## Abstract

The determinism principle, which states that dynamical state completely determines future time evolution, is a keystone of nonlinear dynamics and chaos theory. Since it precludes that two state space trajectories intersect, it is a core ingredient of a topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits embedded in a strange attractor. However, knot theory can be applied only to three-dimensional systems. Still, determinism applies in any dimension. We propose an alternative framework in which this principle is enforced by constructing an orientation-preserving dynamics on triangulated surfaces and find that in three dimensions our approach numerically predicts the correct topological entropies for periodic orbits of the horseshoe map.

## 1. Introduction

Central to chaos theory is the idea that an irregular dynamics can be generated by deterministic dynamical systems, in which perfect knowledge of the present state completely determines future time evolution. In particular, it underlies an essential component of the nonlinear toolbox, the phase-space reconstruction technique, which associates a possibly chaotic time series with a trajectory in a candidate state space whose coordinates are, for example, time-delayed sample values or successive derivatives (Ott 1993; Abarbanel 1996). When the dynamics is generated by deterministic chaos, trajectories in phase space are not randomly organized: neighbouring trajectories have almost identical velocity vectors (figure 1*a*). Indeed, the geometric formulation of determinism in phase space is that two trajectories cannot cross each other, because an intersection point would have two distinct futures.

Constraints imposed by determinism upon phase-space trajectories are also a key ingredient of a topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits (UPO) embedded in a strange attractor (Mindlin *et al*. 1990, 1991; Gilmore 1998; Gilmore & Lefranc 2002). In phase space, UPO are associated with closed orbits. Since these closed orbits cannot intersect each other, they form knots and links which are well defined and do not change type when a control parameter is varied (figure 1*b*). The knot invariants of a UPO are thus genuine fingerprints that carry dynamical information.

What makes this method powerful is that the stretching and squeezing mechanisms which build the chaotic attractor induce a systematic organization in the intertwining of periodic orbits. More precisely, all UPO can be projected to a branched manifold, a *template*, without modifying their topological invariants (Birman & Williams 1983). The template structure describes in a concise way the global topological organization of the attractor. These tools have been successfully used to classify chaotic regimes (Gilmore & Lefranc 2002) and to construct symbolic codings (Plumecoq & Lefranc 2000). Closely related tools based on knot theory have also been used to study bifurcation sequences (Boyland 1994; Hall 1994) and characterize short non-stationary time series (Amon & Lefranc 2004).

However, this approach has an important limitation. It can be applied only to three-dimensional attractors because in higher dimensions, two closed loops can always be deformed into each other. Still, the core principles, determinism and continuity, upon which template analysis is built, apply in any dimension. Extension to higher dimensions thus requires reformulating the determinism principle in a dimension-independent way.

## 2. Determinism and orientation preservation

Non-intersection of trajectories of dimension *d*=1 imposes a non-trivial constraint only in space dimension *D*=3 because only then is the co-dimension *D*−2*d*=1. It has been proposed to base the topological analysis of chaos on the fact that determinism requires that orientation of a phase-space volume element is not modified under action of the flow (Lefranc 2006). In simple terms, the surface of a droplet in a fluid does not intersect itself as it is advected: its interior and exterior remain distinct at all times (figure 2*a*). This formulation of determinism is to the requirement of non-intersecting curves what in electromagnetics the integral form of Gauss's law is to the corresponding Maxwell equation. It naturally adapts to phase spaces of any dimension because a hypersurface is always of co-dimension 1. When the attractor can be embedded in a phase space (e.g. in forced systems), so that it can be sliced into *n*-dimensional Poincaré sections parameterized by *φ*, it is simpler to apply the orientation preservation theorem to hypersurfaces of a Poincaré section (figure 2*b*).

## 3. Periodic orbits and triangulations

We want topological analysis to apply to experimental, possibly short or non-stationary, time series (Amon & Lefranc 2004). As in template analysis, the most natural way to achieve this is to use as input the trajectory of a periodic orbit extracted from the time series. Given the *p* intersections of a period-*p* orbit with each Poincaré section, we must thus study the evolution of surfaces attached to these *p* points.

