A possible procedure for extracting a template

Defining the surface of section and computing the first-return map

To provide a non-trivial example for a template extraction, we will use the Lorenz system [2]

This system is equivariant under a rotation symmetry \({\cal R}_z (\pi)\) that complexifies the state portrait. To simplify it, we modd out the symmetry by using the \(z\)-induced canonical form or, equivalently, the \(z\)-induced differential embedding \(\mathbb{R}^3 (z, \dot{z}, \ddot{z})\). For \(R = 91.6\), \(\sigma = 10\) and \(b = \frac{8}{3}\), this system produces the chaotic attractor plotted in Fig. 1 using the \(z\)-induced differential embedding, that is, the space \(\mathbb{R}^3 (z, \dot{z},\ddot{z})\). This attractor can also be produced using the \(z\)-induced canonical Lorenz form [3].

The step is now to define a surface of section in this differential space. We selected the surface of section

which corresponds to the red line in Fig. 1. Such a surface avoids the local torsion in the lower part of the attractor and the squeezing mechanism in its right part. The first-return map is then computed as shown in Fig. 2(left) : note that the axes are reversed so that the cross-sectional area points from left to right and from the center to the periphery of the attractor [4]. This first-return map is made of four monotone branches three monotonic branches separated by three critical points. To simplify the next steps, we normalized the first-return map within a unit square ; in that case, the critical points have for abscissa

leading to the symbolic dynamics defined by the partition

The branches are ordered from the center to the periphery of the attractor. Branches "0’’ and "2’’ ("1’’ and "3’’) are associated with an even (odd) local torsion

With this symbolic dynamics, it is then easy to compute the Markov transition matrix

The Shannon entropy can be computed from the probabilities \(P_i\) (\(i \in [0,N_{\rm s}^{N_{\rm l}}\)) where \(N_{\rm s} = 4\) is the number of symbols (or strips) and \(N_{\rm l} = 6\) is the length of successive symbols (intersections with the surface of section) considered. Let

a sequence of \(N_{\rm l}\) consecutive symbols. The index

of this sequence corresponds to this symbolic sequence viewed as a number in base \(N_{\rm l}\), where the index \(i\) is the representation of that number in base 10. The normalized Shannon entropy is thus equal to

where the denominator is equal to the maximum entropy that can be obtained with \(N_{\rm s}\) and \(N_{\rm l}\) when all the sequences are equiprobable. In the present example, we obtained \(S_{\rm h} = 0.42\).

Determining the local torsion of each strip

Using the critical points provided by the first-return map shown in Fig. 2, the edges of the different strips are plotted in Fig. 3 with the oriented crossings allowing to sign the \(\pi\)-twists according to the convention sketched in Fig. 4. These oriented crossings are numerically extracted. Following the flow, a positive (negative) crossing corresponds to a clockwise rotation of the tangent space to the trajectory. [5] In our case, strip "0’’ presents one negative and one positive \(\pi\)-twists, leading to a null local torsion since these two \(\pi\)-twists can be removed by using a Reidemesiter move : the resulting local torsion is therefore equal to 0. Strip "1’’ has a torsion -1, strip "2’’ a null torsion, and strip "3’’ a negative one. The four elements \(L_{ii}\) are therefore now known.

Extracting periodic orbits

Low periodic orbits are numerically extracted by a closed-return technique in a surface of section. The difficulty with this attractor results from the low probability with which strip 0 and strip 1 are visited, \(P_0 \approx 0.002\) and \(P_1 = 0.145\), respectively. It therefore requires a large set of intersections with the surface of section for a good approximation of them.
The code OrbLorimag197.f used for extracting the unstable periodic orbits from the first-return map was written 30 years ago by Pascal Dutertre [6] It is written in Fortran. The inputs

• The data file contains the two coordinates of the intersection between the chaotic trajectory and the surface of section \({\cal P}_{\rm c}\) : in the present case, this is the file lormap2.dat produced by the code Lorimap91n.cpp.

