Continued Fractions and Chaos***

Robert M. Corless
Dept. Applied Math
University of Western Ontario
London, Ontario

Mathactivated text
Other available formats
Related links
Author biography


The theory of continued fractions goes back at least to c. A. D. 500 to the work of ryabhata, and possibly as far back as c. 300 B.C. to Euclid. The theory of chaotic dynamical systems is relatively recent, going back only to the work of Poincaré [20] and Birkhoff [2]. The foundations of the theory of continued fractions, as we know it now, are well established due to the work of Euler, Lagrange, Gauss, and others, while the foundations of chaotic dynamical systems are still evolving. This paper will use the well-established theory of simple continued fractions to explore some current results of the theory of chaotic dynamical systems.
***Previously appeared in the American Mathematical Monthly

Author's Reflections:

Revisiting Continued Fractions and Chaos

I started to write the paper `Continued Fractions and Chaos' (CFC hereinafter) in 1988. It was either my first or second single-author paper, and there are lots of things in it that I am grateful for the opportunity to change. Its companion paper `Chaos and Continued Fractions', co-authored with Greg Frank and Graham Monroe (then grad students at Western), was written at about the same time. The companion paper appeared first, in Physica D, in 1989. It wasn't until 1992 that CFC appeared in the Monthly.

The paper was written because of efforts on my part to learn about chaos. In taking Paddy Nerenberg's course on chaos, I was reminded of efforts to compute the continued fraction for in George Maxwell's number theory class when I was an undergraduate: the behaviour of the algorithm for computing the continued fraction led to just the same sorts of divergences that I was seeing in computing the chaotic maps in the course. The paper was a fairly natural outgrowth of that observation, and I gave my first talk at the Ontario Mathematics Meeting in 1988 on this topic (it was a miserable talk---among other things, I committed the cardinal sin of going over time).

There are plenty of places where stylistic improvements to the paper can be made, and I have taken this opportunity to make some of them. In particular, Nick Higham used one pair of sentences from this paper as a `bad example', in his book `Handbook of Mathematical Writing'. Oops, heh-heh, pardon me while I fix that...

In addition, the paper has been thoroughly read by Dhavide Aruliah, Heinz Bauschke, Greg Fee, and Jennifer Overington, who have all made stylistic comments. Dhavide and Jennifer have taken material I had written for talks and workshops, subsequent to the writing of CFC, and used them in part to create tutorials on continued fractions and on chaos (in particular on Lyapunov exponents). It is my hope that these tutorials will be hooked up in a useful way to this paper. Greg Fee dug out and implemented Floyd's algorithm from Knuth, vol II, for identifying the start of a cycle in a map, and this enables us to investigate the floating-point Gauss map even more thoroughly. All of this has helped to make this electronic version not just an updated and corrected version, but one with more content, as well.

As far as the content goes, I found in re-reading it, and adapting it to this electronic medium, that my ideas and knowledge about chaos have grown and changed considerably. In particular, I know a lot more now about shadowing and its importance for computation.

Nevertheless the ideas expressed in this paper remain, at least in embryo, central to my view of reliable and faithful simulation of chaotic dynamical systems. I have followed this paper with several related ones, but the main candidate to be called a proper sequel to this paper is `What Good are Numerical Simulations of Chaotic Dynamical Systems', which appeared this year (1995) in Computers and Mathematics with Applications.

In `What Good' I am concerned chiefly with the fidelity of numerical methods for solving nonlinear ordinary differential equations, and the ideas brought forth in CFC really help to explain the problem that I'm looking at, and also the remaining unresolved problems. You see, if we had a good general understanding of the methods for solving such problems, then we ought to understand the numerical simulation of the Gauss map; but we don't really understand why it works. In CFC I was groping towards understanding what we don't understand: I say in CFC that `On closer examination, however, we see that if the period of an orbit is long, then the orbit behaves for a long time as if it were aperiodic, reflecting the effect of ``nearby'' initial points that are aperiodic.'

I believe this statement to be true, and there is a theorem of Gora and Boyarsky which helps me with that belief. But the point is still not established rigorously, and it seems to me that this is still the central issue with the fidelity of numerical methods. Floating-point arithmetic is finite, and this has serious consequences for modelling the infinite (or at least it ought to, and sometimes it does). We really need a good theory to explain why these mathematical experiments really work (the computation of in this paper). The trick will be in showing that the statistics (distributions) of the orbits will be mostly right if the local accuracy of the computation of the map is high; note that this is a discrete `local to global' problem, and the results of Gora and Boyarsky are the best in this context to date (they still don't explain the Gauss map results, though). This question must be answered before we can place any faith in the computation of Lyapunov exponents for nonlinear problems, for example. Shadowing helps, but is not enough.

So the connection of this paper to Experimental Mathematics is, in some sense, a question of the validation of the tools that we are using with Experimental Mathematics (at least for the simulation of chaotic maps or odes).

Aside from that, computation with the Gauss map leads directly to almost all of the main ideas in chaotic dynamical systems: ergodicity, Markov partitions, shadowing, Lyapunov exponents, reliability of simulation. So in some sense it provides a `Royal Road' to chaos.

Some other questions that asked themselves of me while I re-read this paper are

  1. What is the distribution of Lyapunov exponents among the quadratic irrationals? We ought to be able to compute them all and see (or at least a lot of them). And how does it compare with the distribution of Lyapunov exponents in floating-point arithmetic (though here the transients play a very important, perhaps crucial, role)?
  2. Is the P-th root of a quadratic irrational? It's not known if it's algebraic, but perhaps this weaker result is known? This is a question that can be `experimentally' answered, at least for polynomial coefficients up to a certain size. Then I could remove the footnote that a last-minute editor or referee put in to my paper about this, and fix a typo which caused the referee to put the footnote there...
  3. Are the partial quotients of Stark's continued fraction bounded?
  4. We can write down an integral for Khinchin's constant (the expected geometric mean of the partial quotients of continued fractions), namely

    We can now compute quickly, using a rather fast series. Can this integral be computed quickly any other way? There are an infinite number of ever-weakening jump discontinuities near the origin.

    And by the way, what is the official spelling of Khinchin? I don't use the t, because it didn't appear on his book. But it is spelled differently in many places in the literature, in English. Now that TeX is available, perhaps we can spell it correctly in the Cyrillic alphabet?

All in all, I am very grateful for the chance to revisit this paper, particularly in such company. The prizes won by the other authors in this volume for expository writing include the Chauvenet Prize (Borwein, Borwein, and Bailey, 1992), the Lester R. Ford Award (Jeffrey Lagarias, 1985; Stan Wagon, 1988; Ronald L. Graham, 1991), and the Carl Allendoerfer Award (Ronald L. Graham, 1990). Doubtless by the time you read this other authors in the volume will have won more---it really is an all-star lineup.