Invariant Manifold Dynamics

Transcription

1 Invariant Manifold Dynamics Advanced Aspects of Spacecraft Control and Mission Design AstroNet-II Summer Training School and Annual Meeting G. Gómez 1 J.J. Masdemont 2 1 IEEC & Dept de Matemàtica Aplicada i Anàlisi. Universitat de Barcelona 2 IEEC & Dept de Matemàtica Aplicada I, Universitat Politècnica de Catalunya. Glasgow, 3rd - 7th June 2013 Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

2 Contents-1 1 Introduction and motivation 2 Discrete and continuous dynamical systems Basic definitions Autonomous and non-autonomous systems Examples Orbit generation 3 Invariant objects: fixed points, periodic orbits, tori,... Fixed points Periodic orbits Cross sections and Poincarà c maps The Hamiltonian case Numerical computations of fixed points and periodic orbits The multiple shooting method Numerical computation of invariant 2D tori Numerical approximation of invariant manifolds Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

3 Contents-2 4 The restricted n and three body problems The n- body problem The n- body problem as a perturbation of Kepler s problem The restricted three body problem Zero velocity surfaces and curves Equilibrium points The n- body problem as a perturbation of the RTBP Dynamical substitutes of the equilibrium points 5 Lindstedt-Poincaré methods The mathematical pendulum Lissajous, halo and quasi-halo orbits 6 Normal form around the equilibrium points Reduction to the central manifold Taxonomy of libration point orbits Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

4 Contents-3 7 Applications Astrodynamical applications Transfer between LPO s: LOEWE Shoot the Moon Transfer to LPO s Station keeping Astronomical applications Moon s craters Mass transport in the Solar System Hopping comets Galaxy models 8 Exercises Computation of a family of periodic orbits Computation of a homoclinic or heteroclinic orbit Computation of a LEO LPO trasfer orbit Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

5 References References G. Gómez, J. Llibre, R. Martínez, and C. Simó. Dynamics and Mission Design Near Libration Points. Vol. I Fundamentals: The Case of Collinear Libration Points; Vol. II Fundamentals: The Case of Triangular Libration Points. World Scientific G. Gómez, À. Jorba, J.J. Masdemont, and C. Simó. Dynamics and Mission Design Near Libration Points. Vol. III Advanced Methods for Collinear Points; Vol. IV Advanced Methods for Triangular Points. World Scientific W.S. Koon, M. W. Lo, J.E. Marsden, and S.D. Ross. Dynamical Systems, the Three-Body Problem, and Space Mission Design. Marsden Books, Online copy: www2.esm.vt.edu/ sdross/books/kolomaro_dmissionbk.pdf J.J. Masdemont and J.M. Mondelo. Notes for the Numerical and Analytical Techniques Lectures. Advanced Topics in Astrodynamics, Barcelona Online copy: K.R. Meyer, G.R. Hall, and D. Offin. Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. Springer, Applied Mathematical Sciences Vol. 90, V. Szebehely. Theory of Orbits. The Restricted Problem of Three Bodies. Academic Press, Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

6 Introduction and motivation Introduction and motivation Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

7 The Soho mission Introduction and motivation Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

8 The Genesis mission Introduction and motivation Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

9 The Artemis mission Introduction and motivation Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

10 Introduction and motivation Properties of Libration Point Orbits 1 They are easy and inexpensive to reach from Earth 2 They provide good observation sites of the Sun 3 The libration orbits around the L 1 and L 2 points of the Sun Earth system always remain close to the Earth, with a near-constant communications geometry 4 Orbits around the L 2 point of the Sun Earth system provide a constant geometry for observation with half of the entire celestial sphere available at all times, since the Sun, Earth and Moon are always behind the spacecraft 5 For missions with heat sensitive instruments, the L 2 environment of the Sun Earth system is highly favourable for non-cryogenic missions requiring great thermal stability, suitable for highly precise visible light telescopes Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

11 Introduction and motivation Properties of Libration Point Orbits 6 The libration orbits around the L 2 point of the Earth Moon system, can be used to establish a permanent communications link between the Earth and the hidden part of the Moon 7 Suitable combinations of libration orbits of the ES and EM systems can provide ballistic planetary captures, such as for the one used by the Hiten mission 8 The libration point orbits provide Earth transfer and return trajectories, such as the one used for the Genesis mission 9 The libration point orbits provide interplanetary transport which can be exploited in the Jovian and Saturn systems to design a low energy cost mission to tour several of their moons (Petit Grand Tour Mission / JIMO / Laplace / EJSM) 10 Formation flight with a rigid shape is possible using libration point orbits Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

12 Introduction and motivation Main Topics related to LPO dynamics 1 The phase space arond the libration points in the CRTBP 2 Analytical and numerical computation of libration point orbits in the CRTBP 3 Computation of hyperbolic invariant manifolds 4 Determination of dynamical substitutes of libration points and LPO in N-body models seen as a perturbation of the CRTBP 5 Computation of nominal libration point orbits in accurate solar system models 6 Station-keeping strategies 7 Transfers from the Earth to libration point orbits 8 Transfers between libration point orbits 9 Low energy transfers and weak stability boundaries 10 Formation flight using libration point orbits Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

13 Discrete and continuous dynamical systems Discrete and continuous dynamical systems Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

14 Discrete and continuous dynamical systems Discrete and continuous dynamical systems Dynamical systems are systems that evolve with time Time may be thought as: Continuous (clock), mathematically t R Continuous dynamical systems Discrete (days, months,... ), mathematically t Z Discrete dynamical systems Continuous dynamical systems are given by a system of Ordinary Differential Equations (ODE). Discrete dynamical systems are given by an smooth, 1 1 map. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

15 Discrete and continuous dynamical systems Continuous dynamical systems Defined by a system of autonomous (i.e. time-independent) ODE. ẋ 1 = f 1 (x 1, x 2,..., x n ), ẋ 2 = f 2 (x 1, x 2,..., x n ),... ẋ n = f n (x 1, x 2,..., x n ). In short, with x = ẋ = f (x), for x R n, f : R n R n. x 1. x n R n, f (x) = f 1 (x). f n (x) R n. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

