A trajectory $ \theta_n : = F^n(\theta_0), n = 0,1,2, \dots $ is quasiperiodic if the trajectory lies on and is dense in some $ d $-dimensional torus $ {\mathbb{T}^d} $, and there is a choice of coordinates on $ {\mathbb{T}^d} $ for which $ F $ has the form $ F(\theta) = \theta + \rho\bmod1 $ for all $ \theta\in {\mathbb{T}^d} $ and for some $ \rho\in {\mathbb{T}^d} $. (For $ d>1 $ we always interpret $ \bmod1 $ as being applied to each coordinate.) There is an ancient literature on computing the three rotation rates for the Moon. However, for $ d>1 $, the choice of coordinates that yields the form $ F(\theta) = \theta + \rho\bmod1 $ is far from unique and the different choices yield a huge choice of coordinatizations $ (\rho_1,\cdots,\rho_d) $ of $ \rho $, and these coordinations are dense in $ {\mathbb{T}^d} $. Therefore instead one defines the rotation rate $ \rho_\phi $ (also called rotation rate) from the perspective of a map $ \phi:T^d\to S^1 $. This is in effect the approach taken by the Babylonians and we refer to this approach as the "Babylonian Problem": determining the rotation rate $ \rho_\phi $ of the image of a torus trajectory - when the torus trajectory is projected onto a circle, i.e., determining $ \rho_\phi $ from knowledge of $ \phi(F^n(\theta)) $. Of course $ \rho_\phi $ depends on $ \phi $ but does not depend on a choice of coordinates for $ {\mathbb{T}^d} $. However, even in the case $ d = 1 $ there has been no general method for computing $ \rho_\phi $ given only the sequence $ \phi(\theta_n) $, though there is a literature dealing with special cases. Here we present our Embedding continuation method for general $ d $ for computing $ \rho_\phi $ from the image $ \phi(\theta_n) $ of a trajectory, and show examples for $ d = 1 $ and $ 2 $. The method is based on the Takens Embedding Theorem and the Birkhoff Ergodic Theorem.
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Figure 1. The fish map (left) and flower map (right). The function $ \gamma:S^1\to \mathbb{R}^2 $ for each panel is respectively Eq. 21 and Eq. 22 and the image plotted is $ \gamma(S^1) $ in the complex plane. These are images of quasiperiodic curves with self-intersections, and we want to compute the rotation rate only from knowledge of a trajectory $ \gamma_n\in {\mathbb{R}}^2 $. The curves winds $ j $ times around points $ P_j $, so $ P_1 $ is a correct choice of reference point from which angles can be measured to compute a rotation rate. If instead we choose $ j\ne 1 $, then the measured rotation rate will be $ j $ times as big as for $ j = 1 $. In both cases, $ P_1 $ is the reference point. $ P_1 = (8.25,4.4) $ and $ (0.5,1.5) $ for the fish map and flower map, respectively. The angle marked $ \Delta_n\in [0,1) $ measured from point $ P_1 $ is the angle between trajectory points $ \gamma_n $ and $ \gamma_{n+1} $. For each point $ \gamma_n $ we can define $ \phi_n $ to be a unit vector $ (\gamma_n - P_1)/\|\gamma_n - P_1\| $. Still using $ P_1 $, we can define a map $ \phi: {\mathbb{T}^d} \to S^1 $ - but since this is a one-dimensional torus, $ {\mathbb{T}^d} = S^1 $
Figure 2. The flower map revisited. Suppose instead of having the function $ \gamma:S^1\to \mathbb{R}^2 $ for the flower Eq. 22 in Fig. 1, we had only one coordinate of $ \gamma $, for example, the real component, $ Re~\gamma. $ Knowing only one coordinate would seem to be a huge handicap to measuring a rotation rate. But it is not. In the spirit of Takens's idea of delay coordinate embeddings explained in detail later, we plot $ (Re~\gamma_n,Re~\gamma_{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 $ \rho_{\phi} $ here using $ P_1 $ is the same as for Fig. 1 right
Figure 3. The angle difference for the fish and the flower maps. Here we plot $ (\phi_n, \Delta_n+ k) $ for every $ n\in\mathbb{N} $ and all integers $ k $, where $ \Delta_n = \phi_{n+1}-\phi_n \bmod1 $. In the left panel (the fish map, the easy case) the closure of the figure resolves into disjoint sets (which are curves $ \subset \mathbb{R}\times 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\subset S^1\times \mathbb{R} $. We can choose any such curve to define $ \hat\Delta_n $, namely we define $ \hat\Delta_n = \Delta_n + k $ where $ k $ is the unique integer for which $ (\phi_n,\Delta_n + k)\in C $. 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 $ \theta $ instead of $ \phi $. That is, we take $ \theta_n $ to be $ n\rho $ and $ \Delta(\theta) = \phi(\theta+\rho)-\phi(\theta) \bmod1 \in [0,1) $ and we plot $ (\theta_n,\Delta_n + k) $ for all integers $ k $ (where again $ \Delta_n = \Delta(n\rho) $), These are points on the set $ G = \{(\theta, \Delta(\theta)+ k):\theta\in S^1, k\in\mathbb 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 $ \theta\in S^1 $ and $ k\in \mathbb Z $ there is exactly one point $ y\in [k,k+1) $ for which $ \theta,y)\in G $. Each connected component of $ G $ is an acceptable candidate for $ \hat\Delta $. Unlike the plots in Fig. 3, $ G $ always splits into disjoint curves. Unfortunately the available data, the sequence $ (\phi_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
