## Lorenz and modular flows: a visual introduction A tangled tale linking lattices, knots, templates, and strange attractors.
## 1.Introduction Sometimes, seemingly unrelated objects turn out to be
related... We would like to present here a mathematical example, exhibiting a
close connection between two dynamical systems, one coming from number theory and
the other from meteorology.
We would like to describe a close Copyright for all films and images is by Jos Leys / Etienne Ghys. ## 2.The Lorenz flow## 2.1 The Lorenz strange attractor and its periodic orbitsThe model discovered by E. Lorenz is described by the following differential equation in 3-space [3]: ; ;
Birman and Williams proved that Click on the knot images below for a movie. (Knots are usually associated to a code like 8.19, meaning the 19th knot among those which can be represented in a diagram with 8 crossings, using an ordering which is more traditional than logical). Among the 1 701 936 (prime) knots with 16 crossings or less , only 21 appear as Lorenz knots (checked using a computer and [17]).
## 2.2 The Lorenz templateAs a matter of fact, it is not difficult to describe
these periodic orbits by an ## 3. The modular flow## 3.1 The space of lattices and its topologyGiven two linearly independent vectors L of R^{2} that they generate:
n
_{1}
, n_{2}Z}.
A subset of the plane of this form is called a cω generate the same lattice._{1}+ dω_{2}The set of lattices is a Think of the plane ; . It is easy to check that these series converge since
the exponents 4 and 6 are bigger than 2. Now, an odd exponent would yield a zero
sum, since the lattice is obviously symmetric with respect to the origin, so
that 4 and 6 are indeed the first cases to consider. As for the 60 and 140,
they are normalizing constants which are not relevant to our discussion. The
main point is that the pair of complex numbers ( L)
, g_{3}(L) ) characterizes the lattice. More precisely, a pair of
complex numbers (g) corresponds to a lattice if and only if the
so-called _{2},
g_{3}discriminant Δ = g
is not zero. See for instance [11],
[16]
for a proof._{2}^{3}− 27g_{3}^{2}Summing up,
The picture on the left represents symbolically C=R^{2}. The vertical green axis corresponds to those
lattices for which g=0; this is another copy of _{2}C=R^{2}. The
yellow curve represents Δ=0, but again this
is a one dimensional curve over the complex numbers, and therefore a surface
from the point of view of real numbers.How can we look at this four dimensional object in a concrete way? If g L) ; g(_{3}kL) = k(^{−
}^{6}g_{3}L),so that for each lattice kL)|(^{2}+ |g_{3}kL)|^{2}= 1.Hence the space of lattices of area 1 is identified with the complement of a trefoil knot in the 3-sphere, which, after deleting one point, is the complement of a trefoil knot in the usual 3-space. So, there is some hope of seeing something! Let’s have a look. To get more topological insight, let’s have a look at some additional structures in the space of lattices of area 1. Now, for the fun of it, let’s have a look at the global picture, including the trefoil, the axis, the Seifert fibers, and the Seifert surfaces, all together! (Click on the picture for a movie) ## 3.2 The Modular dynamicsNow that we have a clear topological idea of the space of lattices of area 1, we can define a dynamical
system on that space. The definition is very simple. For every real number . If Our purpose is to give a visual description of this flow and of its periodic orbits. This flow is a well known example
of an ; . Note that and ,so that if one takes two lattices ## 3.3 Periodic orbits and their linking with the trefoilWe can now start the topological description of the periodic orbits of the modular flow. There is a simple way to describe
these periodic orbits. Consider a 2 × 2 matrix
Clearly, the matrix
for some Hence The following periodic clip (click it to run) shows a periodic orbit in the space of lattices of area 1. Note that each point follows a hyperbola, the orbit of in the plane, so that the trajectories of the points are not periodic, but the trajectory of the lattice, as a part of the plane, is indeed periodic. It is not difficult to see that if one replaces
These periodic orbits have a very old mathematical tradition. One finds them in many different areas, in different
disguises: Each one of these periodic orbits is a closed curve in the space of lattices of area 1, hence defines a For the matrix the knot looks disappointing! It is a small trivial knot. . . For the matrix the knot is still trivial but placed differently with respect to the trefoil. For the matrix the knot is more interesting; it is a trefoil knot. For the matrix it is a torus knot T(4, 5). For the matrices and it is, well, more complicated! Before we discuss the nature of these knots, let us ask a seemingly simpler question:
One should quickly recall the notion of k (orange)
do not intersect, and project them onto a plane in a generic way. These projections need not be disjoint of course._{2}
Attach the sign +1 to the left situation and −1 to the right one, and sum these indices over the set of all crossings
of k. The result is called the _{2}linking number
between k and _{1}k. For instance, in the knot picture on the left, the blue
curve crosses once over the orange one, with a '+1' sign, so that the linking number is '+1'. In the picture on the right,
one has two cross-overs with different signs, so that the linking number is 0._{2}The important point is that this number is independent of the (generic) projection used to compute it, and remains invariant if the knots move continuously without intersecting each other. Now, let us come back to our question, and try to compute the linking number between index of this closed curve with respect to the origin, i.e. the number of turns it makes around the
origin. A way to compute this index is to compute the algebraic intersection with the real axis, counting a '+1' or a '
−1' for each intersection according to whether the curve crosses from negative to positive imaginary part, or the other way.
