Finding the rotation rate of any map from a quasiperiodic torus to a circle

Transcription

1 Finding the rotation rate of any map from a quasiperiodic torus to a circle arxiv: v1 [math.ds] 8 Jun 2017 Suddhasattwa Das, Yoshitaka Saiki, Evelyn Sander, James A Yorke June 9, 2017 Abstract A trajectory u n := F n (u 0 ), n = 0, 1, 2,... is quasiperiodic if the trajectory lies on and is dense in some d-dimensional torus, and there is a choice of coordinates on the torus T d for which F has the form F (θ) = θ + ρ mod 1 for all θ T d and for some ρ T d. There is an ancient literature on computing three rotation rates ρ for the Moon. (For d > 1 we always interpret mod1 as being applied to each coordinate.) However, even in the case d = 1 there has been no general method for computing ρ given only the trajectory u n, though there is a literature dealing with special cases. Here we present our Embedding Continuation Method for computing some components of ρ from a trajectory. It is based on the Takens Embedding Theorem and the Birkhoff Ergodic Theorem. Rotation rates are often called rotation numbers and both refer to a rate of rotation of a circle. However, the coordinates of ρ depend on the choice of coordinates of T d. We explore the various sets of possible rotation rates that ρ can yield. We illustrate our ideas with examples in dimensions d = 1 and 2. Keywords: Quasiperiodic, Birkhoff Ergodic Theorem, Rotation number, Rotation rate, Takens Embedding Theorem, Circular Planar Restricted 3-Body Problem., CR3BP 1 Introduction The goal of this paper is to show how a rotation rate of a quasiperiodic trajectory can be computed from a discrete-time trajectory of a quasiperiodic orbit which is often a projection of the full system. Rotation rates and quasiperiodicity have been studied for millenea; namely, the Moon s orbit has three periods current address: Courant Institute of Mathematical Sciences, New York University Graduate School of Commerce and Management, Hitotsubashi University JST, PRESTO Department of Mathematical Sciences, George Mason University University of Maryland, College Park Department of Mathematics, University of Maryland, College Park 1

2 whose approximate values were found 2500 years ago by the Babylonians [5]. Although computation of the periods of the Moon is an easy problem today, we use it to give context to the problems we investigate. The Babylonians found that the periods of the Moon - measured relative to the distant stars - are approximately 27.3 days (the sidereal month), 8.85 years for the rotation of the apogee (the local maximum distance from the Earth), and 18.6 years for the rotation of the intersection of the Earth-Sun plane with the Moon-Earth plane. They also measured the variation in the speed of the Moon through the field of stars, and the speed is inversely correlated with the distance of the Moon. They used their results to predict eclipses of the Moon, which occur only when the Sun, Earth and Moon are sufficiently aligned to allow the Moon to pass through the shadow of the Earth. How they obtained their estimates is not known but it was through observations of the trajectory of the Moon through the stars in the sky. In essence they viewed the Moon projected onto the two-dimensional space of distant stars. We too work with quasiperiodic motions which have been projected into one or two dimensions. The Moon has three periods because the Moon s orbit is basically three-dimensionally quasiperiodic, traveling on a three-dimensional torus T 3 that is embedded in six (position+velocity) dimensions. The torus is topologically the product of three circles, and the Moon has an average rotation rate i.e. the reciprocal of the period along each of these circles. While the Moon s orbit has many intricacies, one can capture some of the subtleties by approximating the Sun-Earth-Moon system as point masses using Newtonian gravitation. This leads to the study of the Moon s orbit as a circular restricted three-body problem (CR3BP) in which the Earth travels on a circle about the Sun and the Moon has negligible mass. Using rotating coordinates in which the Earth and Sun are fixed while the Moon moves in threedimensions, the orbit can thus be approximated by the above mentioned three-dimensional torus T 3 in R 6. Such a model ignores long-term tidal forces and the small influence of the other planets. As another motivating example, the direction of Mars from the Earth does not change monotonically. This apparent non-monotonic movement is called retrograde motion. Now imagine that once each year the direction φ is determined. How do we determine the rotation period of Mars compared with an Earth year from that data? This problem has been unsolved in full generality even for d = 1. This paper considers a setting more general than just the Moon or on Mars, although both give good illustrations of our setting. In fact, for typical discrete-time dynamical systems, it is conjectured that the three kinds of recurrent motions that are likely to be seen in a dynamical system are periodic orbits, chaotic orbits, and quasiperiodic orbits [3]. Starting with a d-dimensional quasiperiodic orbit on a torus T d for some d and a map φ : T d S 1, we establish a new method for computing rotation rates from a discrete-time quasiperiodic orbit. By discrete time, we mean that the trajectory observations are a discrete sequence φ n, n = 0, 1, 2,, as for example when a Poincaré return map is used for the planar circular restricted three-body problem (CR3BP), and φ n is φ at the n th time the trajectory crosses some specified 2

3 Poincaré surface. In the rest of this introduction, we give a non-technical summary of our results, ending with a comparison to previous work on this topic. We then proceed with a more technical parts of the paper, in which we describe our methods and results in detail and give numerical examples. Quasiperiodicity defined. Let T d be a d-dimensional torus. For convenience, throughout this paper we consider a point of this torus [0, 1] d mod 1, where each copy of [0, 1] is the fraction of revolution around a circle. Therefore θ T d can be treated as a set of d real numbers in [0, 1). A quasiperiodic orbit is an orbit that is dense on a d-dimensional torus and such that there exists a choice of coordinates θ T d := [0, 1] d mod 1 (where mod1 is applied to each coordinate) for the torus such that the dynamics on the orbit are given by the map F (θ) = θ + ρ mod 1 (1) for some rotation vector ρ T d where the coordinates ρ i of the ρ are irrational and rationally independent, i.e. if a k are rational numbers for k = 1,, d for which a 1 ρ a d ρ d = 0, then a k = 0 for all k = 1,, d. We will say such a rotation vector ρ is irrational. We assume throughout this paper each continuous function such as those denoted by F, φ, γ, and ψ, and each manifold is smooth, by which we mean infinitely differentiable (denoted C ). This assures rapid convergence of our numerical methods. Rotation rates for projections to the circle. Rotation rates are key characteristics of any quasiperiodic trajectory, and knowledge of them enables us to construct a Fourier series describing the torus as a subset of a Euclidean space in which it lies. Suppose we have available a map φ : T d S 1 from the dynamical system to a circle, and we have available the image φ n F (θ n ) = θ n+1 = θ n + ρ mod 1 on a torus. := φ(nρ) of a trajectory Define (θ) = φ(θ + ρ) φ(θ) S 1. We say ˆ : T d R is the lift of : T d S 1 if ˆ is continuous and ˆ (θ) mod 1 = (θ). Define ˆ n = ˆ (θ n ). Then there is a well-defined rotation rate ρ φ, N 1 ρ φ = lim ˆ n=0 n. (2) N N The existence of the limit is guaranteed by the Birkhoff Ergodic Theorem. Different choices of the lift ˆ can change ρ φ by an integer, so ρ φ mod 1 is independent of the choice of lift ˆ. The rotation rate is this ρ φ. This limit exists and is the same for all initial θ 0 and is also independent of small changes in φ. Also there is a vector a with integer entries a = (a 1,, a d ) (that depends on the choice of φ) such that ρ φ mod 1 = a ρ mod 1 (3) 3

