Abstract This paper develops Chebyshev-Taylor spectral methods for studying stable/unstable manifolds attached to periodic solutions of differential equations. The work exploits the parameterization method -a general functional analytic framework for studying invariant manifolds. Useful features of the parameterization method include the fact that it can follow folds in the embedding, recovers the dynamics on the manifold through a simple conjugacy, and admits a natural notion of a-posteriori error analysis. Our approach begins by deriving a recursive system of linear differential equations describing the Taylor coefficients of the invariant manifold. We represent periodic solutions of these equations as solutions of coupled systems of boundary value problems. We discuss the implementation and performance of the method for the Lorenz system, and for the planar circular restricted three and four body problems. We also illustrate the use of the method as a tool for computing cycle-to-cycle connecting orbits. Introduction Periodic solutions of differential equations are the basic building blocks of recurrence in nonlinear dynamics. Moreover, hyperbolic periodic orbits and their heteroclinic/homoclinic connections are natural generators of chaotic motions. Since a heteroclinic/homoclinic orbit will approach a periodic solution along its local stable/unstable manifolds, computational methods for studying these manifolds are of great interest. A schematic description of the stable manifold of a periodic orbit, beside an actual stable manifold in the Lorenz system are illustrated in Figure 1 . See any of the works [Canalias and Masdemont, 2006; Font et al., 2009; Jorba, 1999; Jorba and Masdemont, 1999; Jorba and Villanueva, 1998; Krauskopf et al., 2005; Masdemont, 2011; Osinga, 2003; Simó, 1998 Simó, , 1988 Simó, , 1989 for more discussion, but we caution that any such list will hardly scratch the surface of the relevant literature. Introduction The stable/unstable normal bundles of a periodic solution approximate the stable/unstable manifolds to first order, and these bundles are obtained by studying the equations of first variation -or equivalentlyby solving certain periodic eigenvalue problems. Higher order jets of the invariant manifold could be studied via higher order equations of variation, however the complexity of these equations grows exponentially with the order of the jet. More efficient methods for studying the jets are obtained by reformulating the invariant manifold as the solution of an operator equation, and studying the operator equation via numerical methods. Introduction The parameterization method is a general functional analytic framework for studying invariant mani- Fig. 1 . The local stable manifold attached to a periodic orbit γ(t): (LEFT) schematic representation of the stable manifold, i.e. it is the union of all solutions accumulating at γ as t → ∞ (periodic orbit shown here as the green loop). The unstable manifold is the same but with orbits approaching the periodic orbit asymptotically in backward time. (RIGHT) local stable manifold of an actual periodic orbit near the Lorenz attractor. This manifold is computed using the techniques of the present work. In particular, changes in color in the figure indicate the transitions from one Chebyshev domain to another. Introduction folds [Cabré et al., 2003a [Cabré et al., ,b, 2005 Canadell and Haro, 2014; Haro, 2012, 2013; de la Llave, 2006a,b, 2007] whose goal is to find a chart/covering map conjugating the dynamics on the invariant manifold to a simple and well understood model (correctly choosing this model is part of the method). By viewing the conjugacy as an operator equation for the unknown chart, the problem is susceptible to numerical methods. We will see below that the operator equation of interest in the present work is actually a PDE with prescribed periodic data. This operator equation is referred to as the invariance equation. The unknown parameterization is not required to be the graph of a function, and hence is able to follow folds in the embedding. See again Figure 1 . Since the invariance equation is based on a dynamical conjugacy, the parameterization method recovers the dynamics on the manifold in addition to the embedding. By now there is a small industry devoted to the parameterization method, and a review of the literature would take us far afield. We refer the interested reader to the recent book on the subject [Haro et al., 2016] , where many examples and much more complete discussion of the literature is found. Introduction The present work is not the first numerical treatment of parameterized stable/unstable manifolds attached to periodic orbits, and indeed we build on the earlier studies of [Cabré et al., 2005; Castelli et al., 2015; Guillamon and Huguet, 2009; Huguet and de la Llave, 2013] . As in these earlier studies, we make a formal Taylor series arguments which analytically reduces the invariance equation before making any numerical computations. Since the coefficients of the Taylor series are themselves periodic functions, the formal power matching scheme leads to so called homological equations describing the unknown coefficients. In the present case of a periodic orbit, the homological equations are linear ordinary differential equations with periodic coefficients and periodic forcing. We solve these linear homological equations recursively using numerical spectral methods. By computing the formal series solution to high order we obtain an approximate solution which describes the stable/unstable manifold far from the periodic orbit. Introduction In the earlier works just cited the differential equations -describing the periodic orbit, describing the normal bundles, and the homological equations describing the higher order jets -are all solved using Fourier spectral methods. Fourier methods