Lorenz and Modular Flows: A Visual Introduction
Posted November 2006.
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.
In 1801, Carl Friedrich Gauss published Disquisitiones Arithmeticae, his first masterpiece. It deals with the foundations of the theory of number fields. Today, many important aspects of this theory can be expressed in terms of a dynamical system acting in the space of lattices: the modular flow.
In 1963, the meteorologist Edward Lorenz was studying a very simplified numerical model for the atmosphere, which led him to the amazing strange attractor popularized through the famous butterfly effect: the flapping wings of a butterfly might cause some tiny change in the state of the atmosphere which can in turn lead to hurricanes!
We would like to describe a close topological connection between these two mathematical objects.
This article cannot be qualified as “mathematical” since it contains no proof! Our main motivation is in the visualization of some of the marvelous mathematical objects which are involved. We have tried to explain enough mathematics, at a rather elementary level, to comment on the pictures and clips which are the true content of this epaper.
Our story is organized in a very simple way. First we describe the Lorenz attractor (2.1) and its template (2.2), then the space of lattices (3.1), the modular dynamics (3.2) and its periodic orbits (3.3), and finally we establish a connection (4.1 and 4.2) between these two dynamical systems!
This is the result of the collaboration of a mathematician and an artistgeometer. The reader may consult [1] for more mathematics, and [2] for more graphics.
Graphics were made in Ultrafractal [13] and Povray [14]. Data for knot drawings were extracted from Knotplot [12]. Larger versions of the films can be seen at Jos Leys' site.
Note: This epaper is graphics intensive and contains a number of Quicktime movies. These can be seen with most web browsers equipped with the Quicktime plugin. Movies may be opened by clicking the button, or, as the case may be, by clicking on a specific picture. (Linuxbased browsers may be an exception, for which we apologize).
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 orbits
The model discovered by E. Lorenz is described by the following differential equation in 3space [3]:
; ;
When Lorenz plotted the trajectories of this differential equation on his primitive computer, he could see something like the picture on the right (click on the image for a movie):
The amazing fact is that this picture is robust. If we perturb the equation, the phenomenon persists: trajectories tend to approach a set which is now called a strange attractor. We shall not try to discuss here the relevance of this object to fluid dynamics, but it happens that the Lorenz attractor has become one of the most paradigmatic symbols in modern dynamical systems, an icon of “chaos theory” (see for instance [4]).
In the 1980’s, Joan Birman and Bob Williams made the simple but crucial observation that a periodic orbit of a vector field in 3space is a closed embedded curve which therefore defines a knot. They suggested that the study of the topology of the knots appearing in the Lorenz equation could yield some understanding of this important dynamical system [5]. The pictures below show some of the periodic orbits that one finds in the Lorenz equation.
Some of these knots look topologically trivial, but some are nontrivial, like the red orbit which turns out to be a trefoil knot. One of the motivations of Birman and Williams was that these periodic orbits seem indeed to give a good approximation of the attractor. In the picture on the right (click on it for a movie), one sees simultaneously a collection of closed orbits, suggesting that the attractor is some kind of limit of its periodic orbits. In a way, one could think of the attractor as an “infinite link with infinitely many components.
Birman and Williams proved that Lorenz knots are indeed very interesting, at the same time rich enough and very peculiar. Rather than stating technical results concerning Lorenz knots, let us limit ourselves to some “numerical statements”. Recall that a knot is simply a closed curve in space with no double point, and that one says that two knots are the same, or are equivalent, if one can go from the first to the second through a continuous deformation without creating intersection points (see for instance [6], [7], [8]). One crude way of classifying knots is to order them according to their crossing number, which is the minimum number of intersections of a projection of a representative of the given knot. One knows for instance that there are 250 (prime) knots with 10 crossings or fewer. It follows from the work of Birman and Williams that among those 250, only the following 8 knots appear as periodic orbits of the Lorenz attractor.
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]).
Some are rather interesting like the knot 'NonAlternating 12.725' with 12 crossings.


As a sample counterexample, the figure eight knot is not a Lorenz knot.


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2.2 The Lorenz template
How can one prove such facts? By dealing with a combinatoric model of the Lorenz equation which is easier to analyze. First, construct the socalled Lorenz template in a way described by the following pictures.
Consider two bands of paper, and unfold them in space as shown in the clip. This produces a twodimensional object in space which is a branched surface: this is the Lorenz template. The branching locus is the vertical segment that one sees in the middle of the picture. On this template, there is a rather natural dynamical system, which is shown on the following pictures.