4 See Prop If ρ is irrational, then once φ is established, the vector a is unique. For d = 1, Eq. 3 says ρ φ = a 1 ρ where a 1 is an integer. The integer a 1 depends on the choice of φ, so we can get ρ for one choice and ρ for another choice. There are cases where it is easy to compute the rotation rate ρ φ. If the angle always makes small positive increases, we can convert φ n+1 φ n mod 1 into a small real positive number in [0, 1), and we can think of n = φ n+1 φ n as numbers in (0, α), where 0 < α < 1. The limit of the average of n is the rotation rate. We must average real numbers, not angles, and making that transition can be difficult. Determination of a lift ˆ. One essential problem in computing ρ φ is the determination of a lift ˆ for φ. Given a lift, we can compute ρ φ using Eq. 2. While we know the fractional part of ˆ is [0, 1), as we will explain later, we must choose the integer part k n of each ˆ n so that all of the points (θ n, ˆ n ) := (θ n, k n + n ) lie on a connected curve in S 1 R (for d = 1) or a connected surface in in T d R (for d > 1). We must choose these integer parts despite the fact that we do not know which θ n corresponds to n. Even in that case d = 1 there has been no general method for computing the lift in order to find ρ φ, though there is a literature dealing with special cases. See for example [4, 8, 9]. We have established a general method for determining the lift ˆ, as summarized in the Figs Our method is based on the Theorem 1.2, a version of the Embedding Theorems of Whitney and Takens. Rotation rates for planar projections. Assume that we are given a planar projection γ : T d R 2 of F of the original quasiperiodic motion and seek to retrieve a rotation rate from the images γ(θ n ). Thus we need to use γ to establish an associated circle map φ : T d 1 in order to make sense of the term rotation. In particular, fix a reference point P R 2 that is not in the image γ(t d ). We establish a mapping to a circle by measuring angles and rotation by considering polar coordinates in the plane centered at point P. In particular, define φ(θ) [0, 1) mod 1 = S 1 by e 2πiφ(θ) = γ(θ) P γ(θ) P. (4) The winding number around P is W (P ) := 1 0 φ (θ + s) ds, where φ = dφ. Note that W (P ) is an integral over the circle so it does not depend on θ. The value dt of W is piecewise constant and integer-valued. In our examples, it is critical that the projection of our quasiperiodic trajectory into R 2 is such that there exists a point P in R 2 with W (P ) = 1. This is true for generic projections γ but is not always true, as shown in the next paragraph. 4

5 Figure 1: The fish map (left) and flower map (right). The function γ : S 1 R 2 for each panel is respectively Eq. 6 and Eq. 7 and the image plotted is γ(s 1 ). In each panel the curve winds once around the point P 1 exactly once. In both cases, this is the reference point from which angles can be measured to compute a rotation rate. For the fish map, this is the point (8.25, 4.4), and for the flower map this is the point (0.5, 1.5). The angle marked n [0, 1) measured from point P 1 is the angle between trajectory points γ n and γ n+1. The curve winds j times around points P j, j = 0, 1, 2,, 6, so P 1 is a correct choice of reference point. Here is a non-generic example: Let z C, and consider the map given by γ(z) = z 2. For convenience, we use the standard association of R 2 with the complex plane C. The map γ maps the unit circle onto the unit circle and for any value of P C, W (P ) = 0 if the reference point P is outside that circle, and W (P ) = 2 if inside, and W (P ) is not defined if P is on the unit circle. Thus there is no point P such that W (P ) = 1. Two illustrative examples. Figure 1 shows the projections maps γ : S 1 R 2, showing how the winding number differs in different connected components of the figure. On the left panel, every point inside the interior connected region that contains P 1 can act as a reference point for measuring angles and yields the same value of ρ φ. If the map is sufficiently simple, the rotation rate can immediately be computed as the average of these angle differences. However, if the map γ is more complicated, measurement of angle is compounded by overlap of lifts of the angle between two iterates, since they can be represented by multiple values (values differing by an integer). Projections to R. Sometimes we are only provided with a scalar-valued function γ : T d R, and yet we can still construct a two-dimensional map and use the methods described for R 2 projections. For example, Figure 2 shows how we can recover a planar map from only the first component Re γ n of the flower map by considering planar points (Re γ n 1, Re γ n ). This map still gives same rotation rate as 5