Topologically, this means that the Seifert surface is two-sided, and that one wishes to compute the algebraic intersection of
k with the Seifert surface._{A} This is described by the pictures below. For a given matrix kevolves, it intersects the surface from time to time,
positively or negatively. The linking number we are looking for is the sum of these signs. In the first instance, we get two
plus signs, in the second instance, one plus sign, and two plus signs and one minus sign in the final one ._{A }It turns out that there is a nice formula for computing this linking number which relates it to a famous arithmetical function. Consider the two matrices
As it turns out, any integral matrix
The word in
We can only refer to [1] for several proofs and more information, but in the next section we shall provide some interpretation. ## 4 Lorenz and modular knots## 4.1 The modular templateSo far we have discussed the periodic orbits in the Lorenz attractor, the
More precisely, for every modular knot We then extend this one dimensional object in the unstable direction to create a two dimensional template. One would need more technical details to give a full description, but we will limit ourselves to images showing the progressive development of the template. Note however that we are discussing topology, so that this construction is far from being unique, and we have made some specific choices for visual reasons. Now the next task is to deform one of the modular knots in the complement of the trefoil knot, so that after the deformation, the
knot lies on the template exactly like Lorenz knots. Again, we cannot give
details and we refer to [1] for more. The general idea is to compress
the space of lattices of area 1 to a small neighborhood of the one dimensional object, by using an old idea (again of Gauss !):
given a lattice ## 4.2 Two final remarks
It follows that the linking number of U letters minus
the number of the V letters in the word expressing A. This is one approach to the proof of the first
theorem that we mentioned: the linking number of A and the trefoil is Rademacher’s number.
exp(t) and s.exp(t) + i exp(−t). Fixing t and letting s run from 0 to 1,
we get a periodic curve cin the space of lattices of area 1, which is a
periodic orbit of . The following clip shows this curve
_{t }c when _{t}t describes some interval. When t is a large negative number, the curve c is a small trivial loop going once around the trefoil knot. When _{
t}t is a big positive number the curve
c gets longer and in the limit fills the whole space._{t}It is known that the the family of curves Riemann hypothesis, one of the
most enticing open questions in mathematics! [15]## References[1]
GHYS, E.: [2]
LEYS, J.: [3]
LORENZ, E.: Deterministic Nonperiodic Flow. [4]
VIANA, M.: What’s New on Lorenz Strange Attractors. [5]
BIRMAN, J. & WILLIAMS, R.: Knotted periodic orbits in dynamical systems.I. Lorenz’s equations.
[6]
ADAMS, C.: [7]
KAUFFMANN, L.: [8]
SOSSINSKY, A: [9]
STEWART, I : [10]
TUCKER, W.: A rigorous ODE solver and Smale’s 14th problem. [11]
SERRE, J.-P.:
[12]
SCHAREIN, R.: [13]
SLIJKERMAN, F. : [14] POVRAY: Public domain raytracing software http://www.povray.org/ [15]
SARNAK, P.: [16]
McKEAN, H. & MOLL, V.: [17] THE KNOT ATLAS.: http://katlas.math.toronto.edu/wiki/Main_Page |