Mitchell J. Feigenbaum

Presentation functions, fixed points, and a theory of scaling function dynamics

Presentation functions provide the time-ordered points of the forward dynamics of a system as successive inverse images. They generally determine objects constructed on trees, regular or otherwise, and immediately determine a functional form of the transfer matrix of these systems. Presentation functions for regular binary trees determine the associated forward dynamics to be that of a period doubling fixed point. They are generally parametrized by the trajectory scaling function of the dynamics in a natural way. The requirement that the forward dynamics be smooth with a critical point determines a complete set of equations whose solution is the scaling function. These equations are compatible with a dynamics in the space of scalings which is conjectured, with numerical and intuitive support, to possess its solution as a unique, globally attracting fixed point. It is argued that such dynamics is to be sought as a program for the solution of chaotic dynamics. In the course of the exposition new information pertaining to universal mode locking is presented.

Presentation functions and scaling function theory for circle maps

Considering first return maps, a most natural renormalisation group fixed point is determined. From it a simple presentation function is constructed, immediately leading to the thermodynamics of critical rotation. The rotation number is encoded in the topological action of the presentation function and the algebraic singularity of critically in that functions derivatives at its fixed points. Any such presentation function determines a circle map dynamics of that rotation number and index of criticality. These functions are naturally parametrised by a trajectory scaling function. The requirement that the dynamics be smooth leads to a prescription for the calculation of the scaling function and hence the dynamics. The theory is highly constrained and suffers in finite-order approximation from the extra constraint of commutativity, which however can be overcome.

Square root singularity in the viscosity of neutral colloidal suspensions at large frequencies

The asymptotic frequency, ω, dependence of the dynamic viscosity of neutral hard-sphere colloidal suspensions is shown to be of the formη0A(ϕ)(ωτp)-1/2, whereA(ϕ) has been determined as a function of the volume fraction ϕ, for all concentrations in the fluid range,η0 is the solvent viscosity, andτp is the Péclet time. For a soft potential it is shown that, to leading order in the steepness, the asymptotic behavior is the same as that for the hard-sphere potential and a condition for the crossover behavior to 1/ωτp, is given. Our result for the hardsphere potential generalizes a result of Cichocki and Felderhof obtained at low concentrations and agrees well with the experiments of van der Werffet al. if the usual Stokes-Einstein diffusion coefficientD0 in the Smoluchowski operator is consistently replaced by the short-time self-diffusion coefficientDs(ϕ) for nondilute colloidal suspensions.

Dynamics of Finger Formation in Laplacian Growth Without Surface Tension

We study the dynamics of “finger” formation in Laplacian growth without surface tension in a channel geometry (the Saffman–Taylor problem). We present a pedagogical derivation of the dynamics of the conformal map from a strip in the complex plane to the physical channel. In doing so we pay attention to the boundary conditions (no flux rather than periodic) and derive a field equation of motion for the conformal map. We first consider an explicit analytic class of conformal maps that form a basis for solutions in infinitely long channels, characterized by meromorphic derivatives. The great bulk of these solutions can lose conformality due to finite time singularities. By considerations of the nature of the analyticity of these solutions, we show that those solutions which are free of such singularities inevitably result in a single asymptotic “finger” whose width is determined by initial conditions. This is in contradiction with the experimental results that indicate selection of a finger of width 1/2. In the last part of this paper we show that such a solution might be determined by the boundary conditions of a finite body of fluid, e.g. finiteness can lead to pattern selection.

Scaling Function Dynamics

In September 1979, P. Hohenberg gave me a picture which showed the first preliminary results of A. Libchaber’s experiment on liquid helium [1], the power spectrum of a measured signal (Fig. 1). It was immediately clear that the picture had something to do with period doubling, but how it was that one was supposed to understand a one-dimensional theory for a discrete dynamics in order to learn what a fluid was doing was in no way very clear. Over a period of a few months, I tried to understand the picture, and, in the end, was lead to an idea that I have called the scaling function [2]. In these lectures, I shall try to explain what the idea is that came out of this observation and while doing so, discuss the idea that goes under the name of “presentation functions” [3]. I will indicate what these notions mean and explain how from that picture you can determine what is actually the most interesting part of the dynamics.

Pattern selection: Determined by symmetry and modifiable by distant effects

We consider Saffman–Taylor channel flow without surface tension on a high-pressure driven interface, but modify the usual infinite-fluid in infinite-channel configuration. Here we include the treatment of efflux by considering a finite connected body of fluid in an arbitrarily long channel, with its second free interface the efflux of this configuration. We show that there is a uniquely determined translating solution for the driven interface, which is exactly the 1/2 width S–T solution, following from correct symmetry for a finite channel flow. We establish that there exist no perturbations about this solution corresponding to a finger propagating with any other width: Selection is locally unique and isolated. The stability of this solution is anomalous, in that all freely impressible perturbations are stabilities, while unstable modes request power proportional to their strength from the external agencies that drive the flow, and so, in principle, are experimentally controllable. This is very different from the behavior of the usual infinite fluid. We conjecture that surface tension on the efflux interface modifies channel-width λ according to 1−2λ=σ/v (i.e., (2π)2B of the literature) with v the velocity of the high-pressure tip, but σ the surface tension of the efflux. That is, λ is decreased below 1/2 by the effect of smoothing the distant efflux. The perturbation theory created here to deal with transport between two free boundaries is novel and dependent upon a symmetry implied by the equations of motion.

Complicated Objects on Regular Trees

Our goal in these lectures will be to study what kind of description is possible for complicated objects that arise from or can be associated with dynamical systems. In particular we will try to understand the structure of Cantor sets and strange attractors and the focal point will be cruder--thermodynamic understanding.