6 Figure 2: The flower map revisited. Suppose instead of having the function γ : S 1 R 2 for the flower Eq. 7 in Fig. 1, we had only one coordinate of γ. Suppose we only have available the real component, Re γ. Knowing only one coordinate would seem to be a huge handicap to measuring a rotation rate. But it is not. In the spirit of delay coordinate embeddings, we plot (Re γ n, Re γ n 1 ) and choose a point P 1 as before, and the map is now two dimensional. The rotation rate can be computed as before. The rotation rate ρ φ here using P 1 is the same as for Fig. 1 right. obtained by using the map in Figure 1. A similar example occurs with the Moon. The mean time between lunar apogees is days, slightly longer than the 27.3-day sidereal month. Suppose we measure the distance D n between the centers of the Earth and Moon once each sidereal month, n = 0, 1, 2,. Then the sequence D n has an oscillation period of 8.85 years and can be measured using our approach by plotting D n 1 against D n, and the point (D n 1, D n ) oscillates around a point P = (D av, D av ), where D av is the average of the values D n. Small changes in P have no effect on the rotation rate. Yet another case arises from The Moon s orbit being tilted about 5 degrees from the Earth-Sun plane. The line of intersection where the Moon s orbit crosses the Earth-Sun plane precesses with a period of 18.6 years. The plane of the ecliptic is a path in the distant stars through which the planets travel. Measuring the Moon s angular distance from this plane once each sidereal month gives scalar time series with that period of 18.6 years. This example can be handled like the apogee example above. As a last example, see also our treatment of the circular planar restricted three body problem in Section 4.2 where we compute two rotation rates of the lunar orbit, the first by plotting the rotation rate around a central point and the second by plotting (r, dr/dt), deriving the rotation rate from a single variable r(t), the distance from a central point, where t is time. The Birkhoff Ergodic Theorem. This theorem allows the computation of space-averages fdµ when a time series is the only information available. 6

7 Theorem 1.1 (Birkhoff Ergodic Theorem) For a quasiperiodic map F : T d T d, the Birkhoff average of a function f along the trajectory θ n = F n θ 0 is B N (f)(θ 0 ) : = 1 N 1 N n=0 f(θ n). Let µ be an invariant, ergodic measure, and f be integrable. Then lim N B N (f)(θ 0 ) = fdµ for every θ 0 T d. This theorem is essential for the computation of rotation numbers. Delay Coordinate Embeddings. Let ψ : T d M 1 be C 2 where M 1 is a smooth manifold of dimension D such as a torus or Euclidean space R D. For a positive integer K, define Θ ψ K : Td R KD as ) Θ ψ K (ψ(θ), (θ) := ψ(f (θ)),, ψ(f K 1 (θ)) for θ T d. K 1 is referred to as the number of delays. See Discussion, Section 5. In the theorem below, if K = 1, there are no delays so we have a Whitney-type embedding theorem, or if D = 1, a Takens-like result. In our applications the manifold M 1 below is either S 1 or R 2 and ψ is either φ : T d S 1 or γ : T d R 2. Theorem 1.2 [Special case of Theorem 2.5 in [11]] Let M be a d-dimensional manifold. Let F : M M be C 2 and let M 1 be a D-dimensional manifold. Assume F is quasiperiodic and 2d < KD. Then for almost every smooth (C 2 ) function ψ : M M 1, the delay coordinate map Θ ψ K : M M 1 K is an embedding of M; that is, it is a diffeomorphism of M onto Θ ψ K (M). We use this theorem with M = T d and ψ is either of the above mentioned functions φ or γ (in which cases D = 1 or 2 respectively). While this result gives a lower bound on the number K, it is often convenient to choose K much larger than required. Comparison to previous work. We have written previously about computation of rotation rate in the papers [1, 2, 13]. A complete streamlined method for the case d = 1 is provided in Section 3; the embedding continuation method is announced in [2], but this is the first paper in which it is explained. In addition, this paper is the first time that we have applied our methods to cases where d > 1. While we used the example (CR3BP) in [1], there we used a Poincaré return map whereas here in Section 4.2 no return map is used. Our work has been heavily influences by other recent works in the field, most notably [4, 8, 9]. Rather than try to give a detailed comparison to those works here, we compare in the relevant sections of the paper. Our paper proceeds as follows. In Section 2, we illustrate our methods using two one-dimensional examples (d = 1). We refer to these as the fish map (introduced by Luque and Villanueva [9]) and the flower map, based on the shapes of the graphs. We also give a detailed description of our method and how it applies in these examples. See Theorem 3.1. In Section 4 we give two-dimensional (d = 2) examples. We end in Section 5 with a discussion of the physical significance of the rotation rates in the restricted three-body problem. 7

8 Figure 3: The angle difference for the fish and the flower maps. Here we plot (φ n, n + k) for every n N where n = φ n+1 φ n mod 1, and all integers k shown here when n + k [ 1, 2]. In the left panel (the fish map, the easy case) the closure of the figure resolves into disjoint sets (which are curves R S 1 ), while on the right (the flower map, the hard case) they do not. Hence if we choose a point plotted on the left panel, it lies on a unique connected curve that we can designate as C S 1 R. We can choose any such curve to define ˆ n, namely we define ˆ n = n + k where k is the unique integer for which (φ n, n + k) C. Of course there are different choices of C and these differ by a vertical shift by integer. When we compute the average of ˆ n to approximate the limiting value ρ φ, the different choices of C yield different values of ρ φ, differing by an integer. Hence we can obtain ρ φ but its integer part is not determined. That is, we can uniquely determine ρ φ mod 1 in the left panel. A better method is needed to separate the set in the right panel into disjoint curves and that is our embedding method. Figure 4: A lift of the angle difference for the fish and for the flower maps. This is similar to Fig. 3 except that the horizontal axis is θ instead of φ. That is, we take θ n to be nρ and (θ) = φ(θ + ρ) φ(θ) mod 1 [0, 1) and we plot (θ n, n + k) for all integers k (where again n = (nρ)), These are points on the set G = {(θ, (θ) + k) : θ S 1, k Z}. This set G consists of a countable set of disjoint compact connected sets, connected components, each of which is a vertical translate by an integer of every other component. For each θ S 1 and k Z there is exactly one point y [k, k + 1) for which θ, y) G. Each connected component of G is an acceptable candidate for ˆ. Unlike the plots in Fig. 3, G always splits into disjoint curves. Unfortunately the available data, the sequence (φ n ) only lets us make plots like Fig. 3. But the Takens Embedding method allows us to plot something like G and determine the lift in the next figure. 8