If we identify the vertical segment with the interval [0,1], then each orbit traverses the template in such a way that the point with coordinate x comes back to the interval at the point 2x mod Z. Therefore if one looks for periodic orbits, we have to look for periodic points of the “doubling map” from the interval to itself. For instance, the doubling map sends the point 1/3 to 2/3, and then back to 1/3 = 4/3  1. So, the point 1/3 is periodic of period 2, and yields a closed orbit on the template, going once in one ear of the template, and once in the other one. The following pictures show several of these periodic orbits on the template.
As a matter of fact, it is not difficult to describe these periodic orbits by an itinerary, which consists of a finite sequence of symbols, say “left” and “right”. For any such sequence, there is exactly one periodic orbit which follows this itinerary, going successively in the left and right wings of the template, in the order given by the itinerary. Conversely, the periodic orbit defines the itinerary, up to a cyclic permutation which amounts to choosing a starting point. In this way, one can imagine a study of the topology of the Lorenz knot associated to a given itinerary. This is what Birman and Williams did. The fact that this “geometric Lorenz attractor” does describe the original Lorenz equation has been established much more recently by Tucker [9],[10].
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3. The modular flow
3.1 The space of lattices and its topology
Given two linearly independent vectors ω_{1}, ω_{2} in the plane, one can consider the subgroup L of R^{2} that they generate:
L= {n_{1}ω_{1} + n_{2}ω_{2} n_{1} , n_{2} Z}.
A subset of the plane of this form is called a lattice. Of course, the same lattice L can be generated in many ways by two vectors. If a, b, c, d are integers such that ad−bc= ±1, the vectors aω_{1}+ bω_{2 }and cω_{1}+ dω_{2} generate the same lattice.
The set of lattices is a topological space: one says that two lattices are close if they can be generated by pairs of vectors in the plane which are close. This space has been studied for many years by mathematicians, starting with the great Gauss in his Disquisitiones Arithmeticae (1801), in which he discussed in depth the arithmetical theory of integral quadratic forms.
Think of the plane R^{2} as the complex plane C. For each lattice L, one can define two complex numbers
; .
It is easy to check that these series converge since the absolute value of the exponents, 4 and 6, are bigger than 1. 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 ( g_{2}(L) , g_{3}(L) ) characterizes the lattice. More precisely, a pair of complex numbers (g_{2}, g_{3}) corresponds to a lattice if and only if the socalled discriminant Δ = g_{2}^{3}− 27g_{3}^{2} is not zero. See for instance [11], [16] for a proof.
Summing up, the space of lattices can be identified with the complement in C^{2} of the curve Δ = 0.
The picture on the left represents symbolically C^{2}. Keep in mind that C^{2 }is R^{4}, so that this is a fourdimensional picture! The horizontal blue axis corresponds to those lattices for which g_{3}=0; this is a copy of C=R^{2}. The vertical green axis corresponds to those lattices for which g_{2}=0; this is another copy of C=R^{2}. The yellow curve represents Δ=0, but again this is a onedimensional curve over the complex numbers, and therefore a surface from the point of view of real numbers.
How can we look at this fourdimensional object in a concrete way? If L is a lattice and k a nonzero real number, one can look at the lattice kL, which is just the same as L, but rescaled by a factor k. This suggests restricting our attention to those lattices whose fundamental domain has area 1, or still in other words, those generated by two vectors ω_{1}, ω_{2} defining a parallelogram of area 1. Note that
g_{2}(kL) = k^{−}^{4}g_{2} (L) ; g_{3}(kL) = k^{− }^{6}g_{3}(L),
so that for each lattice L one can find a unique k > 0 such that g_{2}(kL)^{2}+ g_{3}(kL)^{2}= 1.
This means that if we want to restrict our study to lattices up to rescaling, or to lattices of area 1, we have to look at the complement in the unit sphere g_{2}^{2}+ g_{3}^{2}= 1 of the zero set of Δ. This unit sphere is 3dimensional, and the zero set intersects it in a onedimensional object, which turns out to be a trefoil knot. The 3sphere is easier to visualize than C^{2}. If one chooses a point in a sphere, one can project from that point to the tangent space at the opposite point: this is the socalled stereographic projection, illustrated in the picture on the left for the twodimensional case (the Riemann sphere). (click on the picture for a movie)
Hence the space of lattices of area 1 is identified with the complement of a trefoil knot in the 3sphere, which, after deleting one point, is the complement of a trefoil knot in the usual 3space. So, there is some hope of seeing something! Let’s have a look.