16 Discrete and continuous dynamical systems Continuous dynamical systems An autonomous system of ODE ẋ = f (x), for x R n, f : R n R n, allows to define the flow (general solution of the system) φ t (x), t R, x R n, that satisfies { d dt φ t(x) = f (φ t (x)), φ 0 (x) = x, and φ t+s (x) = φ s (φ t x)). Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

17 Discrete and continuous dynamical systems Variational equations We often need the derivative of the flow with respect to initial conditions; it will be denoted by Dφ t (x). Dφ t (x) can be computed by solving the variational equations: dx = f (t, x), dt da dt = f (t, x)a, x where x is an n-dimensional vector and A is a n n matrix. If x(t) and A(t) are solutions of the previous system with x(t 0 ) = x 0, A(t 0 ) = I n, then Dφ t t 0 (x 0 ) = A(t). Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

18 Discrete and continuous dynamical systems Variational equations The variational equations can be written as a system of n + n 2 ODE as ẋ 1 = f 1 (t, x 1,..., x n ),. ẋ n = f n (t, x 1,..., x n ), ȧ 1,1 =. ȧ n,1 = n k=1 n k=1 f 1 x k (t, x)a k,1,... ȧ 1,n = f n x k (t, x)a k,1,... ȧ n,n =. n k=1 n k=1 f 1 x k (t, x)a k,n, f n x k (t, x)a k,n. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

19 Discrete and continuous dynamical systems Some terminology Given an initial condition x 0, its orbit is {φ t (x 0 )} t R. A fixed point of a continuous dynamical system is a point x 0 whose orbit is φ t (x 0 ) = x 0, t R, this can only happen if f (x 0 ) = 0. An orbit {φ t (x 0 )} t R is said to be periodic if there is T > 0 such that φ T (x 0 ) = x 0, then T is said to be its period. φ t (x 0 ) x 0, for 0 < t < T, An invariant set A is a set of initial conditions invariant by the flow φ t (x 0 ) A, x 0 A. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

20 Discrete and continuous dynamical systems Example: the restricted three body problem The restricted three body problem is an autonomous Hamiltonian system ẋ = H p x, ẏ = H p y, ż = H p z, p x = H x, p y = H y, p z = H z, with and H(x, y, z, p z, p y, p z ) = 1 2 (p2 x + p 2 y + p 2 z) xp y + yp x 1 µ r 1 µ r 2, r 1 = (x µ) 2 + y 2 + z 2, r 2 = (x µ + 1) 2 + y 2 + z 2. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

21 Discrete and continuous dynamical systems Non-autonomous system of ODE A system of non-autonomous ODE is a system of ODE that depends on time ẋ = f (t, x), t R, x R n, f : R R n R n. In order to consider the associated autonomous dynamical system define y = (y 0, y 1,...y n ) T := (t, x) T, and consider in short ẏ 0 = 1, ẏ 1 = g 1 (y 0, y 1,..., y n ),... ẏ n = g n (y 0, y 1,..., y n ). ẏ = g(y). An non-autonomous system of ODE has never fixed points (g 0 = 1, so g 0). Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

22 Discrete and continuous dynamical systems Example: the bicircular four body problem with ẋ = H p x, p x = H x,... H = 1 2 (p2 x + p 2 y + p 2 z) xp y + yp x 1 µ ((x µ) 2 + y 2 + z 2 ) µ 1/2 ((x µ + 1) 2 + y 2 + z 2 ) 1/2 m P ((x a P cos θ) 2 + (y + a P sin θ) 2 + z 2 ) 1/2 m P ap 2 (y sin θ x cos θ), with θ = ω P t + θ 0, where ω P is the mean angular velocity of the third primary, m P its mass and a P the distance from barycentre of the first two primaries to the third one. We can autonomize it by introducing the new variable θ. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

23 Discrete and continuous dynamical systems Discrete dynamical systems Discrete dynamical systems are defined by dipheomorphisms (smooth 1 1 maps) F : R n R n x F (x) We denote by F 1 the inverse map of F, and use superscript notation for the composition of map F 0 (x) = x, F 1 (x) = F (x), F 2 (x) = F (F (x)), F 2 (x) = F 1 (F 1 (x)), F 3 (x) = F (F (F (x))), F 3 (x) = F 1 (F 1 (F 1 (x))),.. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

24 Some concepts Discrete and continuous dynamical systems Given an initial condition, its related orbit is the set {F i (x)} i Z, that is {..., F 3 (x), F 2 (x), F 1 (x), F 0 (x), F 1 (x), F 2 (x),...}. A fixed point is an initial condition such that its orbit is itself F (x 0 ) = x 0. An n-periodic point is an initial condition x 0 such that F n (x 0 ) = x 0, F i (x 0 ) x 0, i = 1,..., n 1. A set of initial conditions A R n is said to be an invariant set if F n (x) A, n Z, x A. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

25 Discrete and continuous dynamical systems Example: the standard map for x, y T = R \ [0, 2π] ( x F : y ) ( x + a sin(x + y) x + y ) Iterates of the standard map for a = 0.7 Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

26 Discrete and continuous dynamical systems Example: the time-period flow of the bicircular problem Denote by ẏ = F (y), the autonomized differential equations of the bicircular four body problem, where y = (θ, x, y, z, p x, p y, p z ) T = (θ, x) T, and by Φ t (y) its flow d dt Φ t(y) = F (Φ t (y)), Φ 0 (y) = y, y R 7. Then Φ t (y) = Φ t ( θ x ) = ( θ + ωp t Φ θ t (x) ), here Φ θ t (x) is the flow from time 0 to time t of the BCP with phase θ. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

27 Discrete and continuous dynamical systems Example: the time-period flow of the bicircular problem We can consider the time 2π/ω P flow for the BCP equations G(y) = Φ 2π/ωP (y), this is ( θ G x ) = θ + 2π ( ω P ω P = Φ θ 2π/ω P (x) θ Φ θ 2π/ω P (x) ). Since the coordinate θ is invariant, we can skip it and define F θ (x) := Φ θ 2π/ω P (x). Note that a fixed point of F θ is a periodic orbit of the BCP with initial phase θ. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

