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# New Knots in the Lorenz Equations

Communicated by *Notices* Associate Editor Daniela De Silva

In this survey, we aim to provide an overview of some of the current research and open questions concerning the relationship between flows in three-dimensional manifolds and knot theory. We will focus on a significant example: knots that arise in the Lorenz equations. Our goal is to convey the essence of the subject through intuitive explanations, while supplying references for the rigorous proofs.

## 1. Periodic Orbits as Knots

Let be a compact three-dimensional manifold. A *flow* on is an action of on (where is considered as a time coordinate) taking each point at time to a point so that and We will assume the flowlines are continuously differentiable and thus a flow defines a vector field, and vice versa, from the existence and uniqueness of an ordinary differential equation (ODE). Since a solution to a first-order ODE can always be continued for all times on a compact manifold, any continuous vector field gives rise to a flow. .

Given a flow, the first natural way to get knots is by considering periodic orbits. Suppose a point is periodic, i.e., there is a finite time so that Then . is a embedding of into By considering this embedding up to isotopies we define a knot, and we can ask the question what knot types do we get for a given flow . .

In dynamical systems we often don’t have complete information of the system. Thus, for any information we deduce about the flow, like the various knot types it contains as orbits, we ask how sensitive this information is under perturbations of the flow, e.g., what periodic orbits can be destroyed by continuous deformations of the flow? Of course a priori we do not know that any single periodic orbit survives under perturbation. Indeed, one can destroy a given orbit in a number of ways (e.g., “plugs”), or slow down the time along the orbits (i.e., multiply the vector field by a smaller and smaller positive scalar) until all points are fixed points for the flow, and there exists no periodic knot

In contrast to this phenomenon, in two dimensions there are results of isotopy stability for periodic orbits, i.e., for any diffeomorphism of a surface there is a nonempty class of periodic orbits so that these orbits persist under deformations: for any diffeomorphism isotopic to and any , will have a periodic orbit isotopic to This class of essential orbits is obtained by “tightening” the diffeomorphism, and reaching typically a diffeomorphism with expanding and contracting directions, (or atypically a reducible diffeomorphism or one with finite order) .

Our main goal is to explain how these two-dimensional results can be applicable to a three-dimensional flow.

## 2. The Lorenz Equations

Edward Lorenz, a meteorologist studying the long-term unpredictability of weather systems, presented in 1963 the following system of ordinary differential equations defined on

Lorenz obtained this system as a truncation of the series expansion of the equations for circulation of a fluid heated from below. Lorenz himself considered the equation with the so-called *classical parameters* , , but we shall see many interesting phenomena arise when considering , , and , as positive real parameters that can vary.

For the classical parameters, Lorenz showed numerically that the equations are *unstable* in the sense that they exhibit:

- (a)
Sensitivity to initial conditions, i.e., a pair of orbits starting at two close-by initial points have distance growing exponentially in a finite time interval.

- (b)
Bounded motion, i.e., orbits do not escape to infinity but starting at some finite time are contained in a compact domain in (although most orbits are nonperiodic).

Thus, the Lorenz equations demonstrate the existence of deterministic, low-dimensional chaos. Today we know the complexity it bears is prevalent in three-dimensional systems. In this sense, the Lorenz equations are interesting not only for their own sake, but as a generic manifestation of chaos and an acid test for any new techniques aiming to analyze chaotic systems

The equations satisfy at the origin, which is, therefore, a fixed/singular point for the flow defined by the equations. Lorenz observed that for the classical parameters almost every orbit is eventually confined to an invariant set with a butterfly shape (see Figure 1). This shape is called the *Lorenz attractor*, as it attracts all nearby orbits.

By the existence and uniqueness theorem, other orbits cannot reach the origin in finite time. However, by linearizing the flow, one can conclude from Poincaré’s stable manifold theorem that for there is a two-dimensional invariant manifold consisting of all points that limit onto the origin as This set is called the stable manifold of the origin. There also exists a one-dimensional manifold of orbits that approach the origin as . called the unstable manifold of the origin. For , there are two other singular points for the equations, one in the center of each wing of the butterfly attractor. These points have complex eigenvalues, reflecting the fact that orbits spiral around them.