• The interval of the first-return map : Dx\(=1\) for a normalized first-return map.
• The number of points in the surface of section : Npt\(=18900\).
• The number of critical points : Npc\(=3\).
• Abscissa of the 1st critical point : Xc(1)\(= 0.0533\)
• Abscissa of the 2nd critical point : Xc(2\(= 0.1442\)
• Abscissa of the 3rd critical point : Xc(3)\(=0.3041\)
• Maximum error for the close return : ermmax\(=0.10\)
• Largest period of the orbit : Pmax\(=6\)
• Reversing the first-return map : Invert = .false.

With these inputs, the code will extract the unstable periodic orbits from 18900 intersections in the surface of section with a maximum relative error of 1% and up to period 6. The extracted orbits are in the file population.res. The first ones are reported below

For instance, period-2 orbit (32) starts at the 8584th intersection and end at the \(8584+2\)th one. To plot it, it is necessary to use exactly the same code that produced the first-return map : a slight change would induce fatal error due to the sensitivity to initial conditions. A copy of the code Lorimap91n.cpp is therefore saved as Loriupo91.cpp and the latter is cleaned from anything but counting the intersections with the surface of section. The trajectory is then plotted for \(8584 \leqslant i \leqslant 8584 + 2\) as shown in Fig. 5.

Computing Linking numbers

The last step is to compute linking numbers to detemine the six elements \(L_{ij}\) of the linking matrix that are still unknow. Indeed, the linking matrix is symmetric, that is, \(L_{ij} = L_{ji}\). We worked with the four knots plotted in Figs. 6 made of two orbits, o\(_1\) and o\(_2\). This choice allows to have each element required twice. The selected orbits are (30), (31), (32), (312) and (313). For each, we computed the linking number

where \(\eta_m ({\rm o}_1, {\rm o}_2) = \pm 1\) are the oriented crossing between o\(_1\) and o\(_2\) (self-crossings are ignored) defined according to Fig. 4. Linking numbers are integers that are topologically invariant under an isotopy --- that is, under Reidemeister moves --- until orbit o\(_1\) and o\(_2\) are involved in a bifurcation.

Predicting linking numbers from the linking matrix

Seing the locations of the oriented crossings allows to use Human Intelligence that has been trained for 30 years in extracting template to select the right elements \(L_{ij}\). Let us start with the two orbits (\(312\)) and (\(313\)) extracted from the attractor shown in Fig. 1. The linking numbers counted on their \(z\)-\(\dot{z}\) plane projection and using \(\ddot{z}\) to sign the oriented crossings [7] (Fig. 6(d)) is

meaning orbit (\(312\)) encircles twice in the positive direction orbit (\(313\)), and vice versa. Using the procedure to predict the linking number from the linking matrix [8] this linking number is

where \(L_{ij}\) are the elements of the linking matrix \(L_{{\cal A}_4}^{\rm I}\) and \(N_{\rm join}\) is the number of positive crossings counted in the joining graph.

The joining graph is constructed as follows. First, the six circular permutations of the orbital sequences (\(312\)) and (\(313\)) are ordered according to the kneading theory [9] as

thus providing the order with which periodic points are located in the surface of section, from the centre to the periphery of the attractor. This order constitutes the bottom line of the joining graph. The upper line is constructed according to the linking matrix \(L_{{\cal A}_4}^{\rm I}\). Points from strips S\(_1\) and S\(_3\) are permuted in blocks since \(L_{21} = -1\) is odd ; moreover, periodic points in strips S\(_1\) and S\(_3\) are reversed in strip block since \(L_{11} = -1\) and \(L_{33} = +1\) are odd. We thus obtained for the upper line of the joining graph

Periodic points from the upper line are linked to those from the bottom line by using a Bernoulli shift \({\cal B}\). For instance, point 331 is linked to \((331) \stackrel{\cal B}{\longrightarrow} 313\). The resulting graph is drawn in Fig. 7 : by definition, all the crossings are positive in joining graph.[5] We have thus \(N_{\rm join} (312,313) = + 4\). The linking number is therefore

as counted in Fig. 6(d).

This is repeated up to correct prediction of the four linking numbers corresponding to Figs. 6 by the linking matrix that is

The corresponding template is drawn in Fig. 8.

A question ? write to Christophe Letellier