28 Discrete and continuous dynamical systems Orbit generation in a dynamical system Assume we are given: A dynamical system, either continuous or discrete. An initial condition, x 0 R n. We want to numerically generate the orbit of x 0. In the discrete case, we just have to write a routine that implements the map and iterate it. In the continuous case: we need to use numerical methods for systems for ODE. Some families of methods for nonstiff ODEs are Runge-Kutta, Taylor or extrapolation ones. One popular method is the RKF78 method by Fehlberg (1968), developed under contract by NASA for the Apollo missions. We will just comment the use of a RKF routine as a black box. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

29 Discrete and continuous dynamical systems Use of a black-box RKF routine Consider a system of n possibly non-autonomous ODE ẋ = f (t, x), and denote by φ t t 0 (x) its flow from time t 0 to time t d dt φt t 0 (x) = f (t, φ t t 0 (x)), φ t t 0 (x) = x, x R n. Given t 0 R, x 0 R n, h 0 R (small), and a tolerance tol, a RKF routine returns t 1 R, x 1 R n, h 1 R verifying: (a) x 1 φ t 1 t0 (x 0 ) < tol, (b) t 1 is as close to t 0 + h 0 as possible, (c) h 1 is the recommended stepsize for the next call. We will denote a call to such an RKF routine as (t 1, x 1, h 1 ) = RKFstep(t 0, x 0, h 0, f, tol). Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

30 Invariant objects: fixed points, periodic orbits, tori,... Invariant objects: fixed points, periodic orbits, tori,... Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

31 Fixed points Invariant objects: fixed points, periodic orbits, tori,... Consider a differential system ẋ = f (x), x R n Let the general solution of this system be φ t (x). An equilibrium solution φ t (y) is such that φ t (y) = y for all t. Obviously φ t (y) is an equilibrium solution if and only if f (y) = 0. The eigenvalues of f (y) x, are called the characteristic exponents of the equilibrium. If f (y)/ x is nonsingular, or equivalently the exponents are all nonzero, the equilibrium point is called elementary. By the implicit function theorem it follows that elementary equilibrium points are isolated. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

32 Invariant objects: fixed points, periodic orbits, tori,... Stability of fixed points The analysis of stability, bifurcations, etc of equilibrium points starts with an analysis of the linearized equations. For this reason ones shifts the equilibrium point to the origin, and the differential equation is rewritten ẋ = f (x) = Ax + g(x), where f (0) A =, g(x) = f (x) Ax x so g(0) g(0) = 0 and = 0. x As has been said, the eigenvalues of A are called the characteristic exponents. The reason of this is that the linearized equations (g(x) = 0) have solutions which contain terms like where λ is an eigenvalue of A. exp(λt), Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

33 Invariant objects: fixed points, periodic orbits, tori,... Periodic orbits As we have already said, a solution φ t (y) is said to be a periodic solution with period T > 0 (equilibrium solutions are not considered periodic) if φ T (y) = y, φ t (y) y, if 0 < t < T. Let φ t (y) be a periodic solution. The matrix φ T (y), x solution of the variational equations at t = T, is called the monodromy matrix, and its eigenvalues are called the characteristic multipliers of the periodic solution. The reason the eigenvalues λ i of the monodromy matrix are called multipliers is that the linearized Poincaré map takes an eigenvector to λ i times itself. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

34 Periodic orbits Invariant objects: fixed points, periodic orbits, tori,... It is tempting to use the implicit function theorem to find a condition for local uniqueness of a periodic solution. Since for a periodic orbit φ T (y) = y, to be able to do that the matrix φ T (y) x I, would have to be nonsingular, or +1 would not be a multiplier. In fact: f (y) is always an eigenvector of the monodromy matrix corresponding to the eigenvalue +1. Proof: Differentiating φ τ (φ t (y)) = φ t+τ (y) with respect to x, and setting t = 0 and τ = T gives (using the notation: φ t (y) := φ(t, y)) φ x (T, y) φ(0, y) = φ(t, y), φ (T, y)f (y) = f (y). x Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

35 Invariant objects: fixed points, periodic orbits, tori,... Cross sections and Poincaré maps Because of the above property one introduces a cross section. Let φ t (y) be a periodic solution. A cross section to the periodic solution, or simply a section, is a hyperplane Σ of codimension 1 through y and transverse to f (y). For example, Σ would be the hyperplane {x : a T (x y) = 0}, where a is a constant vector with a T f (y) 0. The periodic solutions starts on the section and, after a time T, returns to it. By the continuity of solutions with respect to initial conditions, nearby solutions do the same. If y is close to y on Σ, there is a time τ(y ) close to T such that φ τ(y )(y ) is on Σ. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

36 Invariant objects: fixed points, periodic orbits, tori,... Cross sections and Poincaré maps The Poincaré map is defined as the map P : N Σ y φ τ(y )(y ), which is a map of a neighborhood N of y in Σ into Σ. If this neighborhood is small enough the Poincaré map is smooth. The periodic solution now appears as a fixed point of P. Indeed, any fixed point, z, of P is the initial condition for a periodic solution of period τ(z). A point z N such that P k (z) = z for some integer k > 0 is called a periodic point of P of period k. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

37 Invariant objects: fixed points, periodic orbits, tori,... Poincaré maps: numerical computation We are given: A Poincaré map: P(x) = φ τ(x) (x). A surface of section: Σ = {x g(x) = 0}, to be traversed from {g(x) < 0} to {g(x) > 0} or vice versa. We can numerically evaluate P(x) by the following algorithm: input: do: output: t, y. x, g, f, tol t := 0, y := x, h := tol while (g(y) 0) (t, y, h) := RKFstep(t, y, h, f, tol) while (g(y) < 0) (t, y, h) := RKFstep(t, y, h, f, tol) while ( g(y) < tol) δ := g(y)/(dg(y)f (y)) (t, y, h) := RKFflow(t, t + δ, y, h, f, tol) At the end of the algorithm, y = P(x) and t = τ(x). Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