This picture (click it for a movie) shows the 3sphere, or better, its stereographic projection in 3space, or, even better, the projection to your 2dimensional screen of this stereographic projection. As the film evolves, the point from which the projection is done is moving continuously so that you can admire the whole scene! In blue (resp. green), the coordinate axis g_{2}=0 (resp. g_{3} =0) which intersects the sphere in a circle, which is seen from time to time as a straight line when the source of the projection is on this circle. As for the yellow trefoil knot, it is the intersection of the zero set of Δ with the sphere.
To get more topological insight, let’s have a look at some additional structures in the space of lattices of area 1.
Given a lattice L and angle θ ∈ [0,Π], one can rotate the lattice by θ in the plane. One gets another lattice L_{θ}. As θ goes from 0 to Π, this describes a circle in the space of lattices. Note that if θ ≠ 0, the two lattices L and L_{θ} are different in general, with only two exceptions. If L is a square lattice, it is equal to its image by rotations by Π/2 and if L is a regular hexagonal lattice, it is equal to its image by rotations by Π/3 and 2Π/3. All this implies that the space of lattices of area 1 is filled by circles corresponding to the orbits of these rotations. These circles are only topological circles; in fact they are themselves trefoil knots with the exceptions of the two special cases that we mentioned which are shorter (they close after Π/2 and Π/3 instead of Π) and which turn out to be round circles. As a matter of fact, these circles are just the blue and green circles that we previously met. This decomposition of the 3sphere into a collection of trefoil knots and two circles is an example of a Seifert fibration. See the picture on the right. (click on it for a movie)
There is an additional structure in the 3sphere, so to speak dual to the Seifert fibration. We know that to each lattice is associated a nonzero complex number Δ. If one multiplies a lattice by a real number k > 0, this Δ is multiplied by k^{−12} so that the argument is preserved. This suggests looking at the decomposition of the space of lattices of area 1 according to the value of the argument of the complex number Δ. The set of lattices of area 1 for which Δ has a given argument defines a surface in the 3sphere whose boundary is the trefoil. The picture on the left (click on it for a movie) shows one of these Seifert surfaces.
Of course, we get one of these surfaces for each value of the argument of Δ. If this argument describes a full turn, the corresponding Seifert surfaces will sweep around all 3space, always keeping the same boundary. For some value of the argument, the Seifert surface passes through the pole of the stereographic projection, so that the surface seems to be infinite at that instant. (Click on the picture)
To understand this fibration of the complement of the trefoil knot, let’s delete a part of the Seifert surfaces, and let’s look at the motion in the neighbourhood of the trefoil knot. This clip shows parts of four Seifert surfaces, corresponding to four values of the argument, rotating around the trefoil. Topologists say that the 3sphere is an open book, that the trefoil is the binding, and that the Seifert surfaces are the pages. (Click on the image for a movie)
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 image for a movie)
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3.2 The Modular dynamics
Now 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 t, consider the matrix
.
If L is a lattice of area 1, its image by the matrix is again a lattice of area 1. This defines a dynamical system in the space of lattices of area 1 which is called the modular flow. The trajectory of the lattice L is the curve described in the space of lattices by as the time t describes the real numbers.
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 Anosov flow which means that there are stable and unstable directions. Let us explain this. Introduce two other flows by
; .
Note that and ,so that if one takes two lattices L and (resp. L and ) and if one looks at their future (resp. their past) in the modular flow, i.e. if one looks at their images by with t positive (resp. negative), then the two lattices tend to approach exponentially fast. One says that L and (resp. ) are in the same stable (resp. unstable) curve.
The pictures below explain this situation. At first, one sees an initial point L, and two arcs in the stable and unstable manifolds, i.e. two intervals and for s in some interval [−1 , 1]. Then, when the modular dynamics moves forward, one sees the orbits starting from all these points, which generate two surfaces. The green one corresponds to the stable manifold, and one can see quickly that this stable manifold is so close to the central orbit that one cannot distinguish them any more. The red one is the unstable one and on the contrary expands in space and develops huge surfaces that look like parachutes. If one continued the dynamics further in the future, this red surface would become bigger and bigger, and would tend to be denser and denser in space, so that one could no longer grasp the global situation. Finally, the film stops and the modular dynamics flows backwards into the past, and one sees a similar phenomenon but the green and the red are interchanged. This structure of stable and unstable manifolds creating intricate nets in the phase space has become a central tool of study in dynamical systems: the socalled hyperbolic theory.