9 Figure 5: Lifts over an embedded torus. For θ Td define ΘφK (θ) := (φ(θ), φ(θ + ρ),, φ(θ + (K 1)ρ)) TK. The oval curve in the figures is the set Θ(Td ) := {ΘφK (θ) : θ Td }. The Takens Embedding theorem implies that if K 2d + 1 = 3, then for almost any map φ, the set Θ(Td ) is an embedding of Td into TK ; i.e., ΘφK is a homeomorphism of Td (the circle S 1 when d = 1) onto Θ(Td ). In particular the map is one-to-one. Write U := {(ΘφK (θ), (θ) + k) : θ Td and k Z}. This plot has the vertical axis equal to the angle difference (θ) [0, 1) plus each integer k as in our previous graphs. Given the trajectory φn = φ(nρ), define ΘφK,n = ΘφK (nρ) = (φn, φn+1,, φn+k 1 ) TK. Plot znk := (ΘφK,n, n + k) for all integers k Z and n 0, where we treat n = φn+1 φn as a number in [0, 1). Unlike Fig. 3 but like Fig. 4, U always splits into bounded and connected component curves that are disjoint from each other, provided ΘφK,n is a homeomorphism. Hence U, which is the closure of the set {(ΘφK,n, n + k) : k Z, n = 0,, }, separates into disjoint components each of which is a lift of and each of which is a homeomorphic to Td. For each integer k the set {(ΘφK (θ), (θ) + k) : θ Td } is a component as shown in this figure. See Theorem

10 2 Examples of one-dimensional quasiperiodicity. In this section, we give a detailed explanation of how we compute rotation rates for quasiperiodic maps on one-dimensional tori. In one dimension, Equation 3 relating ρ φ mod 1 to ρ ρ φ mod 1 = aρ mod 1 (5) where a is a positive or negative integer depending on γ and on the reference point P in Eq. 4. Given a map γ for a d = 1 quasiperiodic process, we can only determine ±ρ mod 1 from a trajectory. Changing ρ by adding an integer does not change γ n, so it is only possible to determine ρ mod 1. Writing γ 1 (θ) := γ( θ), we see that γ(nρ) = γ 1 (n(1 ρ)) so both maps give the same trajectory when using ρ for γ and 1 ρ for γ 1. Hence we cannot distinguish between the two maps if we only know the trajectory and so we cannot distinguish ρ mod 1 from 1 ρ mod 1 (which is also ρ mod 1) using only the trajectory (cf. Section 3.3 for the d > 1 case). Example 1. The fish map. Luque and Villanueva [9] addressed the case of a quasiperiodic planar curve γ : S 1 C and introduced what we call the fish map, depicted in the left panel of Fig. 1. Let γ(θ) := ˆγ 1 z 1 + ˆγ 0 + ˆγ 1 z + ˆγ 2 z 2, (6) where z = z(θ) := e i2πθ and ˆγ 1 := 1.4 2i, ˆγ 0 := i, ˆγ 1 := i, ˆγ 2 := i. (See Fig. 5 and eq. (31) in [9]). They chose the rotation rate ρ = ( 5 1)/ for the trajectory γ n = γ(nρ) for n = 0, 1, so we also use that ρ. The method in [9] requires a step of unfolding γ, which our method bypasses. We measure angles with respect to P 1 = i, where the winding number W (P 1 ) = 1. Example 2. The flower map. We have created an example, the flower map in Fig. 1, right, to be more challenging than the fish. Let γ 6 (θ) := (3/4)z + z 6 where z = z(θ) := e i2πθ. (7) We use the same ρ = ( 5 1)/2 as above. The choice of a reference point P 1 for which W (P 1 ) = 1 is shown in the right panel of Fig. 1. For our computations, we use P = P 1 := i. Points P j with W (P j ) = j for j = 0 through 6 are also shown. For ρ γ6 to yield ρ or 1 ρ mod 1 is essential to choose a point P where W (P ) = 1. In this example the values of n are dense in S 1, and max θ ˆ (θ) minθ ˆ (θ) 1.2. For both examples, Figure 1 shows two successive iterates γ n and γ n+1, and the angle n between these two iterates, computed with respect to a reference point P 1. It was computed by finding φ n, the angle of γ n with respect to P 1 as in Eq. (4). Using this, n = φ n+1 φ n [0, 1) S 1. On the left, in 10

11 the fish map case, if we choose ˆ 0 := 0 (or alternatively := 0 + m for some m), then we have selected the component in which all ˆ n must lie. This is what is referred to below as the easy case. Choose some k, write J k := [a, b]. Choose m n is the integer for which n + m n J k. It is not as easy to do this for the flower map on the right. Figure 3 right shows that the possible lifts when plotted against φ form a tangled mess which does not resolve into bounded components, while when plotted against θ we obtain components that are diffeomorphic to S 1. Figures 3 and 4 show the possible lift values ˆ n of the angle difference n plotted with respect to angle θ in Fig. 3 and φ in Fig. 4. For the fish map on the left, we see that we can set [a, b] [ k, k] for any integer k. Furthermore, we investigated the rotation rate of the signal viewed from P 1 = 7 + 4i. Using the Weighted Birkhoff Average (see Appendix), we observe that the deviations of the approximate rotation from ρ falls below when the iteration number exceeds N = 20, 000, and since we know the actual rotation rate, we can report that the error in the rotation rate is then below Once we have found a proper lift for the flower map, we can do the same procedure. The next section explains how we go about finding a lift in this more complicated case. 3 Embedding Continuation Method. In this section, we describe how we find the lift of a map using our embedding continuation method. A schematic of these ideas is depicted in Figure 5.???THIS IS NOT RELATED! The case d = 1. Let φ be the angle in Eq. (4). Note that while an angle φ(s) is in S 1, its derivative φ (s) is a real number. A traditional approach to the computation of ρ φ uses the lift ˆφ : R d R of φ : T d S 1, which when d = 1 can be defined as ˆφ(θ) := θ 0 φ (s)ds. (8) Hence ˆφ : R R, and it is a smooth function with the property that ˆφ(θ mod 1) = ˆφ(θ) mod 1. One then argues that the result ˆφ(θ) is independent of the path since ˆφ(θ) mod 1 = φ(θ) is independent of the path. Define the lift of as follows. ˆ (θ) := θ+ρ θ φ (s)ds (9) In other words ˆ (θ) = ˆφ(θ+ρ) ˆφ(θ). As with ˆφ above, in the d > 1 case, one can define ˆ by interpreting the above equation as a path integral from θ to θ + ρ in R d where the result is independent of the path. Hence ˆ (θ) is a smooth function of θ T d. The Birkhoff limit for d = 1. By the Birkhoff Ergodic theorem (Lemma. 1.1) and the definition of 11