We therefore represent the dynamics in a triangulated space whose nodes are *p* periodic points *P*_{i} in a Poincaré section (with *F*(*P*_{i})=*P*_{i+1}, *F* being the Poincaré return map). In this space, points *P*_{i} are 0-cells, line segments joining two points are 1-cells, triangles are 2-cells, etc. (figure 3*a*). More generally, the analogue in this discrete space of an *m*-dimensional surface is a collection of contiguous simplicial cells on *m*+1 periodic points. As Poincaré sections are swept, periodic points *P*_{i} move in a section plane and so do cells attached to them (figure 3*b*). Our goal is that the dynamics induced on *m*-cells reflects faithfully that of *m*-dimensional phase-space surfaces under the action of the chaotic flow, and that it carries the signature of the stretching and squeezing mechanisms operating in the original phase space. We require that an *m*-dimensional map *F*_{m} induced by the flow mimics the original return map *F* by being invertible, satisfying determinism, and being the result of a continuous deformation of triangulation facets, just as *F* is a continuous deformation of identity. As we see below, this does not imply that facets are trivially advected between sections, because degeneracies sometimes occur, and action must be then taken to preserve orientation.

## 4. Orientation-preserving dynamics on triangulations

In the following, we confine to the three-dimensional case. We shall see that requiring orientation preservation selects a unique formalism that predicts correct entropy values for periodic orbits of the horseshoe map (Lefranc 2006). An extension to higher dimensions is not yet fully understood and is postponed to a future work.

The natural volume element of a two-dimensional triangulation of periodic points in Poincaré sections of a three-dimensional flow is the triangle joining three periodic points *P*_{i}, *P*_{j}, *P*_{k}. If *P*_{i}(*φ*) denotes the position of *P*_{i} in section *φ* (with *P*_{i}(0)=*P*_{i} and ), the natural evolution of in successive Poincaré sections is(4.1)which however is not uniformly valid as a 2-cell. Indeed, one of the three points (say *P*_{k}(*φ*)) can pass between the other two at some *φ*=*φ*_{0} and change the orientation of the candidate 2-cell *T*(*φ*) given by (4.1) (figure 4). As we have stressed above, this violates determinism, and the dynamics must thus be modified.

Determinism seems to be violated when triangle *T*(*φ*_{0}) in figure 4 becomes degenerate because its exterior and interior, as defined by the outer normals, seem to be exchanged as the triangle sides go through each other like a balloon that would be flattened and then inverted. At that point, we must keep in mind that only the node motion is physical, from which the dynamics of higher-dimensional facets is reconstructed.

As illustrated in figure 5, we can restore orientation, and hence determinism, by preventing the opposing sides that are about to collide (represented as a solid and a dashed line, respectively) from doing so. More precisely, we construct the edge dynamics so that the left (solid line) and right (dashed line) sides remain at the left and right, respectively. Since the left (respectively, right) side consists of itinerary (respectively, ) before degeneracy and of itinerary (respectively, ) after degeneracy, their relative positions are preserved through triangle inversion by transforming paths along periodic points as(4.2a)(4.2b)These rules also apply to reverse paths (e.g. ).

For computational purposes, itineraries visiting edges in a given order are mapped to words over alphabet , and (4.2*a*) and (4.2*b*) to a transformation that in a given *w* replaces each occurrence of the letter *e*_{ij} by the string *e*_{ik}*e*_{kj} and each occurrence of *e*_{ik}*e*_{kj} by *e*_{ij} (hence is invertible, as ). For example,In the dynamics generated by , the image of an itinerary along periodic points depends on how these points rotate around each other. To show that this dynamics provides a faithful description of the action of the flow around the orbit, we now show that this formalism correctly predicts the entropies of periodic orbits of the horseshoe map. The entropy of a periodic orbit is an invariant defined as the minimal topological entropy of a flow containing this orbit (Boyland 1994; Bestvina & Handel 1995); a positive-entropy orbit is a powerful indicator of chaos (Mindlin *et al*. 1991; Amon & Lefranc 2004; Thiffeault 2005).

## 5. Orbit entropy and invariant word

Assume that when following motion of periodic points *P*_{i}(*φ*) in section plane as *φ* is swept from 0 to 2*π*, we detect *l* triangle inversions . The induced return map that transforms an itinerary *w* into another *w*′ is(5.1)where . As an example, we consider the periodic orbit 00111 of a suspension of the standard horseshoe map equipped with the usual symbolic coding (Gilmore & Lefranc 2002; figures 3*b* and 7*a*). Figure 6 shows how the five periodic points rotate around each other in section plane over one period and how the itinerary starting from *e*_{15} is transformed over three periods.