38 Poincaré maps Invariant objects: fixed points, periodic orbits, tori,... Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

39 Poincaré maps Invariant objects: fixed points, periodic orbits, tori,... Poincaré maps can display local properties around a reference orbit Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

40 Poincaré maps Invariant objects: fixed points, periodic orbits, tori,... Poincaré maps can display the global structure of the phase space Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

41 Invariant objects: fixed points, periodic orbits, tori,... Poincaré maps: differential The differential of P(x) is DP(x) = d dx φ τ(x)(x) = d dτ φ τ(x)(x) + Dφ τ(x) (x) = f (P(x))Dτ(x) + Dφ τ(x) (x), so we need the differential of the time-return map Dτ(x). From g(p(x)) = g(φ τ(x) (x)) = 0, we get Dτ(x) = Dg(P(x))Dφ τ(x)(x) Dg(P(x))f (P(x)). DP(x) = f (P(x)) Dg(P(x))Dφ τ(x)(x) Dg(P(x))f (P(x)) + Dφ τ(x) (x). Note that Dφ τ(x) (x) is given by the variational equations Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

42 Invariant objects: fixed points, periodic orbits, tori,... Fixed points of Poincaré maps If the multipliers of the periodic solution are 1, λ 2,..., λ n, then the multipliers of the corresponding fixed point of the Poincaré map are λ 2,..., λ n. A periodic orbit of period T is isolated if there is a neighborhood L of it with no other periodic orbits in L with period near to T. A periodic orbit is isolated if and only if the corresponding fixed point of the Poincaré map is an isolated fixed point. There may be periodic solutions of much higher period near an isolated periodic orbit. A periodic orbit is called elementary if none of its nontrivial multipliers are +1. By the implicit function theorem it follows that elementary fixed points and elementary periodic orbits are isolated. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

43 Invariant objects: fixed points, periodic orbits, tori,... The Hamiltonian case We have already seen that the monodromy matrix of a periodic solution has +1 as a multiplier. If the differential equations of motion are Hamiltonian where ( q z = p then the monodromy matrix would be simplectic ) ( 0 I, J = I 0 ż = J H(z), φ T (z ), z ), H = ( φt (z ) T ( ) φt (z ) ) J = J. z z H/ z 1. H/ z 2n, Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

44 Invariant objects: fixed points, periodic orbits, tori,... The Hamiltonian case The simplectic character of the monodromy matrix implies that the algebraic multiplicity of the eigenvalue +1 would be even and so at least equal to 2. Actually, this is simply due to the fact that an autonomous Hamiltonian system has a first integral. In fact if φ t (z ) is a periodic solution of the Hamiltonian system, the row vector H(z )/ z is a left eigenvector of the monodromy matrix corresponding to the eigenvalue +1. There is a good geometric reason for this property: the periodic solution lies in a (2n 1) dimensional level set of the Hamiltonian, and typically in nearby level sets of the Hamiltonian there is a periodic orbit. So, in Hamiltonian systems, periodic orbits are not isolated. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

45 Invariant objects: fixed points, periodic orbits, tori,... The Poincaré map in the Hamiltonian case Consider the Poincaré map P : N Σ, where N is a neighborhood of z in Σ (initial conditions of a periodic orbit of the system). We can apply the Flow Box theorem, so there are coordinates (u 1, u 2,..., u 2n ) such that the system of differential equations, in these new coordinates becomes: and H(u) = u 2. u 1 = 1, u 2 = 0,..., u 2n = 0, In these coordinates, we may take Σ to be u 1 = 0. Since u 2 is the integral in these coordinates, P maps the level sets u 2 = constant into themselves; so, we can ignore the u 2 component of P. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

46 Invariant objects: fixed points, periodic orbits, tori,... The Poincaré map in the Hamiltonian case Let u 2 = h, Σ h be the intersection of Σ and the level set H = h and let y 1 = u 3,...,y m 2 = u m be coordinates in Σ h. Here h is considered as a parameter (the value of the integral). In these coordinates the Poincaré map P is a function of y and the parameter h. So P(h, y) = (h, Q(h, y)), where for fixed h, Q(h, ) is a mapping of a neighborhood N h of the origin in Σ h into Σ h. Q is called the Poincaré map in an integral surface (it is a symplectic map). Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

47 Invariant objects: fixed points, periodic orbits, tori,... Cilinders of periodic orbits The eigenvalues of Q(0, 0), y are called the nontrivial multipliers of the fixed point in the integral surface. If the multipliers of a periodic solution of a Hamiltonian system are 1, 1, λ 3,..., λ 2n, then the multipliers of the fixed point in the integral surface are λ 3,..., λ 2n. If none of the nontrivial multipliers are 1, then we will say that that the periodic solution is elementary. If we apply the implicit function theorem to Q(h, y) y = 0, we can proof that an elementary periodic orbit lies in a smooth cylinder of periodic solutions parametrized by the energy H. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

48 Invariant objects: fixed points, periodic orbits, tori,... The Lyapunov Center Theorem Theorem (The Lyapunov Center Theorem) Assume that a Hamiltonian system has an equilibrium point with exponents ± 1ω, λ 3,..., λ n, λ 3,..., λ n, where 1ω 0 is pure imaginary. If λ j 1ω / Z, j = 3, 4,..., n, then there exist a one parameter family of periodic orbits emanating from the equilibrium point. Moreover, when approaching the equilibrium point along the family, the periods tend to 2π/ω and the nontrivial multipliers tend to exp(2πλ j /ω), j = 3,..., n. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

49 Invariant objects: fixed points, periodic orbits, tori,... Numerical computations Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

50 Invariant objects: fixed points, periodic orbits, tori,... Numerical computation of fixed points We have two cases to consider. In the case of a flow ẋ = f (x), we look for p such that G(p) := f (p) = 0. In the case of a map, x F (x), we look for p such that G(p) := F (p) p = 0. In both cases the problem is reduced to look for a zero of a function G : R R. We can use Newton s method in several variables Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