are both efficient and accurate when applied to periodic solutions of moderate length. This efficiency is due in part to the fact that differentiation is a diagonal operation in the transform domain, and in part to the fact that the FFT speeds up evaluation of nonlinearities. However, the decay rate of the Fourier coefficients gets increasingly slow as the period/harmonic complexity of the orbit grow. In practice this means that it is necessary to compute more and more Fourier coefficients, and for long enough orbits the Fourier approximation becomes impractical. Introduction There is much recent interest in numerical methods based on Chebyshev spectral approximation of solutions to boundary value problems. We refer the interested reader to [Driscoll et al., 2008; Platte and Trefethen, 2010; Trefethen, 2007 Trefethen, , 2013 and the references discussed there in. The present work builds on techniques developed by a number of authors which use Chebyshev spectral methods to compute long periodic solutions of differential equations [Gameiro et al., 2016; van den Berg and Sheombarsing, 2016] . The merit of this approach is that the Chebyshev spectral methods posess many of the advantages of Fourier series -for example differentiation is a tri-diagonal operation in the transform domain and the fast cosine transform is available for evaluating nonlinearities -but Chebyshev series apply to non-periodic boundary value problems. Treating a periodic solution as a series of coupled boundary value problems -on smaller domains -facilitates control of the decay rates of the coefficients. Motivated by these developments, the present work applies Chebyshev methods not only the periodic orbit -but also to the computation of the normal bundles and the homological equations for the higher order jets. The result is a computational method for finding Chebyshev-Taylor expansions of the local stable/unstable manifolds attached to periodic orbits. Our method applies to more complicated orbits and their attached invariant manifolds than could be studied using only the Fourier-Taylor approach. Introduction Remark 1.1 [Connecting orbits and extensions of local stable/unstable manifolds]. Of course computing local stable/unstable manifolds attached to periodic orbits is only a means to an end. In applications we are often interested in either using the local manifolds to compute connecting orbits, or to grow larger portions of the invariant manifold in order to study the global dynamics. While in the present work we do consider a number of example computations for connecting orbits, we do not make any serious attempt to numerically grow larger local manifolds. This is because the literature on computational methods for growing invariant manifolds is extensive and well developed. The interested reader will want to consult the review paper for a thorough overview of the literature, and will find other powerful methods and fuller discussion in [England et al., 2005; Osinga, 2000 Osinga, , 2003 . We only note that the methods developed in the present work could be combined with existing continuation methods for even better results. This is especially true for methods which exploit the curvature or other differential geometric properties of the manifold. Remark 1.2 [Automatic differentiation and polynomial nonlinearities]. Multiplication of Taylor and Chebyshev series is straight forward thanks to the Cauchy product in the former case and the discrete cosine convolution operation in the later. Then formal series manipulations for polynomial nonlinearities are especially transparent in Chebyshev-Taylor bases. In the present work we are interested in applications coming from celestial mechanics which involve non-polynomial vector fields. In order to simplify matters we exploit methods of automatic differentiation and transform to the polynomial setting, albeit in a higher dimensional phase space. This is discussed in detail in Section 2.3. Remark 1.2 [Automatic differentiation and polynomial nonlinearities]. Multiplication of Taylor and The use of automatic differentiation is a convenience rather than a necessity, as the FFT could be used to evaluate general nonlinearities. In fact, even after automatic differentiation we use the fast cosine transform to evaluate higher order polynomial nonlinearities in the present work. Nevertheless, the use of automatic differentiation in the present work simplifies the implementation details of our algorithms -as all our computations are reduced to Newton's method for large polynomial systems. Automatic differentiation also simplifies a-posteriori error analysis for the method, which when followed to its logical conclusion provides mathematically rigorous validated error bounds for the parameterizations. Review of the parameterization method As already mentioned in the introduction, the parameterization method is much more general than what we actually use in the present work. We refer the reader again to the book [Haro et al., 2016] . In the following section we review some basic notions in the very simple setting of an orientable local manifold associated with one stable/unstable Floquet exponent. Generalities such as multiple stable/unstable exponents, complex conjugate exponents, and non-orientable bundles are discussed in detail in [Castelli et al., 2015] . The methods of the present work apply in these more general setting with only obvious modifications. We focus on the one dimensional case to simplify the exposition. Review of the parameterization method Fig. 2. Geometric meaning of Equation (1): (LEFT) the model space for the stable manifold is the cylinder Review of the parameterization method which we endow with the model vector field t = 1, σ = λσ. Here λ is the stable Floquet exponent of the periodic orbit γ(t). Now if P is any mapping from the cylinder into R M , then g