Coding each orbit by a sequence of letters and according to the order in which the orbit passes through the left and right wings of the butterfly yields a natural way to assign to each infinite orbit an infinite sequence in and such that a periodic orbit corresponds to a periodic sequence. In practice this can be formally defined (and computed) as recording whether the orbit passes through the first or third , quadrant whenever it changes from positive to negative See .

There are also orbits that travel around the attractor for a while and then limit onto the singularity at the origin. These are part of the stable manifold of the origin and correspond to finite sequences of and s Thus the stable manifold of the origin is forced to wind around the attractor, separating any two different symbols, and yet allowing orbits to cross from one wing to the other without intersecting it s.

## 3. The Geometric Model

The Lorenz equations have proven to be extremely difficult to study, and many questions regarding their behavior remain unanswered. Around the end of the 1970s, two independent groups considered a simplified flow encapsulating the chaotic behavior of the Lorenz equations, in order to better understand the formation of chaos in three dimensions. Afraimovich, Bykov, and Shilnikov *cross section*) that all orbits intersect, bar some that limit onto the singularity. Thus, there is a map from the rectangle to itself, which takes a point to the next point where its flowline intersects with This map is defined for all points in the rectangle . except for those on a one-dimensional line within the intersection of the stable manifold of , and This line consists of points flowing only downwards from . until reaching (note that the stable manifold of contains other points that hit a number of times before hitting it along this one-dimensional line and then continuing to One defines the flow so that the image of a vertical segment in the rectangle such as the dashed segment in Figure ).2 is a shorter vertical segment somewhere in .

The geometric model has been a fruitful area of study in its own right. Its analysis is ongoing and has motivated many new techniques. It is “singular hyperbolic,” i.e., has expanding directions and contracting directions except at the singularity, and can be precisely proven to be chaotic (see for example *admissible* sequences, i.e., those that correspond to some orbit in the attractor, and nonadmissible ones. For instance, considering Figure 2, the sequence is nonadmissible, as an orbit starting in the right wing will slowly shift to the left until it switches to the left wing at some point. The exact set of admissible sequences depends on how the triangles are glued, particularly on the points and where the tips of the triangular “butterfly wings” are attached when constructing Each of these points thus corresponds to the first intersection point of one side of the one-dimensional unstable manifold of the origin. For any specific choice one gets a specific geometric model and a set of admissible sequences. Moreover, the correspondence between orbits that do not end at the singularity and admissible sequences is a bijection. .

In 1978, Williams introduced a topological representation of the Lorenz attractor called the Lorenz template. The template is the branched surface with a semi-flow depicted in Figure 3. It is only a semi-flow as at the branchline, two orbits coincide into one and, thus, once an orbit crosses through the branchline, it does not know what wing it came from. Starting from a maximal geometric model, i.e., when and are on the boundary of the rectangle, the template is obtained by collapsing each interval of the stable direction (parallel to the dashed line in the figure) to a single point. Different periodic orbits (or different points of the same orbit) cannot pass through the same stable segment. Thus, there is a one-to-one correspondence between periodic orbits that can appear in the geometric model and the periodic orbits on the template. Furthermore, a periodic orbit in the geometric model can be brought to the template by an isotopy, sliding it along the stable direction, and thus it has the same knot type as the corresponding orbit in the template. Williams explored the topological properties of periodic orbits in the template that were considered as knots, i.e., as embeddings up to isotopy. The template with its embedding into facilitates the study of all possible knots for any choice of geometric model. One can see directly from the template all Lorenz knots can be drawn so that the crossings all have the same sign (strands coming from the left pass beneath strands coming from the right wing) and for many knots such a drawing does not exist. Birman and Williams proved many other knot properties, e.g, Lorenz knots are all prime and fibered. Pierre Dehornoy found that out of the first 1.7 million knots (ordered by crossing number), precisely 20 are Lorenz knots, and characterized their Alexander polynomial. For some of their surprising geometric properties see