51 Invariant objects: fixed points, periodic orbits, tori,... Numerical computation of fixed points We can implement Newton s method to look for a zero of G : R R, as follows: input: p 0, G, tol, maxit do: p := p 0 for it from 1 to maxit do if ( G(p) < tol) return p solve DG(p) p = G(p) for p p := p p error (maxit exceeded) output: p (if OK) Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

52 Invariant objects: fixed points, periodic orbits, tori,... Linear behavior around a fixed point: flows Consider a flow ẋ = f (x), with a fixed point p f (p) = 0. The Taylor expansion of f around p up to order one is f (x) = f (p) + Df (p)(x p) + O( x p 2 ) := A(x p) + O( x p 2 ), so that the linearized flow around p is ẋ = A(x p). The eigenvalues of A are the exponents of the fixed point p. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

53 Invariant objects: fixed points, periodic orbits, tori,... Linear behavior around a fixed point: flows Assume λ Spec(A), λ 0, Av = λv, v 0. If λ R, consider ϕ(t) = p + e λt v. Then: ϕ(t) satisfies the system of ODE of the linearized flow, ϕ (t) = λe λt v = e λt (λv) = e λt (Av) = A(e λt v) = A(ϕ(t) p). If λ > 0, then ϕ(t) p when t +, so that it gives a stable manifold of the linearized flow. If λ < 0, then ϕ(t) p when t, so that it gives a unstable manifold of the linearized flow. The extistence of a stable or unstable manifold of the full (nonlinear) dynamical system is ensured by the stable manifold theorem for flows. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

54 Invariant objects: fixed points, periodic orbits, tori,... Linear behavior around a fixed point: flows Assume λ Spec(A), λ 0, Av = λv, v 0. If λ = iω, for ω R, let v 1 + iv 2 be a corresponding eigenvector, with v 1, v 2 R n. Then Therefore, if we define we have Av 1 + iav 2 = A(v 1 + iv 2 ) = iωv 1 ωv 2. ϕ γ (t) = p + γ ((cos ωt)v 1 + (sin ωt)v 2 ), ϕ γ(t) = A(ϕ γ (t) p). so that ϕ γ (t) satifies the linearized system of ODE and gives a one-parametric family of 2π/ω-periodic solutions of the linear system. The Lyapunov center theorem guaranties, under suitable non-resonance conditions, the existence of a family of periodic orbits for the full nonlinear system, with limiting period 2π/ω. The case λ = a + iω, for a, ω R, a, ω 0 is not possible for a Hamiltonian system and will be not considered. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

55 Invariant objects: fixed points, periodic orbits, tori,... Linear behavior around a fixed point: maps Consider a discrete dynamical system given by x F (x), with a fixed point p. The Taylor expansion of F around p up to order one is F (x) = F (p) + DF (p)(x p) + O( x p 2, so, if A := DF (p), the linearized dynamical system around p is x L F (x) := p + A(x p). The eigenvalues of A are also called the multipliers of the fixed point p. For a symplectic map, λ SpecA = 1/λ SpecA. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

56 Invariant objects: fixed points, periodic orbits, tori,... Linear behavior around a fixed point: maps Assume λ Spec(A), λ 0, Av = λv, v 0. If λ R, λ 1, consider ϕ(ξ) = p + ξv. Then, L F (ϕ(ξ)) = ϕ(λξ), so that {ϕ(ξ)} ξ R parametrizes an invariant set of the linear approximation. Moreover, λ < 1 then L n F (ϕ(ξ)) = ϕ(λ n ξ) ϕ(0) = p, when n +. Therefore, ϕ parametrizes a stable manifold of p for the linearized map. λ > 1 then L n F (ϕ(ξ)) = ϕ(λ n ξ) ϕ(0) = p, when n. Therefore, ϕ parametrizes a unstable manifold of p for the linearized map. The extistence of a stable or unstable manifold of the full (nonlinear) dynamical system is ensured by the stable manifold theorem for maps. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

57 Invariant objects: fixed points, periodic orbits, tori,... Linear behavior around a fixed point: maps Assume λ Spec(A), λ 0, Av = λv, v 0. If λ C, λ = 1, assume λ = cos ρ + i sin ρ and let v 1 + iv 2 be an associated eigenvector We define ϕ(ξ) = p + γ ((cos ξ)v 1 (sin ξ)v 2 ). Then, if we apply the linearized map to ϕ(ξ), we get ϕ(ξ) ϕ(ξ + ρ). so ϕ parametrizes an invariant closed curve of the linear flow We therefore have a one-parametric family (with parameter γ) of curves, with rotation number ρ, invariant by the linearized map Under generic nonr-degeneracy conditions, there is a one-parameter family of invariant curves for the full (nonlinear) map with varying rotation number, that converges to ρ as we approach the fixed point Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

58 Invariant objects: fixed points, periodic orbits, tori,... Linear behavior around a periodic orbit Consider an autonomous Hamiltonian system An initial condition x 0 of a T periodic orbit is also a fixed point of φ T Although this is not useful to numerically find the p.o., as we have already seen, it is useful to study the linear behaviour around it Consider the RTBP (or any autonomous Hamiltonian system), and let x 0 be an initial condition of a T periodic orbit. Then, its monodromy matrix, has 1 as double eigenvalue. M := Dφ T (x 0 ) Moreover, M = Dφ T (x 0 ) is a symplectic matrix, which implies that: if λ is an eigenvalue of M, then 1/λ is also eigenvalue. Then, We will assume that λ i λ 1 i Spec(M) = {1, 1, λ 1, λ 1 1, λ 2, λ 1 2 } Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

59 Invariant objects: fixed points, periodic orbits, tori,... Linear behavior around a periodic orbit If Spec(M) = {1, 1, λ 1, λ 1 1, λ 2, λ 1 2 }, the linear behaviour around the p.o. is better studied in terms of its stability parameters, s 1 and s 2, which are defined as It is easy to check that s 1 = λ 1 + 1/λ 1, s 2 = λ 2 + 1/λ 2 s i R, s i > 2 λ i R \ {±1} s i R, s i 2 λ i C, λ i = 1 s i C \ R, λ i C \ R, λ i 1 Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