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3.3 Periodic orbits and their linking with the trefoil
We 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 A with integral coefficients and determinant 1:
_{.}
Clearly, the matrix A preserves the standard square lattice Z^{2} in R^{2}. Suppose that A is hyperbolic, which means that a + d > 2, in which case A is diagonalizable over the real numbers. In this case, there is a 2×2 matrix P such that
_{}
for some t (note that the product of the two eigenvalues of A is 1). So, if we define the lattice L to be the image of Z^{2} by P, one finds that L is fixed by . In this way, for each integral matrix with determinant 1, we find a fixed point for for some t , i.e. a periodic orbit of the modular flow. Note that the period of this periodic orbit is t , which is the logarithm of the absolute value of an eigenvalue of A.
Hence every hyperbolic matrix A defines a periodic orbit of the modular flow.
This 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 A by ± BAB^{−1} where B is some other integral matrix with determinant 1, one gets the same periodic orbit.
There is a natural bijection between periodic orbits of the modular flow and conjugacy classes of hyperbolic integral matrices of determinant 1, up to sign.
These periodic orbits have a very old mathematical tradition. One finds them in many different areas, in different disguises: closed geodesics on the modular surface, equivalence classes of integral indefinite quadratic forms, ideal classes in quadratic number fields, continued fractions etc.
Each one of these periodic orbits is a closed curve in the space of lattices of area 1, hence defines a knot in the complement of the trefoil knot. We wish to investigate these modular knots from the topological point of view. Let us first look at some examples.
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:
Given a hyperbolic integral matrix A with determinant 1, denote by k_{A }the associated knot. Can one compute the linking number between k_{A} and the trefoil knot?
One should quickly recall the notion of linking number of two disjoint oriented knots in 3space (also introduced by Gauss in the very different context of electromagnetism). Suppose that two oriented knots k_{1} (blue) and k_{2} (orange) do not intersect, and project them onto a plane in a generic way. These projections need not be disjoint of course.
Let us consider the points where the projection of k_{1}, the blue knot, crosses once over the projection of k_{2}, the orange knot. These crossing points may have two local behaviors, as shown in the pictures below.
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_{1} over k_{2}. The result is called the linking number between k_{1} and k_{2}. 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 crossovers with different signs, so that the linking number is 0.
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 k_{A} and the trefoil knot. Recall that the trefoil knot is the boundary of a Seifert surface where the argument of Δ is equal to some fixed value, for instance 0 (which corresponds to Δ being a positive real number). Given a closed oriented curve in the complement of the trefoil knot, one can consider its image by the Δ map, and one gets a closed curve in the complex plane which avoids the origin. Clearly, the linking number between and the trefoil is the 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 twosided, and that one wishes to compute the algebraic intersection of k_{A} with the Seifert surface.
This is described by the pictures below. For a given matrix A, one draws at the same time a Seifert surface and the knot k_{A}. As the construction of k_{A }evolves, 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.
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
U= ; V= .
As it turns out, any integral matrix A with determinant 1 is conjugate, up to sign, to a product of U’s and V’s. For instance:
A= = UVVVUUVVVUUUUV
The word in U and V is uniquely defined up to a cyclic permutation. For each matrix A, one can therefore define R(A) as the number of factors equal to U, minus the number of factors equal to V. This invariant of the matrix A will be called the Rademacher invariant of A. One of the results in [1] is the following:
Theorem: The linking number between k_{A } and the trefoil knot is equal to the Rademacher invariant of A.
We can only refer to [1] for several proofs and more information, but in the next section we shall provide some interpretation.
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4 Lorenz and modular knots
4.1 The modular template
So far we have discussed the periodic orbits in the Lorenz attractor, the Lorenz knots, and the periodic orbits of the modular flow, the modular knots.
Theorem: Lorenz knots and modular knots coincide.
More precisely, for every modular knot k_{A}, one can deform it in 3space to make it coincide with one of the periodic orbits of the Lorenz attractor, and conversely. Unfortunately, we cannot provide a proof here, and we will only show these deformations using some images and movies!
The first step of the proof consists in the construction of a modular template inside the space of lattices of area 1, which looks like the Lorenz template. We begin by constructing a onedimensional object consisting of two parts. The space of hexagonal lattices of area 1 is a circle, that we have already met as one of the axes in our description of the space of lattices. The second part consists of all lattices with a fundamental domain having the shape of a horizontal rhombus with inner angles between Π/3 and 2Π/3.
Stated differently, one looks at lattices generated by two vectors which are symmetric with respect to the xaxis and which make an angle between Π/3 and 2Π/3 (and which are of determinant 1). This defines an interval since the angle can vary in [Π/3, 2Π/3]. Note that when this angle is equal to Π/3 or 2Π/3, the corresponding lattice is hexagonal so that this interval connects two points of the circle of hexagonal lattices. As a matter of fact, this interval is nothing more than a diameter. (Click the image at right for a movie).