However, whether these results are at all relevant to the original system of equations remained an open question for a long while. In 1999, Steve Smale compiled a list of 20 problems for the next century

The 14th problem was the first to be solved, by Warwick Tucker

## 4. The Relation to the Modular Surface

In 2006, Étienne Ghys gave an International Congress of Mathematicians (ICM) talk in which he studied the set of knots arising from a different three-dimensional system; the geodesic flow on the modular surface. The modular surface is the quotient of the hyperbolic plane by the action of The geodesic flow is defined on the unit tangent bundle to it, i.e., the space . of a point on the surface together with a direction.Footnote^{1} The geodesic flow is naturally defined on the complement of a trefoil knot in a three-dimensional sphere.Footnote^{2} In a surprising twist of events, he discovered that all the periodic orbits of the geodesic flow are carried by Williams’s Lorenz template in Figure 3. Therefore, the set of closed modular geodesics is exactly the set of Lorenz knots. In particular, using Tucker’s result, the set of closed orbits for the Lorenz equations at the classical parameters is a subset of the closed modular geodesics.

^{1}

The geodesic flow is defined on points on a surface, and a tangent direction, by taking the unique geodesic through the point tangent to the given direction, and moving along it a distance to a new point and new tangent direction.

^{2}

The modular surface is not compact but is a finite volume surface with a puncture/cusp. The cusp corresponds to the trefoil in .

This is remarkable, as the two systems are very different; for example, the Lorenz flow is dissipative while the geodesic flow is volume preserving. Another remarkable fact about the relationship between these systems is that the geodesic flow is a hyperbolic flow, possessing contracting and expanding directions, opening the door to relating the Lorenz flow to hyperbolic systems such as the geometric Lorenz model.

## 5. The Lorenz Equations with Different Parameters

Everything discussed above is true for a specific point in parameter space, the classical parameters that Lorenz considered. However, dynamical systems research usually focuses on how the behavior of a family of systems changes when one varies the parameters.

Several authors addressed the question of varying the parameters in the Lorenz system. Colin Sparrow

Another question concerning varying parameters is whether there exist points in the parameter space where the system has either a *homoclinic orbit*, which is an orbit limited onto a singular point both for and or a ,*heteroclinic orbit*, an orbit limited onto one singular point in backward-moving time and onto another when These are always nongeneric points. But they are still interesting as they typically influence the behavior of the equations for nearby parameters as well. .

The field of numerically studying the parameter space has been subject to continuous improvement over the last 30 years. One of the invariants one can try to define for the Lorenz flow is called the *kneading sequence*, i.e., the symbol corresponding to the attractor boundary. In

These regions are separated by curves in the parameter space along which there exist homoclinic orbits. The homoclinic orbit connecting to the origin as limits to or is a loop embedded in existing all along the curve separating two adjacent regions. The study of these curves of existence of homoclinics was initiated in Shilnikov in the eighties and expanded lately in ,

In addition to the curves of homoclinics separating the regions in the parameter plane, there are many (likely infinitely many) curves of existence of homoclinics within each region. Two of these are depicted in Figure 5 computed by Creaser, Kerkhoff, Osinga, in

The curves , and , in Figure 5 indicate the existence of homoclinic orbits of a certain topological type shown in Figure 6. They are central to the system’s global topology: The two-dimensional stable manifold of the origin, also called the *Lorenz manifold*, is extremely intricate, winding through the attractor while preserving the symmetry of the system. Moreover, as other orbits cannot intersect it, it determines much of the motion of the orbits in the three-dimensional space. For beautiful renderings of this manifold see