60 Invariant objects: fixed points, periodic orbits, tori,... Linear behavior around a periodic orbit In an autonomous Hamiltonian system, let x 0 be such that φ T (x 0 ) = x 0, with M = Dφ T (x 0 ), Spec(M) = {1, 1, λ 1, λ 1 1, λ 2, λ 1 2 } and s 1 = λ 1 + 1/λ 1, s 2 = λ 2 + 1/λ 2 Hiperbolic case: s i R, s i > 2 λ i R \ {±1} There is a stable manifold of the fixed point of φ T, tangent to the λ i eigendirection of M at x 0. There is a unstable manifold of the fixed point of φ T, tangent to the λ 1 i eigendirection of M at x 0. In terms of the periodic orbit: There is a stable manifold of the periodic orbit whose section through the λ i, λ 1 i eigenplane is tangent to the λ i eigendirection. There is an unstable manifold of the periodic orbit whose section through the λ i, λ 1 i eigenplane is tangent to the λ 1 i eigendirection. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

61 Invariant objects: fixed points, periodic orbits, tori,... Linear behavior around a periodic orbit Elliptic case: s i R, s i 2 λ i C, λ i = 1. Let λ i = cos ρ + i sin ρ ( s i = 2 cos ρ), and let v 0 be such that Mv = λ i v = λ i (v 1 + iv 2 ) There is a continuous, one-parametric family of closed curves invariant by the linearization of φ T around x 0 in the {x 0 + α 1 v 1 + α 2 v 2 } α1,α 2 R plane, with rotation number ρ Under generic non-degeneracy conditions, there is a Cantorian family of invariant curves around x 0, with limiting rotation number ρ. When transported by the flow, these invariant curves generate 2 dimensional invariant tori. Rational values (times 2π) for ρ also give rise to bifurcated p.o., with period 2π/ρ. The particular rational values ρ = 2π (s i = 2) and ρ = π (s i = 2), are known as the parabolic case. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

62 Invariant objects: fixed points, periodic orbits, tori,... Numerical computation of a family of periodic orbits In a rotating, barycentric, dimensionless coordinate system, the differential equations of motion for the circular three dimensional restricted problem can be written as Ẍ 2Ẏ = Ω X, Ÿ + 2Ẋ = Ω Y, Z = Ω Z, with Ω = 1 2 (X 2 + Y 2 ) + (1 µ)r µr 1 2, and { r 2 1 = (X µ) 2 + Y 2 + Z 2, r2 2 = (X µ + 1) 2 + Y 2 + Z 2. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

63 Invariant objects: fixed points, periodic orbits, tori,... Numerical computation of a family of periodic orbits From the inspection of the terms appearing in the differential equations it follows that if for a fixed value of the mass parameter µ, (x(t), y(t), z(t)) is a particular solution, then (x( t), y( t), ±z( t)) is also a solution for the same value of µ. This property is useful when we compute periodic solutions with some symmetry. We shall look, for instance, for orbits symmetrical with respect to the (x, z)-plane (halo orbits). Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

64 Invariant objects: fixed points, periodic orbits, tori,... Numerical computation of a family of periodic orbits Consider an initial point P 0 = (x 0, y 0, z 0, x 0, ẏ 0, ż 0 ) = (x, y, z, ẋ, ẏ, ż) 0, with y 0 = ẋ 0 = ż 0 = 0. Let P f = (x f, y f, z f, x f, ẏ f, ż f ) = (x, y, z, ẋ, ẏ, ż) f, with y f = 0, be the first intersection of the orbit which passes by P 0 with y = 0. In order to have a periodic orbit, due to the symmetries, it is sufficient that ẋ f = f 1 (x 0, z 0, ẏ 0 ) = 0, ż f = f 2 (x 0, z 0, ẏ 0 ) = 0, We define X = (x, z, ẏ) T and F = (f 1, f 2 ) T, then the solution of the above nonlinear system of equations are the zeros of F(X 0 ) = 0. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

65 Invariant objects: fixed points, periodic orbits, tori,... Numerical computation of a family of periodic orbits Assume that the rank of the Jacobian matrix of F, G = DF, is two. Then F(X 0 ) = 0 describes a curve, usually called the characteristic curve, in the space (x, z, ẏ) 0. In order to compute a family of periodic orbits we must follow the characteristic curve of the family using, for instance, a continuation method. The idea of that method is to integrate that curve along the arc parameter s, defined in this case by ds 2 = dx 2 + dz 2 + dẏ 2. That means to integrate the following system of differential equations: where dx ds = A 1 A 0, dy ds = A 2 A 0, dz ds = A 3 A 0, A 1 = fz 1 f 2 ẏ f 1 ẏ f z 2, A 2 = (fx 1 f 2 ẏ f 1 ẏ f x 2 ), A 3 = fx 1 fz 2 fz 1 fx 2, A 0 = (A A2 2 + A2 3 )1/2, and f i α are the partial derivatives of the functions f i with respect to their arguments. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

66 Invariant objects: fixed points, periodic orbits, tori,... Numerical computation of a family of periodic orbits If we have available some points on the characteristic curve, X 0, X 1,...,X n, the integration of the above equations, with some linear multistep method such as an Adams method, produces a new point Xn+1 0 near the curve. Due to the errors of the numerical integration, this point must be refined in order to obtain a new periodic orbit, X n+1. To do that we must use a modified Newton method because the number of variables, 3, is higher than the number of equations, 2. So if X k 1 is an approximation of the solution, we define the following iteration X k = X k 1 (DF(X k 1 )) 1 F (X k 1 ) = X k 1 + X k 1, with the additional condition that the euclidean norm of the correction X k 1 be minimum. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

