration levels ranging from 15 microg kg(-1) to 300 microg kg(-1) according to the maximum residue limit of each quinolone. This method is also suitable for porcine, bovine, ovine and fish muscle tissue.     

The liapunov dimension of strange attractors

Many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange. In this paper the question of estimating the dimension of the attractor is addressed. While more general conjectures are made here, particular attention is paid to the idea that if the Jacobian determinant of a map is greater than one and a ball is mapped into itself, then generically, the attractor will have positive two-dimensional measure, and most of this paper is devoted to presenting cases with such Jacobians for which the attractors are proved to have non-empty interior.     

Structure and crises of fractal basin boundaries

We discuss the structure of fractal basin boundaries in typical nonanalytic maps of the plane and describe a new type of crisis phenomenon.     

Is the dimension of chaotic attractors invariant under coordinate changes?

Several different dimensionlike quantities, which have been suggested as being relevant to the study of chaotic attractors, are examined. In particular, we discuss whether these quantities are invariant under changes of variables that are differentiable except at a finite number of points. It is found that some are and some are not. It is suggested that the word dimension be reversed only for those quantities have this invariance property.     

Indecomposable continua in dynamical systems with noise: Fluid flow past an array of cylinders

Standard 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 time-1 Poincare map. Then we add a small amount of noise, so that on each iteration the Poincare map is perturbed smoothly, but differently for each time cycle. Fix an x coordinate x(0) and an initial time t(0). We discuss when the set of initial points at a time t(0) whose trajectory (x(t),y(t)) is semibounded (i.e., x(t)>x(0) 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. (c) 1997 American Institute of Physics.     

Metamorphoses: Sudden jumps in basin boundaries

In some invertible maps of the plane that depend on a parameter, boundaries of basins of attraction are extremely sensitive to small changes in the parameter. A basin boundary can jump suddenly, and, as it does, change from being smooth to fractal. Such changes are calledbasin boundary metamorphoses. We prove (under certain non-degeneracy assumptions) that a metamorphosis occurs when the stable and unstable manifolds of a periodic saddle on the boundary undergo a homoclinic tangency.This research was supported in part by grants and contracts from the Defense Advanced Research Projects Agency, the Consiglio Nazionale delle Ricerche (Comitato per le Matematiche), and the National Science Foundation     

Crinkled changes of variables for maps on a circle

Nonlinear Dynamics - Take a piece of paper, crush it into a ball, and pound it flat onto a surface. The map from the original piece of paper to the surface is a non smooth change of variables...