Preturbulence: A regime observed in a fluid flow model of Lorenz

This paper studies a forced, dissipative system of three ordinary differential equations. The behavior of this system, first studied by Lorenz, has been interpreted as providing a mathematical mechanism for understanding turbulence. It is demonstrated that prior to the onset of chaotic behavior there exists a preturbulent state where turbulent orbits exist but represent a set of measure zero of initial conditions. The methodology of the paper is to postulate the short term behavior of the system, as observed numerically, to establish rigorously the behavior of particular orbits for all future time. Chaotic behavior first occurs when a parameter exceeds some critical value which is the first value for which the system possesses a homoclinic orbit. The arguments are similar to Smale's horseshoe.Research supported by NSF Grant MCS 76-24432     

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...     

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.     

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.

     

Book review: Nonlinear oscillations,dynamical systems,and bifurcations of vector fields

Journal of Statistical Physics -     

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.