12 the winding number, ρ φ = lim N B N( ˆ )(θ) = = 1 0 ρ 1 ˆ (θ)dθ = 1 ρ 0 0 φ (θ + s)dθ ds = W (P ) φ (θ + s)ds dθ (10) ρ ds = W (P )ρ. 3.1 d 1, the general case. A major difficulty in evaluating ρ φ is that ˆ (θ n ) is not known even though ˆ (θ) mod 1 = (θ). This is because ˆ (θ) R is a lift of (θ) S 1 ; i.e., they differ by an (unknown) integer m(θ) := ˆ (θ) (θ). The key fact is that from its definition, ˆ (θ) is continuous and since it is defined on a compact set it is uniformly continuous. We describe in Steps 1 and 2 below how to choose the integer part of (θ n ) consistently, that is, so that (θ n ) is continuous on S 1. They collectively constitute our Embedding Continuation Method. Step 1. The embedding. Let N be given; we imagine N 10 5 or 10 6 if d = 1. Choose K 2 and define the delay coordinate embedding Θ(θ):= Θ γ K (θ) := (γ(θ), γ(θ + ρ),, γ(θ + (K 1)ρ)). Since γ(θ) R 2, we have Θ(θ) R 2K. By our version of the Takens Embedding Theorems, Theorem 1.2, if 2d < 2K, i.e. d < K, for almost every smooth function γ, the map Θ : T d R 2K is an embedding. In particular, there are no self intersections i.e., if Θ(θ 1 ) = Θ(θ 2 ), then θ 1 = θ 2. That implies Γ 0 defined by Γ 0 (θ) = (Θ(θ), ˆ (θ)) is also an embedding of T d, this time into R 2K+1. Write U ={(Θ(θ), (θ) + k) : for all θ T d and all k Z}. The minimum distance ɛ between components of U. For each j Z, define Γ j (θ) = (Θ(θ), ˆ (θ) + j), and write Γ j := Γ j (T d ). Then U is the union of all Γ j. These sets are vertical translates of Γ 0 by an integer j, i.e. translates in the second coordinate. These are all disjoint from each other (since Θ 0 (T d ) is assumed to be an embedding). Let ɛ = inf{ p 1 p 2 : p 1, p 2 U and are in different Γ j }, where is the Euclidean norm on R 2K+1. Then ɛ > 0 and ɛ is the minimum distance between points on different components of U. In general ɛ is unknown from just the time series γ n, so we have to fix a threshold δ > 0, assuming that δ < ɛ. Then if p 1, p 2 U and p 1 p 2 < δ, it follows that p 1 and p 2 are in the same component of U. Step 2. Extending by δ-continuation. Write Θ n := Θ(nρ). The goal is to choose integers m n 12

13 so that all of the points (Θ n, n + m n ) for n = 0,, N 1 are in the same component. This may be impossible if N is not large enough. The point (Θ 0, 0 ) is in some component and we choose m 0 = 0 which determines a component. Let A be the set of n {0,, N 1} for which m n has an assigned value. This set A changes as the calculation proceeds. Initially m n is assigned only for n = 0 so at this point in the calculation the set A contains only 0. Each time we assign a value to some m n, that subscript n becomes an element of A. If there is an n 1 A and an n 2 / A and an integer k such that (Θ n1, n1 + m n1 ) (Θ n2, n2 + k) < δ, then the two points are in the same component and we assign n 2 = k, which adds one element, n 2 to the set A. Keep repeating this process (if possible) until all m n are assigned values. For N sufficiently large, all can be assigned values, in which case we define ˆ n = n + m n for all n {0,, N 1}. Define ρ N γ := N 1 ˆ n=0 n. N Theorem 3.1 For a d-quasiperiodic map assume d < K. For almost every smooth γ : T d R 2, for each P / γ(t d ), and for δ sufficiently small, for all sufficiently large N (depending on δ), the above value ρ N γ is well defined (since all m n are defined). Let φ be the angle defined in Eq. 4. Then lim N ρn γ = ρ φ. See Prop. 3.2 for a characterization of how ρ φ is related to ρ. Of course we need an efficient algorithm to carry out this process of assigning values to the m n, and we describe an algorithm later. Choosing K. For the flower we use an embedding dimension of K 10. Our Embedding Continuation Method applied to flower yields ρ with 30-digit precision at N = 200, 000 when used with our Weighted Birkhoff Averages. 3.2 The method for higher-dimensional quasiperiodicity For a trajectory θ n = nρ mod 1 on T d where ρ is an irrational vector, we select a smooth composite map γ : T d R 2 and assume we know γ n := γ(nρ). As in Eq. (4), define φ(θ) for any P R 2 where P / γ(t d ). Note that φ : T d S 1 is smooth since γ is. The rotation rate that can be computed depends on the choice of γ and to some extent P, and thus on φ. Let ˆφ : R d R be the continuous lift function, defined analogously to Eq. (8). Define the ρ φ - 13

14 rotation rate as ρ φ := lim N ˆφ(θ + Nρ) ˆφ(θ). (11) N This number depends on the map projection to R 2, since this map determines φ. To write a formula for ρ φ relating it to the rotation vector of the map, we first give a way of relating ˆφ and φ. For each coordinate j, write e j for the vector that is 1 in the j th coordinate and zero elsewhere. Define a j := ˆφ(θ + e j ) ˆφ(θ). Each a j is an integer since ˆφ(θ + e j ) mod 1 = ˆφ(θ) mod 1, and it is also independent of θ. Hence a := (a j ) is a constant integer vector that is independent of θ. The integer a j is the number of times the image under φ of the j th coordinate circle is wrapped around the range circle by γ and φ. Next let θ = (θ 1,, θ d ) and define g : R d R as the bounded function g(θ) := ˆφ(θ) d j=1 a jθ j. Hence ˆφ(θ) = d a j θ j + g(θ), (12) j=1 and this representation is unique when we restrict g to be continuous and bounded and g(0) = 0. Note that g(θ + e j ) = g(θ) for each j, meaning that it is periodic with period 1 in each variable. It follows that φ(θ) = ( These representations satisfy the key property ˆφ(θ) mod 1 = φ(θ mod 1). d a j θ j + g(θ mod 1)) mod 1. (13) j=1 Proposition 3.2 Let a = (a 1,, a d ) be the integer vector in Eq. (12). Write ρ := (ρ 1,, ρ d ). Then ρ φ satisfies ρ φ = d a j ρ j. (14) The proof of this proposition follows from the fact that g is bounded, implying that j=1 g(θ + Nρ) g(θ) lim N N = 0 so that Eq. (11) becomes ρ θ = lim N d j=1 a j Nρ j N = d j=1 a j ρ j. We discuss the set of possible values for (ρ φ1,..., ρ φd ) in the next section. 14