The picture on the right shows this part of the construction. It starts with two rotations (one in 4space and one in 3space ) to place the trefoil knot in a convenient position, and then shows the construction of the circle and its diameter, that we have just described. (Click on the image)
We then extend this onedimensional object in the unstable direction to create a twodimensional 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 onedimensional object, by using an old idea (again of Gauss !): Given a lattice L, one seeks the shortest nonzero vector v in L, which is usually unique up to sign, unless the lattice has a fundamental domain with the shape of a rhombus. Then one looks for the shortest vector w in L which is not proportional to v. These two vectors v ,w define a preferred basis of L. Now, one shortens w progressively while increasing the length of v, keeping the determinant constant equal to 1. Rather than going into details, let us look at some images showing the deformation of several modular knots until they are in a “Lorenz position”.
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4.2 Two final remarks
A final comment. We know that a Lorenz periodic orbit is coded by a finite list of symbols “left” and “right” expressing the itinerary of the orbit, visiting successively the left and right ears of the Lorenz template. One can check that if one associates to an orbit the product A of the matrices U and V in the same order as the “left” and “right”, then the above deformation identifies the Lorenz knot with the modular knot k_{A}. Now, one sees that the right ear has linking number +1 with the trefoil and the left ear has linking number −1.
It follows that the linking number of k_{A} with the trefoil is equal to the 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.
A final treat. The modular flow is not the only remarkable dynamical system on the space of lattices of area 1. One can also look at the dynamics generated by the matrices that we have already met as stable manifolds. This flow is classically called the horocyclic flow. One would need many clips to picture the wonderful dynamics of this flow, but we shall restrict ourselves to just one, describing the periodic orbits. Fix real numbers t and s, and consider the lattice L_{s,t }generated by the two complex numbers exp(t) and s.exp(t) + i exp(−t). Fixing t and letting s run from 0 to 1, we get a periodic curve c_{t }in the space of lattices of area 1, which is a periodic orbit of . The following clip shows this curve c_{t} when t describes some interval. When t is a large negative number, the curve c_{ t} is a small trivial loop going once around the trefoil knot. When t is a big positive number the curve c_{t} gets longer and in the limit fills the whole space.
It is known that the the family of curves c_{t} tends to fill the space in a “uniform” way but the quantitative estimate of the velocity of this phenomenon is equivalent to the famous Riemann hypothesis, one of the most enticing open questions in mathematics! [15]
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References
[1] GHYS, E.: Knots and Dynamics, to appear in the proceedings of the International Congress of Mathematicians, Madrid 2006.
[2] LEYS, J.: Mathematical Imagery. http://www.josleys.com
[3] LORENZ, E.: Deterministic Nonperiodic Flow. J. Atmos. Sci.20, (1963), 130141.
[4] VIANA, M.: What’s New on Lorenz Strange Attractors. Math. Intell. 22,(2000), 619.
[5] BIRMAN, J. & WILLIAMS, R.: Knotted periodic orbits in dynamical systems.I. Lorenz’s equations. Topology 22(1983), no.1, 47–82.
[6] ADAMS, C.: The knot book. An elementary introduction to the mathematical theory of knots. Revised reprint of the 1994 original. American Mathematical Society, Providence, RI, 2004. xiv+307 pp.
[7] KAUFFMANN, L.:On knots. Annals of Mathematics Studies 115 Princeton University Press, Princeton, NJ,(1987).
[8] SOSSINSKY, A: Knots. Mathematics with a twist. Translated from the 1999 French original by Giselle Weiss. Harvard University Press, Cambridge, MA, 2002. xxii+127 pp.
[9] STEWART, I : The Lorenz Attractor Exists. Nature 406 (2000), 948949.
[10] TUCKER, W.: A rigorous ODE solver and Smale’s 14th problem. Found.Comput. Math. 2 (2002), no. 1, 53–117.
[11] SERRE, J.P.: A course in arithmetic. Translated from the French. Graduate Texts in Mathematics, No. 7. SpringerVerlag New York, Heidelberg, 1973. viii+115 pp.
[15] SARNAK, P.: Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series. Comm. Pure Appl. Math. 34 (1981), no. 6, 719–739.
[16] McKEAN, H. & MOLL, V.: Elliptic curves. Function theory, geometry, arithmetic.Cambridge University Press, Cambridge, 1997.
NOTE: Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some these materials. Some of the items above can be accessed via the ACM Portal, which also provides bibliographic services.