67 Invariant objects: fixed points, periodic orbits, tori,... Numerical computation of a family of periodic orbits This is a problem of conditioned extrema with Lagrange function L = ( X k 1 ) T X k 1 + λ T (F (X k 1 ) + DF (X k 1 ) X k 1 ), whose solution is X k 1 = (DF (X k 1 )) T ( DF (X k 1 ) (DF (X k 1 )) T ) 1 F(X k 1 ). After some iterations of the procedure we get the periodic orbit with a given precision. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

68 Invariant objects: fixed points, periodic orbits, tori,... The multiple shooting method Sometimes, due to high instabilities, it is not possible to refine a periodic orbit using the standard correction method (single shooting). In other situations, a reference orbit is computed in some simplified model (RTBP) and must be "refined" in a more realistic force field model for a relatively long time span. In both cases, the use of a multiple shooting method is required. In the multiple shooting method, the total time span (the period, if we want to refine a periodic orbit) is splitted into a number of shorter subintervals selecting, for instance, N equally spaced points t 1, t 2,..., t N. Different time intervals cal also be used. We will denote by t = t i+1 t i and by Q i = (t i, x i, y i, z i, ẋ i, ẏ i, ż i, ) T, i = 1, 2,..., N the points on the initial reference orbit. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

69 Invariant objects: fixed points, periodic orbits, tori,... The multiple shooting method Let φ(q i ) be the image, after the time interval t, of the point Q i under the flow associated to the equations in which the refinement must be done. As, in this way, the epochs t i are fixed, (t i = t 1 + (i 1) t), we can write Q i as Q i = (x i, y i, z i, ẋ i, ẏ i, ż i, ) T. If all the points Q i would belong to the same orbit, then φ(q i ) = Q i+1, i = 1,..., N 1. If this is not the case, a change of the starting values is needed in order to fulfill the matching conditions. In this way, one must solve a set of N 1 nonlinear equations, which can be written as F Q 1 Q 2. Q N = φ(q 1 ) φ(q 2 ). φ(q N 1 ) Q 2 Q 3. Q N = Φ Q 1 Q 2. Q N 1 Q 2 Q 3. Q N = 0. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

70 Invariant objects: fixed points, periodic orbits, tori,... The multiple shooting method A Newton s method is used to solve the above system If Q (j) = ( Q (j) 1, Q(j) 2 F(Q 1,..., Q N ) T = 0. ) T,..., Q(j), denotes the j-th iterate of the procedure, Newton s equations can be written as N DF (Q (j) ) (Q (j+1) Q (j) ) = F(Q (j) ), where the differential of the function F has the following structure A 1 I A 2 DF = with DΦ = diag(a 1, A 2,..., A N 1 ). I A N 1 I, Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

71 Invariant objects: fixed points, periodic orbits, tori,... The multiple shooting method Since each of the transition matrices, A i, that appear in DΦ are 6 6, at each step of the method we have to solve a system of (N 1) 6 equations with 6 N unknowns, so some additional conditions must be added. As additional equations we could fix some initial and final conditions at t = t 1 and t = t N. In this case one must take care with the choice because, since the matrix DF (Q) can have a very large condition number, the problem can be ill conditioned from the numerical point of view. To avoid this bad conditioning, we can choose a small value for t, but in this case, as the number of points Q i increases (if we want to cover the same time span) the instability is transfered to the procedure for solving the linear system. Also, extra boundary conditions can force the solution in a non natural way giving convergence problems when we try to compute the orbit for a long time interval. To avoid this, we can apply Newton s method directly. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

72 Invariant objects: fixed points, periodic orbits, tori,... The multiple shooting method As the system has more unknowns than equations, we have (in general) an hyperplane of solutions. From this set of solutions we try to select the one closer to the initial orbit used to start the procedure. This is done by requiring the correction to be minimum with respect to some norm (i.e. the euclidean norm). Denoting by Q (j) Q (j) = Q (j+1) Q (j), requiring Q (j) 2 to be minimum, one gets the Lagrange function L( Q, µ) with (vector) multiplier µ from which we get L( Q, µ) = Q T Q + µ T (F (Q) + DF(Q) Q), Q (j) = DF (Q (j) ) T [ DF (Q (j) ) DF (Q (j) ) T ] 1 F(Q (j) ). (1) Since the matrix DF(Q (j) ) is usually very big, a special factorization in blocks is suitable to get the solution (1) in a computationally and efficient way. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

73 y y y Invariant objects: fixed points, periodic orbits, tori,... The multiple shooting method x x x (x, y) projections of the orbits obtained with the multiple shooting procedure at different steps. The figure on the left is the orbit of the RTBP, computed with LP expansions, from which the initial points Q i are taken. The orbit with large jumps discontinuities is the one obtained after the first two iterations. The figure on the right is the orbit computed after 8 iterations. The initial orbit is a quasihalo orbit with β = 0.2 and γ = Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

74 Invariant objects: fixed points, periodic orbits, tori,... Numerical computation of invariant 2D tori We develop the methodology for an autonomous Hamiltonian system. We look for a parametrization of a 2D torus, ψ : R 2 R 6 (θ 1, θ 2 ) ψ(θ 1, θ 2 ) with ψ being a 2π periodic function in θ 1, θ 2, solving ψ(θ 1 + tω 1, θ 2 + tω 2 ) = φ t (ψ(θ 1, θ 2 )), t R, θ 1, θ 2 [0, 2π], where ω 1, ω 2 are the frequencies of the torus. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

75 Invariant objects: fixed points, periodic orbits, tori,... Numerical computation of invariant 2D tori To reduce the dimension of the problem, we observe that ϕ(ξ) = ψ(ξ, 0), is an invariant curve invariant of φ 2π/ω2, and satisfies for ρ = 2πω 1 /ω 2. ϕ(ξ + ρ) = φ 2π/ω2 (ϕ(ξ)), Once we have ϕ, we can recover ψ using ψ(θ 1, θ 2 ) = φ θ 2 2π 2π ω 2 ( ϕ(θ 1 θ 2 2π ρ) ) Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