15 3.3 The many possible representations for rotation vectors. While the definition of quasiperiodicity requires that the map have a coordinate system that turns the map into Eq. 1, that requirement by itself does not determine ρ. Let ψ = Aθ where ψ T d and A is an integer-entried matrix with determinant deta = 1, then in this new coordinate system Eq. 1 becomes ψ ψ + Aρ mod 1. (15) Hence Aρ is also a rotation vector. If d > 1, then in Sec. 3.3 we point out that the set of such rotation representations Aρ of the rotation vector is dense in T d. Instead of having a well-defined rotation vector for a quasiperiodic torus, we have a dense set of rotation representations. In Eq. (14) we gave a dense set of rotation rates ρ φ that can be obtained via projection. Now suppose there is a map φ : T d T d with rotation vector ρ for which we can measure the rotation rates of each coordinate in the image. We now give a full description of the set of obtainable rotation vectors, showing that they are dense in the torus, and can be obtained via changes of coordinates of the torus. See Section 5 for a discussion of the physical ramifications of these many rotation rates in the context of the restricted three-body problem. For the analogue of Eq. (12), the lift ˆφ : R d R d has the form ˆφ(θ) = Aθ + G(θ) where A is an integer-entry matrix and G is bounded. As in the previous section, the observed rotation rates depend only on A. If we let θ n+1 = θ n + ρ and y = ˆφ(θ), then we get y n+1 = y n + Aρ. Let S denote the set of integer-entried d d matrices with determinant ±1. Observe that for any B S, B 1 S. A matrix in S can be viewed as a change of variables on the torus, since it preserves volume. Therefore we call a vector ρ R d a rotation representation of ρ R d if ρ = Aρ for some A S. We ask: Assuming the vector ρ is irrational, what are all the possible rotation vectors (i.e., rotation representations), assuming A S? We show that the set of all possible rotation representations are dense in the torus. We have defined ρ in Eq. (1) in terms of a given coordinate system. Proposition 3.3 For an irrational rotation vector ρ, the set of its rotation representations is Sρ (meaning {Aρ : A S}), and Sρ mod 1 is dense in T d. Proof. To simplify notation we prove only the case of d = 2. The proof for d > 2 is analogous. See [14]. Write ρ = (ρ 1, ρ 2 ). Note that the matrices B m := 1 m and C k := 1 0 are in S 0 1 k 1 for all integers m and k, as is A := B m C k. Then the vectors (ρ 1, y k ) = C k (ρ 1, ρ 2 ) mod 1 are vertical 15

16 translates (translates in the direction (0, 1)) of (ρ 1, ρ 2 ) mod 1, where {y k } is a dense set in S 1. When we similarly apply B m for all m to each (ρ 1, y k ) we obtain a dense set of horizontal translates of (ρ 1, y k ) and thereby obtain a dense set in T 2. Every coordinate of every point in that dense set is of the form k 1 ρ 1 + k 2 ρ 2 mod 1 where k 1 and k 2 are integers. 4 Higher-dimensional quasiperiodic examples We develop a higher-dimensional method to compute the rotation vector ρ purely from knowledge of the sequence θ n := F (θ n ). The question of how to compute the rotation vector is actually two questions. Question 1: If we compute a rotation vector, what are the possible values? Question 2: How do we compute any of the possible values for the rotation vector in difficult cases? Figures 7, 8, and 9 demonstrate that like in one dimension, in d dimensions we are able to use d independent planar projections combined with a higher-dimensional version of our embedding continuation method in order to find a lift, each projection leading to one component of a d-dimensional rotation vector. In fact, these rotation vectors are not unique. In this section, we give a detailed discussion of our higher-dimensional method, describing the possible values we can achieve in calculating a rotation vector. We then illustrate our method for three examples: the fish torus, the flower torus, and the restricted three-body problem. 4.1 Two examples in a higher dimension: fish and flower tori T 2. We use the fish and flower maps from the previous section in order to create 2-dimensional torus maps. Let ρ := ( 5 1)/2 and φ := 3/2, and define (θ n, y n ) := (nρ mod 1, nφ mod 1) T 2 (16) Let γ be either the fish or the flower map defined in the previous section. Define the torus-version f T of the γ map(s) as follows. Let Re( ) and Im( ) denote the real and imaginary components of a complex number, and let f T : T 2 R 3. Write f T (θ n, y n ) = (f 1, f 2, f 3 )(θ n, y n ), where f 1 (θ n, y n ) = Re(γ(θ n ) + 2) cos(2πy n ) (17) f 2 (θ n, y n ) = Re(γ(θ n ) + 2) sin(2πy n ) (18) f 3 (θ n, y n ) = Im(γ(θ n )). (19) The +2 is just for convenience so that the torus image can wrap around the origin rather than having to wrap it around some other point. For each γ, the map f T takes a quasiperiodic trajectory into R 3. 16

17 Figure 6: The fish and flower torus. The top figures show two views of the fish torus, and the bottom two views of the flower torus. These figures can be thought of as projections of tori onto the plane represented by the page. The three coordinate axes are presented here to clarify which two-dimensional projection is being used. The projections of the tori on the left are simply connected so there is no way to choose a point P that would yield a non-zero rotation rate. The projections on the right yield images of the tori that are annuli with a hole in which P can be chosen to yield non-zero results. Each is a plot of N = iterates. The red circle is the initial point. 17