induces a vector field on the image of P by restriction. (RIGHT) Review of the parameterization method The derivative of P pushes forward vector fields defined on the cylinder. The goal of the parameterization method is to find a mapping P so that the push forward of the model dynamics is equal on the image of P to the vector field induced by g. If these vector fields are equal then they have the same dynamics. But the dynamics on the cylinder are completely understood. See Figure 3 . Parameterization of stable/unstable manifolds attached to periodic orbits Let Ω ⊂ R M be an open set and g : Ω → R M be a real analytic vector field. Suppose that γ : R → R M is a T -periodic solution of the first order ordinary differential equatioṅ Parameterization of stable/unstable manifolds attached to periodic orbits that is we assume that γ (t) = g(γ(t)) with γ(t + T ) = γ(t) for all t ∈ R. Suppose also that γ has one stable Floquet exponent λ ∈ R, with λ < 0, so that (by the stable manifold theorem) there exists a two dimensional manifold of solutions which converge exponentially fast to the periodic orbit γ. Let v : R → R M denote the stable normal bundle of γ(t), associated with the exponent λ. We assume that v is an orientable bundle, so that v(t) is T periodic as well. We note that (v, λ) solve the eigenvalue problem Parameterization of stable/unstable manifolds attached to periodic orbits subject to some normalization, perhaps v(t) = 1 for all t ∈ R (though in numerical applications we will choose other normalizations). The goal of the parameterization method is to find a smooth function P : Parameterization of stable/unstable manifolds attached to periodic orbits subject to the first order constraints Parameterization of stable/unstable manifolds attached to periodic orbits and Parameterization of stable/unstable manifolds attached to periodic orbits Then geometric content of Equation (1) is illustrated in Figure 2 , but one easily checks that the image of P is a stable manifold. Parameterization of stable/unstable manifolds attached to periodic orbits Indeed, let P be a smooth solution of Equation (1) subject to the first order constraints. Choose any σ 0 ∈ (−1, 1) and define the curve x : [0, ∞) → R M by Parameterization of stable/unstable manifolds attached to periodic orbits as e λt σ 0 ∈ (−1, 1) for all t ≥ 0. Then x(t) is a solution curve for the differential equation having x(0) = P (0, σ 0 ). Moreover for any t 0 ∈ [0, L], since P is continuous we have that Parameterization of stable/unstable manifolds attached to periodic orbits that is, a point on the image of P accumulates at the periodic orbit γ with asymptotic phase γ(t 0 ). In particular, the image of P is a local stable manifold for γ. One can actually prove more. For example if P solves Equation (1) subject to the first order constraints, then P actually satisfies the flow conjugacy Parameterization of stable/unstable manifolds attached to periodic orbits for all t ≥ 0. Here Φ is the flow generated by the vector field g. The meaning of this flow conjugacy is illustrated in Figure 3 . The proof of the flow conjugacy is given for example in [Castelli et al., 2015] . In the same reference it is shown that solutions of Equation (1) are unique up to the choice of the eigenfunction v(t). (Any constant multiple of v is a parameterization of the stable normal bundle, but up to this choice of scaling the solution is unique). If the solution P exists, it is as regular as g. In this case P is real analytic if g is [Cabré et al., 2003b [Cabré et al., , 2005 . Moreover, in the case of one stable exponent, there exists a choice of scaling small enough that the solution P exists, and is analytic. See [Cabré et al., 2003a [Cabré et al., , 2005 Huguet and de la Llave, 2013] . Formal series and the reduction to homological equations Since P exists and is analytic it makes sense to seek a power series solution Formal series and the reduction to homological equations where σ ∈ (−1, 1) and the functions A α : R → R M are analytic and T -periodic. Plugging the power series into Equation (1), expanding the nonlinearities, and matching like powers leads to equations for the unknown Taylor coefficient functions. Formal series and the reduction to homological equations Example: the Lorenz system in the specific case of the Lorenz field we have that M = 3, and that the vector field g is given by Formal series and the reduction to homological equations Flow conjugacy generated by Equation (1): Suppose that P solves the invariance equation (1). Then the push forward of the model dynamics match the vector field induced on the image of P by g, hence the map P takes orbits on the cylinder to orbits on the image of P , i.e. P conjugates the flow on the cylinder to the flow generated by g and the above diagram commutes. Since the flow on the cylinder is known, we obtain the conjugacy given in Equation (4). In particular, since P maps the zero section in the cylinder to γ, and since all orbits in the cylinder accumulate to the zero section, the image of P is a local stable manifold for γ. Formal series and the reduction to homological equations where β, ρ, σ are positive constants. Suppose that γ(t) = (x(t), y(t), z(t)) is an analytic period-T orbit with stable (or unstable) exponent λ and that v(t) = (v 1 (t), v 2 (t), v 3 (t)) is an analytic T -periodic parameterization of the stable normal bundle. We look for Formal series and the reduction to homological equations and Formal series and the reduction to homological equations equating and matching like powers of s leads to Formal series and the reduction to homological equations define the functions g α by Formal series and the reduction to homological equations We write g α (P ) for short. Now we seek A α (t) the T -periodic solution of the equation Formal series and the reduction to homological equations which