Specifically, four triangle inversions occur when point 4 successively crosses the four edges *e*_{15}, *e*_{13}, *e*_{25} and *e*_{23}. The induced return map for edge itineraries is then . For example, *e*_{15} is transformed as follows:Edges that are not crossed by point 4 are trivially modified (e.g. ). This leads to the closed set of transformation rules(5.2)for the four edges in the invariant set of *F*_{1} (the image of each edge in this set can be written as an itinerary over these four edges or their reverses).

Table 1 displays iterates computed using (5.2). Their length diverges exponentially as *m*→∞, which indicates that trajectories in the neighbourhood of the orbit are continuously stretched apart by the flow. The growth rate(5.3)should be equal to the entropy *h*_{T}(*P*) of orbit *P* (i.e. the minimal topological entropy of a map containing *P*; Boyland 1994). It is obtained as the logarithm of the leading eigenvalue of the transition matrix (*M*′_{ee}) whose entries count occurrences of edge *e*′ or of its reverse in *F*_{1}(*e*) given by (5.2). Here, . In general, there are paths in the *F*_{1}-invariant set of a typical orbit that trigger a ‘squeezing’ rule (4.2*b*); for example, *e*_{16}*e*_{67}→*e*_{17} for horseshoe orbit 0010111 and a new set of basis paths must be defined before entropy can be computed from a transition matrix (Morant *et al*. 2006, unpublished data). Table 1 also shows that (*p* is the orbit period) converges to an infinite word *w*_{∞} satisfying , which is the analogue of the infinitely folded unstable manifold of the periodic orbit.

For all 746 periodic orbits of the horseshoe map up to period 12, the growth rate (5.3) has been found numerically to *agree to machine precision with topological entropy obtained by the train-track algorithm* (Boyland 1994; Bestvina & Handel 1995), as illustrated in table 2. This suggests that in three dimensions our approach provides a faithful description of the dynamics.

Given that transformations (5.1) are *invertible*, it is remarkable that the asymptotic dynamics is *singular*. Indeed, consider the itinerary , which is the shortest subpath of *w*_{∞} visiting the four edges in (5.2), and which is displayed in the right bottom corner of figure 6. The image can be decomposed as the concatenation of a subpath of *w*_{0} and of a reverse copy of *w*_{0}. *Thus, this path is folded onto itself by the dynamics* and the action on the orbit of the two-dimensional invertible return map can be described by a singular one-dimensional map of an interval (which is *w*_{0}, unfolded) into itself, as shown in figure 7*b*. This also holds for subsequent iterates , hence for the infinite word *w*_{∞}. This expresses the fact that to each invertible return map (e.g. Hénon map) is associated an underlying lower-dimensional non-invertible map (e.g. logistic map) describing the dynamics along the unstable manifold, a keystone of the Birman–Williams construction (Birman & Williams 1983; Gilmore & Lefranc 2002). Note that the symbolic name 00111 can be recovered from figure 7*b* using the usual coding for orbits of one-dimensional maps, which makes the new formalism promising for constructing global symbolic codings from topological information (Plumecoq & Lefranc 2000). Moreover, the description of how segments along *w*_{0} are folded over each other and of how neighbouring cells are squeezed provide us with a signature of the stretching and folding mechanisms from which the simplest template carrying the periodic orbit studied could be determined.

## 6. Conclusion

To conclude, we have proposed that orientation preservation is a more general formulation of determinism than non-intersection of trajectories. In three dimensions, we find that enforcing it on a triangulation of periodic points induces a non-trivial dynamics on paths along periodic points. A promising result is that despite its simplicity this formalism numerically predicts the correct entropies for periodic orbits of the horseshoe map. Preliminary calculations also suggest that it leads to a combinatorial description of the folding of the invariant unstable manifold over itself, yielding information about the symbolic dynamics of the orbit. It now remains to prove the validity of the approach in three dimensions and to try to extend it to higher dimensions.

## Footnotes

One contribution of 14 to a Theme Issue ‘Experimental chaos II’.

- © 2007 The Royal Society