18 Figure 7: Projections of the fish torus and the flower torus. The coordinates used to find angle 1 (left) and angle 2 (right) for the fish torus (top) and the flower torus (bottom). The red circle shows the initial condition. The shows the point with which the angle is measured. Note that for the the fish torus, the point from which the angle is measured is very close to the edge torus image. For angle 2, points are projected onto a tilted plane that makes angle 0.05π with the horizontal. See section 4.1 for a full description of these projections. 18

19 Figure 8: Lifts of the angle difference for the fish torus and flower torus. This figure shows the angle versus angle difference lift for fish torus angle 1 (top left) and angle 2 (top right) and the flower torus angle 1 (bottom left) and angle 2 (bottom right), using the projections depicted in the previous figure. 19

20 Figure 9: Angle differences for the fish torus and flower torus. The angle versus angle difference for angle 1 (left) and angle 2 (right) for the fish (top) and flower (bottom). These figures show three possible viable angle differences, each differing by an integer, for the same projections as were depicted in the previous two figures. 20

21 Two projections of a torus for two rotation rates. Figure 7 shows two independent projections of f T to R 2. For the first rotation rate, we project f T to (f 1, f 2 ) in the plane. Then we measure the angle φ from a refernce point P which is not in the image of the torus. In particular, P = (0, 1.5) for the fish torus, and (0, 0.1) for the flower torus. For both maps, this projection gives a rotation rate of φ/2π (the denominator 2π comes from the fact that we are measuring angles in [0, 1]). For a second rotation rate, let R α be the rotation matrix that tilts by angle α = 0.05π in the f 2 f 3 plane. Namely R α = cos α sin α 0 sin α cos α Set h = R 0.05π f. Define r = h h 2 2. Then our projection is to the value (r, f 3 ). We measure the angle of this projection relative to the point (8.25, 4.4) for the fish torus, and (2.6, 1.4) for the flower torus. For both maps, this projection gives a rotation rate of MISSING ROTATION RATE and 1 ρ. Why the tilt by 0.05π rather than use value of r with respect to the original coordinates? Because without the tilt (ie α = 0), the projection would be a curve rather than a thick strip, which would not give a true test of our embedding continuation method in two dimensions. In both cases, we get a map whose image has at least one hole (in which the winding number = ±1), and we can measure angles φ and angle differences compared to a point inside one of the holes, as long as the torus has a winding number W (p) = 1 with respect to points in this hole. Thus just as for the one-dimensional case, we compute the lift, and then compute the rotation rate for these two different projections. Figure 8 shows the computed lift, and Fig. 9 shows the original values of the angle difference from which this lift was computed. Note that fish torus lift is easy to compute while the flower torus requires and embedding. As mentioned earlier, rather than using Birkhoff Averages, we achieve more rapid convergence using our Weighted Birkhoff Average, denoted WB N. See the Appendix for details. Define the ρ approximation ρ N := WB N ( n). Fluctuations in ρ N fall below for N > 20, 000. Since we know the actual rotation rate, we can report that the error ρ ρ N is then below

22 4.2 The circular planar restricted three-body problem (CR3BP) CR3BP is an idealized model of the motion of a planet, a moon, and an asteroid governed by Newtonian mechanics Poincaré [15, 17] introduced his method of return maps using this model. In particular, we consider a circular planar three-body problem consisting of two massive bodies ( planet and a large moon ) moving in circles about their center of mass and a third body ( asteroid ) whose mass is infinitesimal, having no effect on the dynamics of the other two. This model can also (simplistically) represent the Sun-Earth-Moon system discussed in the introduction though the parameter µ has to be changed, and the Moon is the body that is assumed to have negligible mass. All three travel in a plane. We assume that the moon has mass µ and the planet mass is 1 µ where µ = 0.1, and writing equations in rotating coordinates around the center of mass. Thus the planet remains fixed at (q 1, p 1 ) = ( 0.1, 0), and the moon is fixed at (q 2, p 2 ) = (0.9, 0). In these coordinates, the satellite s location and velocity are given by the generalized position vector (q 1, q 2 ) and generalized velocity vector (p 1, p 2 ). Define the distance of the asteroid from the moon and planet are d 2 moon = (q µ) 2 + q 2 2 d 2 planet = (q 1 + µ) 2 + q 2 2. The following function H is a Hamiltonian (see [18] p.59 Eqs ) for this system H = 1 2 (p2 1 + p 2 2) + p 1 q 2 p 2 q 1 1 µ d planet µ d moon, (20) where p 1 = q 1 q 2 and p 2 = q 2 + q 1 The terms in the square brackets are resp. the kinetic energy, angular momentum, and the potential. We get the equations of motion from dq i dt dp i dt = H pi, = H qi. (21) 22

23 Figure 10: Two views of a two-dimensional quasiperiodic trajectory for the restricted three-body problem described in Section 4.2. That is, the equations of motion are as follows: dq 1 dt dq 2 dt dp 1 dt dp 2 dt = p 1 + q 2, = p 2 q 1, = p 2 µ q µ d 3 moon = p 1 µ q 2 d 3 moon (1 µ) q 1 + µ, (1 µ) dplanet 3 q 2 d 3 planet, (22) For fixed H, Poincaré reduced this problem to the study of the Poincaré return map for a fixed value of H, only considering a discrete trajectory of the values of (q 1, p 1 ) on the section q 2 = 0 and dq 2 /dt > 0. Simplifying assumptions. The CR3BP equations assume (i) distance between planet and moon = 1 (ii) gravitational constant G = 1 (iii) sum of masses of planet and moon = 1 We now scale time so that the line between the two massive bodies rotates at 1 radian per unit time which, a unit of time which we denote as sec, so the moon rotates at a rate of r moon = 1/2π rev/sec in fixed coordinates, and its period of revolution is 2π. We again measure angles as a fraction of a full rotation and not in terms of radians. The asteroid s orbit in rotating coordinates is shown in Fig. 10. Here time is continuous so we can measure the total angle through which a trajectory travels, retaining the integer part. The first rotation rate 23