we refer to as the homological equation for P . Note that this is a linear inhomogeneous first order ordinary differential equation with periodic coefficients, and that the right hand side is independent of A α . Indeed, g α depends recursively on lower order terms. The Floquet theory guarantees that our homological equation has a unique periodic solution for each α ≥ 2. Then we recursively solve the equations to order N and have the approximate solution Formal series and the reduction to homological equations Remark 2.1 [A-posteriori error analysis]. Truncation error analysis is treated carefully in [Castelli et al., 2017] . Note that the analysis in that reference is independent of the basis used to represent A α (t): only the implementation exploits that these functions are given as Fourier series. Then the methods of the work just cited apply directly to the expansions used in the present work. The key to the analysis in [Castelli et al., 2017] is that the approximation P N have small defect. In the present work we only check the defect numerically, and postpone to an upcoming work more careful analysis of the errors for our Chebyshev-Taylor approximations. Formal series and the reduction to homological equations The calculations above generalizes to any polynomial vector field in the obvious way, and we have that the A α satisfy homological equations of exactly the form given in Equation (6). Only the term Dg(γ(t)) and the form of the recursive functions g α (P ) depend explicitly on the form of the vector field g. In the other examples in the present work we simply write down the correct homological equations and leave the derivations as an exercise for the interested reader. A dynamical perspective on automatic differentiation The term automatic differentiation refers to a whole suite of methods for managing the complexity of problems involving high order derivatives. These methods exploit the fact that differentiation and multiplication are related through the chain rule. For a general introduction and an overview of the literature we refer to [Rall and Corliss, 1996; Bücker and Corliss, 2006] , though in the present work our use of the term is much closer to that of Chapter 2 of [Haro et al., 2016] and also [Jorba and Zou, 2005] . See also Chapter 4.7 of [Knuth, 1998] . The discussion of the literature in this last reference is especially illuminating, though the reference does not use the "automatic differentiation" terminology. A dynamical perspective on automatic differentiation While automatic differentiation can be viewed as a technique for computing transcendental functions of polynomials (where the result should again be expressed as a polynomial), automatic differentiation can also bee seen as a technique transforming non-polynomial into polynomial problems by appending additional polynomial differential equations. The solutions of the appended differential equations are required to give the non-polynomial terms of the original problem. When viewed this way it is possible to use automatic differentiation to solve problems involving Fourier and Chebyshev rather than just Taylor bases. Such extensions are discussed for example in , where automatic differentiation is simply viewed as a change of variables which transforms the given problem to a polynomial problem. A dynamical perspective on automatic differentiation In general a change of variables will disturb the orbit structure of a differential equation, unless the change of variables induces a dynamical (semi)-conjugacy. This issue is considered at length in [Kepley and Mireles James, 2017] , where a dynamical systems interpretation of automatic differentiation is introduced. We adopt this interpretation here as well. So, suppose that U ⊂ R d is an open set and that f : U → R d is a real analytic vector field with non-polynomial nonlinearity. From the perspective of the present work, the goal of automatic differentiation is to find a mapping R : U → R D with D > d and a polynomial vector field F : A dynamical perspective on automatic differentiation is the projection into the lower dimensional space, and that A dynamical perspective on automatic differentiation for all x ∈ U. A dynamical perspective on automatic differentiation This leads to the following observations. A dynamical perspective on automatic differentiation • Since A dynamical perspective on automatic differentiation so that the original field is recovered by projection. A dynamical perspective on automatic differentiation • Equation (7) is an infinitesimal conjugacy, and implies that R maps orbits of f to orbits of F . Moreover it says that graph of R is invariant under the flow generated by F . A dynamical perspective on automatic differentiation • Then when we solve the differential equations given by F with initial conditions on the graph of R, we can recover the orbits of f simply by projecting. A dynamical perspective on automatic differentiation The procedure for choosing R and F is best illustrated through examples, but it is worth noting that R will have the same domain as f . So even though F is polynomial and hence entire, the composition of F • R will have the same singularities as f . A dynamical perspective on automatic differentiation Example: Consider the Kepler problem A dynamical perspective on automatic differentiation We introduce the new variable A dynamical perspective on automatic differentiation and note that away from the origin one has A dynamical perspective on automatic differentiation Then taking A dynamical perspective on automatic differentiation one easily checks the conjugacy Equation (7), and also that the first two components of F • R recover f . Solution curves of the vector field F with initial conditions on the graph of R recover solutions of the Kepler problem as long as they remain on the graph of R. A dynamical perspective on automatic differentiation , and A dynamical perspective on automatic differentiation is a solution of the Kepler problems as long as x(t), y(t), z(t) is on the graph of R. Moreover, the trajectory leaves the graph of R if and only if there is a collision in the the Kepler problem. This example is typical of automatic differentiation for problems in celestial mechanics. A thorough discussion -from the dynamical systems point of view -of automatic differentiation for the circular restricted four body problem is found in [Kepley and Mireles James, 2017] . Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) The periodic orbit γ, its stable/unstable normal bundle v(t), and the higher order Taylor coefficients A α (t) for 2 ≤ α ≤ N are all periodic solutions of non-autonomous differential equations of the forṁ Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) with period τ , and where h : Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) In each case we look for a solutions v : [0, τ ] → R M satisfying the boundary value problem Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) We develop a Chebyshev scheme to solve this class of problems. Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) To begin, break the solution into sub-pieces using the following mesh. Let 0 = t 0 < t 1 < t 2 < . . . < t D = τ , and Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) Thus, each v i (t) is a solution ofu(t) = h(u(t), t). Moreover, at each of the points t i , 1 ≤ i ≤ D − 1, two different pieces are defined and they must agree. Then we impose the boundary conditions Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) for i = 1, and to impose periodicity Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) We want to expand each piece v i (t) using Chebyshev polynomials. To do so, we first rescale the problem to the interval [−1, 1]. First, note that since there are no time dependence in g (the original system for which we compute the periodic orbit) we can translate the time Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) , so that the solution satisfieṡ Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) for all i = 1, . . . , D. Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) For the time varying case, i.e. the case of the bundles and the homological equations, the solution will depend on the lower power α of the coefficient in question. Since the bundles and Taylor coefficients of the parameterization are all periodic with the same period, we choose a fixed mesh for all the problems. That is, the number of subdomains D will be the same at every step. Moreover each L i is a fixed proportion of the global period. So L i = p i L for p i ∈ (0, 1] some given constant. In applications unless otherwise specified we use a uniform mesh, so that Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) where T k is the k−th Chebyshev polynomial, which are defined as follows. Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) Definition 3.1. The Chebyshev polynomials T k : [−1, 1] → R, k = 0, 1, 2, . . . are defined by T 0 (t) = 1 and T 1 (t) = t and the recurrence relation Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) It is well known that these polynomials satisfy T k (cos θ) = cos(kθ), a fact which can be used to prove further results relating the Chebyshev series to results from Fourier analysis. Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) Definition 3.2. We denote the set of unknown Chebyshev coefficients of the full periodic orbit v by Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) So that for every i = 1, 2 . . . , M , A (i) represent the set of coefficients of the i−th piece of v defined on the interval Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) k ∈ R M since each has an image included in R M , which are all expanded using distinct Chebyshev expansions. Finally, a (i,j) k denotes the k−th coefficient of the j−th dimension of v i . Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) To rewrite the system as an operator defined on the set A we integrate (10) from −1 to t and obtain Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) Note that h : [−1, 1] → R M , since it depends on the lower and current term of the expansion of the parameterization which all are defined on [−1, 1] after rescaling time. Thus it can also be expanded using Chebyshev polynomials. We set Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) and substitute both Chebyshev expansions in (12) to get an equation whose only time dependence is in the Chebyshev polynomials themselves/the integral. We use the recurrence formulas for the integral of the Chebyshev polynomials and rewrite the initial condition so that after simplification the Chebyshev coefficients need to satisfy a set of conditions defined in the space of Chebyshev coefficients. That is, for all i = 1, . . . , D, j = 1, . . . , M and k ≥ 0, we define Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) Definition 3.3. We write c k±1 to denote c k±1 := c k+1 − c k−1 . Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) We omit the derivation of the operators f Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) . For further details, and to see why the term k = 0 does not depend on the vector field but only on the boundary condition, we refer to . Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) The boundary condition provides the initial component for the new problem, that is for the case f Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) . So we use the form given in Equation (13) for every subintervals except for the case i = 1, for which we use Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) The operator involves coefficients of the Chebyshev expansion of h, which we need to write in terms of the unknowns A. Since the Chebyshev expansion of a sum (or difference) of two functions f and g will be given by the sum (or difference) of the Chebyshev coefficients of f and g, and since we assume h to be polynomial, the only other case to handle is a product. We use the following Lemma. Lemma 1. If f (t) and g(t) are expanded with Chebyshev series so that Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) Therefore, we can completely rewrite each f (i,j) k without the coefficients c k . To compute an approximation of the solution, we consider a finite dimension approximation Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) Note that the boundary conditions introduce a dependence between each subdomain, hence we solve the equations simultaneously. In the case α ≥ 2 one can directly use Newton's method to findĀ such that F (Ā) ≈ 0, whereĀ is a finite dimensional approximation of the unknowns A with same dimension as F . Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) The first order data: In the case α = 0 or α = 1, we have additional unknowns hence we also need to add phase conditions. For the periodic orbit γ, the period τ is an unknown. For the eigenvalue equation defining the stable/unstable bundles it is the eigenvalue/Floquet exponent which is unknown. In both cases we must balance the equations. Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) For the periodic orbit γ(t), we replace h(v(t), t) by the given vector field g(v(t)). Note that for any s, the time translated solution γ s (t) = γ(t + s) is still a solution of the problem since it is periodic and satisfiesγ s (t) = g(γ s (t)). In order to isolate a solution we impose a Poincaré conditioṅ Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) for some p 0 ∈ R M withṗ 0 := g(p 0 ). This condition translates into a condition on v 1 (−1) after applying the proper change of variable to Chebyshev coefficients aṡ Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) The normal bundle satisfies the eigenvalue probleṁ Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) where Dg(γ(t)) is the periodic matrix given by the derivative of g evaluated at the periodic orbit. It follows that any rescaling kv(t) of the bundle is again a solution associated to the same eigenvalue λ. To isolate a solution we fix v(0) = K, where we are free to choose K. In term of the Chebyshev coefficients Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) For simplicity we truncate this condition to Chebyshev Expansion for Periodic Solutions ofu(t) = h(u(t), t) which still isolates an eigenfunction. Examples We introduce the following operator to simplify the expansion of the operator F in each example. Examples Definition 4.1. Let (j 1 , . . . , j n ) be a set consisting of hyperscript corresponding to component of the solution. That is 1 ≤ j k ≤ M for all k = 1, . . . , n, we denote their Cauchy product of convolutions by Examples αn,kn . Examples Note that the case α = 0 simply returns the convolution product. The example of the Lorenz system All of our numerical computations use the classical parameter values β = 8 3 , ρ = 27 and σ = 10. The operator defining the unknowns A α (t) is given by The example of the Lorenz system with k ≥ 1. The formula for k = 0 is omitted since it is already explicitly given in (13). Note that C involves the lower order terms A β for β ≤ α but we are only solving for A α with the lower order terms fixed. The example of the Lorenz system Only the α = 0 problem is nonlinear in the unknown Chebyshev coefficients and this requires a good initial guess to obtain a periodic orbit. In the present work we use the data from [Viswanath, 2003] as the input for a Newton method. For each α we truncate and consider the finite dimensional problem The example of the Lorenz system The corresponding truncated operator is such that F : R 3mD+1 → R 3mD+1 for α = 0, 1 and F : R 3mD → R 3mD in the remaining cases. The example of the Lorenz system In Figure 5 we computed the stable manifold for two periodic orbits. For these computations we use D = 10 Chebyshev domains, m = 50 Chebyshev coefficients per domain, and N = 50 Taylor nodes. The time rescaling in the operator (i.e. half the period) is L ≈ 1.1530 for the shorter orbit and L ≈ 1.9101 for the longer one. Each color represent a different subdomain of the Chebyshev time decomposition. The example of the Lorenz system In Figure 6 we extended the manifold on the right of figure 5 by integrating backward in time 100 points evenly distributed on the boundary of the parameterization. The orbits were computed by integrating backward in time for t = 0.5. Using the conjugacy relation (4), we have that one would need to integrate for t = 2.5212 to go from σ ≈ 10 −16 to the boundary σ = 1. Thus, allowing the utilization of much smaller time lapse to get a good extension of the attractor for this orbit. The discretization of the continued manifold is very coarse, but the example is only included to show that nearly all the "slow" dynamics of the manifold is captured by the parameterization. Once we start integrating the local manifold orbits move away very rapidly. The example of the Lorenz system A-posteriori error analysis: Since computing a local invariant manifold with the parameterization method does not necessarily involve a small parameter, its not always clear what it means to talk about "the order" of the error. This is especially true when polynomials of high degree are used to approximate the manifold in a large neighborhood of the periodic orbit. In this case it is more natural to use the notion of defect to quantify errors, and we exploit the fact that the parameterization satisfies a conjugacy equation. The example of the Lorenz system That is, since the equation (4) must be satisfied, we define and t 0 is some fixed test time. Sampling points in [0, τ ] with σ = ±1 leads to a useful and numerically accessible estimate. Some heuristics are also