## 6. Heteroclinic Knots

The vortices of the spirals in Figures 4 and 5 are called T-points. At each of these points in parameter space it is found numerically that there exists a heteroclinic orbit of the Lorenz flow, i.e., an orbit limiting onto the singularity at the origin for and to one of the wing centers for As the Lorenz flow possesses a symmetry . there is always a second orbit at each such parameter connecting the origin to the other wing center. ,

Since orbits are repelled from infinity for the Lorenz equations, it makes sense to add infinity as a fourth singular point for the equations. It is shown in

Like homoclinic points, the heteroclinic orbits may exhibit different topologies at different parameters, as shown in Figure 7 for the two first heteroclinic parameters. It appears to be the case that infinitely many different knot types, i.e., all twist knots, appear as heteroclinic knots in different T-parameters

## 7. Two-Dimensional Isotopies for Three-Dimensional Flows

The heteroclinic knots have two important properties. The first is that they pass through all four singular points. As the heteroclinic knot is invariant so is its complement, and so one may remove the knot with the singularities and obtain a nonsingular flow on a knot complement. The boundary behavior on a small neighborhood of the knot can be determined from the local linearization at the fixed points.

Another property is that the triangle tips in Figure 2 that are part of the unstable manifold of the origin, do not hit the interior of the cross section at a parameter where there’s a heteroclinic knot, as they hit the wing centers which are on the boundary of the cross section. This means, for example for the trefoil knot on the top of figure 7, that the two triangles must each stretch from side to side of the rectangle, and so the freedom in moving the triangle tips within the cross section is lost.

Using this, it is shown in

The gist of the proof is as follows: The existence of a parameter for which there is a heteroclinic trefoil uses a result about existence of homoclinic parameters

Consider the cross section with the return map at a trefoil parameter. We have little information about the return map, save the fact that that its first intersection with the stable manifold of the origin divides it into two halves, and each half of the cross section contains a fixed point at one wing center and points that limit onto the other wing center. Thus, the image of each half of the section passes it from side to side when it returns. This yields a way to define a diffeomorphism on a surface where , contains a subregion corresponding to the cross section and the diffeomorphism contains the return map from to itself as part of its action on .

Now the two-dimensional theory allows us to identify a class of periodic orbits that must exist for any diffeomorphism in the isotopy class of In particular they must exist for . itself even though we only have partial information about its action. The subset of periodic orbits contained in must therefore exist for the Lorenz equations.

Note that the question of whether or not the Lorenz equations are hyperbolic at such a parameter (when removing the heteroclinic connection with the singular points) is completely open. However it follows from the proof that the equations could be brought to a hyperbolic system by a deformation essentially collapsing trivial parts of the dynamics as induced by the two-dimensional isotopy of the return map. Thus, this suggests a new view of Smale’s 14th problem, namely that the equations and the geometric model are not equivalent, but the geometric model is a tightening or of the original equations.

Yara Hatoom has recently proved similar results for the heteroclinic connection at the second T-point in Figure 7. Namely, for such a parameter, there exists a hyperbolic flow defined on the complement of the figure-eight knot, so that by a similar two-dimensional analysis all its periodic orbits also appear as periodic orbits in the Lorenz equations at that parameter. This this set of infinitely many periodic orbits is as far as we know different from the set of Lorenz knots, and does not contain the figure-eight knot itself (which is the union of singularities and heteroclinic connections and not a periodic orbit).

Conjecturally, for each T-point one can find a corresponding hyperbolic flow on a knot complement, and the corresponding templates get more complicated. This implies more and more knot types appear as periodic orbits at subsequent points, and also in the open regions in Figure 4 around each T-point, but this is yet to be proven.

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## Credits

Figures 1–3, Figure 6, Figure 7, and photo of Tali Pinsky are courtesy of Tali Pinsky.

Figure 4 is courtesy of Roberto Bairo and Andrey Shilnikov.

Figure 5 is courtesy of Jennifer L. Creaser, Bernd Krauskopf, and Hinke M. Osinga.