Figure 5. Lifts over an embedded torus. Let $ \Theta : = \Theta_K^\phi $ be as in Eq. 15 and let $ \theta_n = n\rho $ be a trajectory on $ {\mathbb{T}^d} $. Assume $ K\ge 3 $. By Theorem 1.2 for almost any map $ \phi $, the set $ \Theta( {\mathbb{T}^d} ) $ is an embedding of $ {\mathbb{T}^d} $ into $ {\mathbb{T}} ^{K} $; i.e., $ \Theta $ is a homeomorphism of $ {\mathbb{T}^d} $ (the circle $ S^1 $ when $ d = 1 $) onto $ \Theta( {\mathbb{T}^d} ) $. In particular the map is one-to-one. The smooth (oval) curve is the set $ (\Theta( {\mathbb{T}^d} ),0) $. As in our previous graphs, the vertical axis shows the angle difference $ \Delta ( \theta ) \in [0,1)+k $ for all integers $ k $. Write $ \mathbb{U} : = \{(\Theta(\theta),\Delta(\theta)+k):\theta\in {\mathbb{T}^d} \mbox{ and } k\in\mathbb{Z}\} $. Unlike Fig. 3 but like Fig. 4, $ \mathbb{U} $ always splits into bounded, connected component manifolds that are disjoint from each other. Hence $ \mathbb{U} $, which is also the closure of the set $ \{(\Theta(\theta_n),\Delta_n + k):k\in \mathbb Z,n = 0,\cdots,\infty\} $, separates into disjoint components each of which is a lift of $ \Delta $ and each of which is homeomorphic to $ {\mathbb{T}^d} $. For each integer $ k $ the set $ \{(\Theta(\theta),\Delta(\theta)+k):\theta\in {\mathbb{T}^d} \} $ is a component as shown in this figure. See Theorem 2.1
Figure 6. Illustrating a chain of points on a rigid rotation on the torus. $ x_n = n\sqrt{3} (\bmod 1), y_n = n\sqrt{5} (\bmod 1) $ for $ n = 0, \cdots, N-1 $ are plotted with the origin indicated by $ 0 $ at the center on the panel. Each point $ \theta_n = (x_n,y_n) $ is labeled with its subscript $ n $. Here $ N = 100 $ (left) and $ = 20,000 $ (right). Only the neighborhood of the origin is shown for the right panel. In the left panel, $ \theta_4 $ and $ \theta_{93} $ (ⅰ) are near the origin and (ⅱ) their subscripts are relatively prime and (ⅲ) the total of the subscripts is less than $ N $. On the right points with subscripts $ 4109 $ and $ 11,700 $ play the corresponding role. In each case it follows that there is a chain of points starting from $ 0 $ and ending at any desired $ \theta_m $ where $ 0 < m < N $. This chain is a series of steps, each achieved by either adding one of the two subscripts or subtracting the other. See Prop. 2 and the algorithm sketched in its proof. In the left panel such a chain - adding $ 93 $ or subtracting $ 4 $ at each step - is shown that ends at $ \theta_{90} $
Figure 8. 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 $ \times $ 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 $ 0.05 \pi $ with the horizontal. See Section 4.1 for a full description of these projections
Figure 9. Angle differences for the fish torus and flower torus. Each panel shows three possible angle differences, each differing by an integer, for the same projections as were depicted in Fig. 8. The angle versus angle difference for angle 1 (left) and angle 2 (right) for the fish torus (top) and flower torus (bottom). In the final panel, the picture cannot be separated into separate components
Figure 10. Lifts of the angle difference for the fish torus and flower torus. Here one of the possible lifts has been selected from each panel in Fig. 9. Each panel 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 Fig. 8
Figure 7. 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 nonzero results. Each is a plot of N = 50090 iterates. The red circle is the initial point
Figure 13. Convergence to the rotation rates for the CR3BP. For these two figures, we used differential equation time step $ dt = 0.00002 $ and we compute the change in angle after 50 such steps, that is, in time "output time" $ Dt = 0.001 $. We show the convergence rates to the estimated rate of $ 0.001\times \rho^{*}_{\theta} $ (left) and of $ 0.001 \times \rho^{*}_{\phi} $ (right). For both cases rotation rates are calculated using the Weighted Birkhoff averaging method $ \mbox{WB} ^{[2]}_{N} $ in Eq.13 and show fast convergence
Figure 12. Plots of the circular planar restricted three-body problem in $ r-r' $ coordinates. As described in the text, we define $ r = \sqrt{(q_1+0.1)^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 $ \rho^{*}_\phi $ is calculated by measuring from $ (r,r^{\prime}) = (0.15,0) $ in these coordinates
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The fish map (left) and flower map (right). The function
The flower map revisited. Suppose instead of having the function
The angle difference for the fish and the flower maps. Here we plot
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
Lifts over an embedded torus. Let
Illustrating a chain of points on a rigid rotation on the torus.
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
Angle differences for the fish torus and flower torus. Each panel shows three possible angle differences, each differing by an integer, for the same projections as were depicted in Fig. 8. The angle versus angle difference for angle 1 (left) and angle 2 (right) for the fish torus (top) and flower torus (bottom). In the final panel, the picture cannot be separated into separate components
Lifts of the angle difference for the fish torus and flower torus. Here one of the possible lifts has been selected from each panel in Fig. 9. Each panel 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 Fig. 8
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 nonzero results. Each is a plot of N = 50090 iterates. The red circle is the initial point
Two views of a two-dimensional quasiperiodic trajectory for the restricted three-body problem described in Section 4.2
Convergence to the rotation rates for the CR3BP. For these two figures, we used differential equation time step
Plots of the circular planar restricted three-body problem in