helpful. For example we find that choosing N and K so that the norm of the last Taylor component is around machine precision leads to excellent results. A useful norm for making this assessment is The example of the Lorenz system as this involves only sums of the known coefficients. The example of the Lorenz system We test the conjugacy for t 0 = 10 −5 and t 0 = 1 with 200 different starting points evenly distributed on the parameterization of the manifold and then took the average of the resulting errors. The results are displayed in table 1. From this table, one can note that the choice of N and K are such that the conjugacy error remains considerably small for longer period of time while the norm of the last Taylor sequence is not too far beyond machine precision. Short Connecting orbit Following the convention established in we say that there is a short connection from γ 1 to γ 2 if the local parameterization of the unstable manifold of γ 1 intersects the local parameterization of the stable manifold of γ 2 . (Here we are assuming that γ 1,2 come equipped with some choice of local manifold parameterizations. In fact it is more correct to say that the connection is short relative to these fixed parameterizations). In this case one can establish the existence of a connection without the use of any numerical integration. The fact this is possible provides another illustration of the fact that our manifold computations are in some sense not local. In Figure 7 we display the stable manifold of the orbit AB and the unstable manifold of ABB in Lorenz and observe several intersection of the two manifolds. The boundary of the unstable manifold crosses the stable manifold giving a connecting orbit. Let P : [0, 2L 1 ]×[−1, 1] → R 3 denote the parameterization of the local stable manifold of the periodic orbit AB, whose period is 2L 1 . Similarly, let Q : [0, 2L 2 ] × [−1, 1] → R 3 be the parameterization of the unstable manifold of the periodic orbit ABB, whose period is given by 2L 2 . Once the unstable manifold is restricted to one of its boundary circles only 3 unknowns remain. We set x = (θ s , σ s , θ u ) so that the desired intersection is a zero of Short Connecting orbit Therefore S : R 3 → R 3 and it is possible to apply Newton's method to obtain an approximation of the Then the connecting orbit can be computed from those coordinates in the parameter space. Since the value σ s of x is already quite small we used the conjugacy relation (4) with t = 2 and it is already sufficient. For the other half of the connecting orbit, we use (4) again to determine the "time of flight". In this case, the unstable eigenvalue is λ ≈ 0.9947, so one would need to integrate backward in time for t ≈ 34.7246 to obtain σ smaller than 10 −15 . Lets stress this point again: computing this orbit using the linear approximation of the stable/unstable manifolds would require almost 35 units of time integration to cross from the unstable to the stable normal bundles. Using the parameterized manifolds no integration is necessary and the entire orbit is represented "locally". Short Connecting orbit These results are displayed in figure 8 . The green curve is the one obtained using the stable manifold while the trajectory in red correspond to the one using the unstable manifold and the time t = 34.7246. The Circular Restricted Three Body Problem The circular restricted three body problem (CRTBP) describes the evolution of a massless particle moving in the gravitational field of two other massive bodies, called the primaries. It is assumed that the primaries move in circular orbits about their center of mass. In this work we focus on the case that the third body moves in the plane of the primaries. In co-rotating coordinates the equations of motion are The Circular Restricted Three Body Problem Here µ is the mass ratio of the primary bodies. In the rotating frame: Fig. 9 . Schematic representation of the CRTBP: two primary bodies with mass ratio µ move on circular orbits about their center of mass. After changing to a co-rotating frame the more massive primary is located at x = −µ, y = 0 while the smaller primary is fixed at x = 1−µ, y = 0. The CRTBP studies the motion of a third and massless particle moving in the gravitational field of the primaries. The co-linear libration points L j , j = 1, 2, 3 are shown on the x axis. The libration points L 4,5 form equilateral triangles with the primaries but play no role in remainder of the discussion hence are not shown in the figure. (1) the center of mass is at the origin, (2) the motion of the primaries is fixed and they sit on the x−axis at −µ and 1 − µ. This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. The CRTBP is much studied as an example in Hamiltonian dynamics and celestial mechanics going back to the work of Poincaré. It is a useful model of the motion of a satellite or astroid influenced by a two body system such as Earth/Moon, Sun/Earth, or Sun/Jupiter. It is also one of the simplest N -body systems which admits chaotic motions, hence is not integrable. For further discussion of this and more general N −body problems, we refer the reader to the books of [Belbruno, 2004 [Belbruno, , 2007 Gómez et al., 2001; Jorba and Masdemont, 1999; Meyer et al., 2009] . See also [Alessi et al., 2009; Belbruno et al., 2013; Belbruno, 1981; Canalias and Masdemont, 2006; Font et al., 2009; Gómez et al., 2004; Koon et al., 2001; Llibre et al., 1985; Llibre and Simó, 1980; Martínez and Simó, 2014; Masdemont, 2005] . This list of references constitutes only the barest introduction to the relevant literature, but much more complete discussion is found in the books and papers just cited. This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. The system has five relative equilibrium points, also referred to as libration points, and we denote these by L j with j = 1, 2, 3, 4, 