24 Figure 11: Plots of the circular planar restricted three-body problem in r r coordinates. As described in the text, we define r = (q ) 2 + q 2 2 and r = dr/dt. This figure shows r versus r for a single trajectory. The right figure is the enlargement of the left. One of the two rotation rates ρ φ is calculated by measuring from (r, r ) = (0.15, 0) in these coordinates. Figure 12: Convergence to the rotation rates for the CR3BP. For these two figures, we used differential equation time step dt = and we compute the change in angle after 50 such steps, that is, in time output time Dt = On the left, we estimate the rotation rate ρ θ. The figure compares the convergence to the estimated value using our Weighted Birkhoff Averages WB 1,N and WB 2,N and the Birkhoff average B N. (Section our Appendix) of On the right, we show the convergence rate to the estimated value of ρ φ calculated using WB 2,N. Notice that on the left, even at the highest N value, the error is still at

25 ρ θ of the asteroid s orbit is its average rate of rotation about the planet, that is, the average rate of change of the angle θ measured from (q 1, q 2 ) = ( 0.1, 0). We compute that ρ θ = rev/sec, that is, about -2.5 θ-revolutions per unit time Fig. 12 (left) shows the error in convergence to the value 0.001ρ θ using the Birkhoff Average (B N) is approximately 1/N, which if extrapolated, would mean that we would require N to obtain an error of In contrast, the convergence is much faster when we apply the Weighted Birkhoff Average WB 1,N, and WB 2,N is even faster than WB 1,N. Thus for WB p,n, p = 2 is expected to be the best choice of integer p in quadruple precision calculations. Note that the rotation rate in the fixed coordinate frame is ρ θ + 1/(2π). The second rotation rate ρ φ measures the oscillation in the distance r from the planet. In particular, we project to the (r, r ) plane, where r := dr/dt. That is, define r = (q ) 2 + q 2 2 and r = dr = ((q dt ) dq 1 dq + q 2 dt 2 )/r, as shown in Fig. 11. The angle φ is measured from (r, dt r ) = (0.15, 0). The fast convergence to the value 0.001ρ φ by WB 2,N is seen in Fig. 12 (right), where ρ φ = rev/sec. The period of time between perigees is the reciprocal, or about 0.43 time units. We used the order-8 Runge-Kutta method in Butcher time steps of h = [16] to compute trajectories of CR3BP with The meaning of rotation rates for the CR3BP. In [1], we investigated the same asteroid orbit of the CR3BP as is studied here, but instead of the continuous-time trajectory that lies on a two-dimensional torus as presented above, there we used a Poincaré map. The coordinates of the asteroid were recorded each time the asteroid crossed the line q 2 = 0 with dq 2 /dt > 0. The map trajectory is thus a trajectory on a closed curve. Hence there is only one rotation rate, a much simple situation. Choosing a point inside the closed curve, we computed a rotation rate, namely the average angular rotation per iteration of the Poincaré map. The rotation rate ρ P per Poincaré map on the Poincaré surface q 2 = 0 (or equivalently, θ = 0) around (q 1, p 1 ) = ( 0.25, 0) was computed as We felt that the issues of rotation rates could be clarified if we computed the trajectory as a continuous orbit as we do here. The two rotation rates computed here ρ φ and ρ θ and our previous result ρ P relation to our previous results: ρ P = ± ρ φ ρ θ mod 1. bear the following ρ P = = ± See the caption of Fig. 12. We solved the differential equation using an 8 th -order Runge-Kutta method using quadruple precision. approach. Both approaches are based on rotating coordinates, but there is another 25

26 The orbit as a slowly rotating ellipse. The asteroid rotates about the planet at a rate of ρ θ revolutions per unit time when viewed in the rotating coordinate in which the moon and planet are fixed. The sidereal rotation rate (as viewed in the coordinates of the fixed stars) is ρ θ + 1/(2π). We can think of the orbit as an approximate ellipse whose major axis rotates and even changes eccentricity (being more eccentric when the asteroid apogee is aligned with the planet moon axis). Without the moon the asteroid orbit would be perfectly elliptical with its major axis fixed in position, but the moon causes the ellipse to rotate slowly. The angle φ(t) tells where the asteroid is on its roughly elliptical orbit; Fig. 11 shows that the apogee occurs when when the distance from the planet r 0.27 and the perigee when r 0.05, with some variation. The time between successive perigees averages 1/ρ φ. Note that the difference in these rates satisfies ρ φ [ρ θ + 1/(2π)] / Hence relative to the fixed stars, that is, in non-rotating coordinates, the asteroid s ellipse s major axis precesses slowly. Its apogee point returns to its original position (in non-rotating coordinates) after the asteroid passes through its apogee approximately 1639 times. 5 Discussion and conclusions One key difference between d = 1 and d > 1 is that in the higher-dimensional case, there is a dense set of rotation vectors, as we describe this set in Section 3.3. We can restate this as follows. For any quasiperiodic system with d = 2 and any ɛ > 0 and any numbers ρ 1, ρ 2 [0, 1), there is a linear choice of coordinates for which the rotation vector is (ρ 1, ρ 2 ) where (ρ 1, ρ 2 ) and ( ρ 1, ρ 2 ) differ by less than ɛ in each coordinate. Hence computing ρ 1 and ρ 2 makes no sense unless we can find coordinates that have special meaning. But the rotation rates ρ φ or ρ γ are significant, as in the case of the Moon where some measurement in R or R 2 has particular significance. For d = 1, we can obtain ρ mod 1 and 1 ρ mod 1 depending on the choice of orientation on S 1. Finally, in Section 4.2, we apply our method to the quasiperiodic torus occurring for a 4-dimensional circular restricted 3-body problem, depicted in Fig. 10. In particular, we explain the relationship between the two rotation rates obtained from the original differential equation system and the rotation rate which was previously obtained from the Poincaré map. The Appendix discusses our method for greatly accelerating the convergence of the Birkhoff average. Notes on delay coordinate embedding theorems. H. Whitney [10] showed that a topologically generic smooth map Γ from a d-dimensional smooth compact manifold M into R D where 2d + 1 D is a diffeomorphism on M; in particular the map Γ : M F (M) is an embedding of M. Sauer et al [11] modified Takens result in two ways. First, it replaced topologically generic by almost every (in the sense of prevalence ) in Theorems 26