5. Three of these lie on the x-axis (the co-linear libration points). These are denoted L 1,2,3 . See Figure 9 for a schematic representation of the problem. This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. Each of the co-linear libration points are of saddle × center stability type. By an application of the Lyapunov center theorem, each of the centers gives rise to one parameter families of periodic orbits. These are known as Lyapunov families. We compute invariant manifolds attached to some Lyapunov orbits below. First we transform to a polynomial problem. This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. Rewriting (18) as a vector fieldu =ĝ(u) giveŝ This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. This system is non-polynomial and we propose a related polynomial system through the use of automatic differentiation. Following Kepley and Mireles James, 2017; Lessard et al., 2016] and the discussion in Section 2.3 we derive the polynomial vector field This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. with the additional (initial condition) constraints This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. That is, we take R defined by This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. where u = (u 1 , u 2 , u 3 , u 4 ). Then the conjugacy of Equation (7) is satisfied. Moreover, one easily checks that π 4 (g(R(u))) =ĝ(u). Of course R is defined only on the complement of the collision set of the CRTBP. Since periodic orbits in the CRTBP occur in one parameter families parameterized by energy, we fix L and look for a periodic orbit with this half period. Moreover, we use the well known reversible symmetry of the problem to formulate a different boundary condition that will still provide periodic solutions. For example, orbits in a Lyapunov family of one of the co-linear libration points have no velocity in the x direction when they cross the x axis -and the time between the two crossing of the x axis are separated by exactly the half orbit. So that a periodic solution γ starting on the x axis with frequency 2L will satisfy γ 2 (0) = 0, γ 3 (0) = 0, γ 2 (2L) = 0 and γ 3 (2L) = 0. This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. Remark 4.1. Note that the last two conditions of (21) could also be γ 2 (L) = γ 3 (L) = 0. This would allow us to compute only the half orbit, the symmetry of the problem providing the solution for the second half of the trajectory. The use of Chebyshev expansion would allow such a choice. However, since the periodic orbit is not the final product of our computations, but only an input into the higher order equations we compute full orbits in this work. This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. For the remaining two conditions, we simply rewrite the initial condition on v 5 and v 6 after imposing the symmetry, so that This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. Due to the choice of the boundary condition, time translation of the solution does not satisfy this system. Thus we drop the Poincaré condition previously given and obtain a system that is still fully determined and with isolated solutions (since we don't solve for the period we dispense with one scalar equation). Two possibilities arise from that remark, one could fix L to a given value and still find an orbit as previously mentioned. The other choice is to use L as a variable and obtain a fully determined system by adding a condition in which the energy level is fixed to a chosen constant. Such a choice is necessary for example if we want to compute heteroclinic connecting orbits. We summarize the discussion in the following lemma. This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. Lemma 2. Let v 1 (t), v 2 (t), v 3 (t), v 4 (t) be periodic function with same period ω and such thatv 1 = v 2 , This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. and This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. Then v 5 (t) and v 6 (t) are periodic with period ω. This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. Proof. We first note that This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. is a solution of (22). But by unicity of the solution it follows that v 5 (t) = w 5 (t). Moreover, we have that w 5 (t) is periodic with period ω since v 1 (t) and v 3 (t) are periodic with period ω. Thus v 5 (t) is periodic with period ω, as desired. For v 6 (t), the proof is similar using the fact that This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. is a solution of (23). This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. To fix the energy we have to use an integral of the CRTBP, namely This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. known as the Jacobi integral. In our system of coordinates, this is This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. Since E is constant along any orbit of the system, we evaluate it at the endpoint of the first piece of the Chebyshev decomposition. This leads to a new phase condition that can replace the one previously exhibited at equation (16). In terms of the Chebyshev coefficients it is given by This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. We now focus on expanding the operator we need to solve in order to obtain the coefficients of the Chebyshev expansion of the i−th component of the solution. In the case k ≥ 1, each case is given by This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. (1, 6, 6, 6) This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. α,k±1 (6, 6, 6) , This choice of coordinates introduces the Coriolis effect, that is the coordinates are non-inertial. α,k±1 (3, 6, 6,
{'paper_id': '4945864', 'title': 'Chebyshev-Taylor parameterization of stable/unstable manifolds for periodic orbits: implementation and applications', 's2_url': 'https://api.semanticscholar.org/CorpusID:4945864', 'source': 'S2ORC'}