76 Invariant objects: fixed points, periodic orbits, tori,... Numerical computation of invariant 2D tori Note that ϕ(ξ + ρ) = φ 2π/ω2 (ϕ(ξ)), s a functional equation: we have "infinite equations" (one for each value of φ [0, 2π) ) and "infinite unknowns" (we cannot describe a general function φ by a finite number of parameters). We discretize space of functions by looking for ϕ as a truncated Fourier series N f ϕ(ξ) = A 0 + (A k cos(kξ) + B k sin(kξ)) k=1 We will discretize parameter space by looking for ϕ satisfying ϕ(ξ i + ρ) = φ 2π/ω2 (ϕ(ξ i )), for 2π ξ i = i, i = 1,..., 2N f N f Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

77 Invariant objects: fixed points, periodic orbits, tori,... Indeterminations We have two indeterminations to cope with: Invariant curve indetermination: Assuming there exists a parametrization of a 2D torus ψ(θ 1 + tω 1, θ 2 + tω 2 ) = φ t (ψ(θ 1, θ 2 )), t R, θ 1, θ 2 [0, 2π], not only ϕ(ξ) = ψ(ξ, 0) satisfies ϕ(ξ + ρ) = φ 2π/ω2 (ϕ(ξ)), but any ϕ(ξ) = ψ(ξ, η 0 ), for η 0 [0, 2π) also does. This indetermination can be avoided by fixing a curve on the torus. This can be done by prescribing a value for a coordinate of A 0. It must be chosen by geometrical considerations. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

78 Invariant objects: fixed points, periodic orbits, tori,... Indeterminations Phase shift indetermination: If ϕ(ξ) satisfies then, for any ξ 0 R, ϕ(ξ + ρ) = φ 2π/ω2 (ϕ(ξ)), ϕ ξ0 (ξ) = ϕ(ξ ξ 0 ), also does. This indetermination can be avoided by prescribing a coordinate of A 1 to be zero. Assume that A 1 = (A 1 1,..., A 6 1), B 1 = (B1, 1..., B1). 6 If (A k 1, B1 k ) (0, 0), since ( ) A k 1 cos(k(ξ ξ 0)) + B1 k sin(k(ξ ξ 0)) = A k 1 cos(kξ 0) B1 k sin(kξ 0) cos(kξ) + ( ) A k 1 sin(kξ 0) + B1 k cos(kξ 0) sin(kξ) we can always choose ξ 0 such that Ãk 1 = 0. := Ãk 1 cos(kξ) + B k 1 sin(kξ) Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

79 Invariant objects: fixed points, periodic orbits, tori,... The system of equations We want to set a system of equations such that: We want to prescribe values of the energy (Hamiltonian) For that, we add an additional equation We want to overcome high instability For that, we will implement multiple shooting Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

80 Invariant objects: fixed points, periodic orbits, tori,... The system of equations We look for ϕ 0,..., ϕ m satysfiying H(ϕ 0 (0)) h = 0, ϕ j+1 (ξ i ) φ δ/m (ϕ j (ξ i )) = 0, j = 0,...m 2, i = 0,..., 2N f ϕ 0 (ξ i + ρ) φ δ/m (ϕ m 1 (ξ i )) = 0, i = 0,..., 2N f where ξ i = (i 2π)/(1 + 2N f ), i = 1,..., 2N f, δ = 2π/ω 2 and the unknowns are with h, δ, ρ R, A j i, Bj i R 6 and h, δ, ρ, A 0 0, A 0 1, B 0 1,..., A 0 N f, B 0 N f,..., A m 1 N f, B m 1 N f, N f ϕ j (ξ) = A j 0 + ) (A jl cos lξ + Bjl sin lξ. This system is (1 + 6m(1 + 2N f )) (3 + 6m(1 + 2N f )). l=0 Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

81 Invariant objects: fixed points, periodic orbits, tori,... Computation of a torus The tori we are looking for are embedded in 2 parametric families, which can be paramtrized by 2 parameters among h, δ, ρ. Therefore, in order to compute a torus, we eliminate one coordinate of A 0 0, in order to fix a curve on the torus, set a coordinate of A 0 1 equal to zero and eliminate it, in order to get rid of the phase shift indetermination. eliminate two unknonws among h, δ, ρ, in order to fixate a particular torus. When applying Newton s method, we end up with a (1 + 6m(1 + 2N f )) (3 + 6m(1 + 2N f ) 4) system of linear equations, with unique solution but which has more equations than uknowns. This is not a problem, as long as we use the general rutine we have described, specifying the kernel dimension to be zero. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

82 Invariant objects: fixed points, periodic orbits, tori,... Numerical approximation of invariant manifolds Numerical approximation of invariant manifolds We start with the simplest example in the case of invariant manifolds of a fixed point of a diffeomorphism. Assume that p = (x, y ) is a fixed point of P : U R 2 R 2, with characteristic exponents λ > 1, µ (0, 1). We want to compute W u (p) (for W s (p) just use P 1 ). Let (x(t), y(t)) be a parametric representation of the manifold around p, with p = (x(0), y(0)) and P(x(t), y(t)) (x(λt), y(λt)). This can be as simple as the linear approximation (x(t), y(t)) = (x + tv 1, y + tv 2 ), where v = (v 1, v 2 ) is a normalized eigenvector of the eigenvalue λ or any higher order approximation. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

83 Invariant objects: fixed points, periodic orbits, tori,... Numerical approximation of invariant manifolds Numerical approximation of invariant manifolds First we check up to which value of t the difference P(x(t), y(t)) (x(λt), y(λt)) is less than some tolerance. Let t 0 be this value. In general it is not needed to be extremely restrictive concerning the tolerance because the small errors tend to decrease (at least locally) due to the compression along the stable direction. Then we take a fundamental domain generated by t ranging in (t 0 /λ, t 0 ). A simple method generates a big piece of W u (p) by taking n points in this domain and performing iterates of those points. Some care must be taken also to prevent from large bendings of the manifold. This can be done choosing adequately a step, t, on the fundamental domain (which obviously can not be kept fix on the full domain) and performing a given number of iterations, say k, with this step. When the value of t is greater than t 0, then both the parameter t and the step, t, must be divided by the eigenvalue and k can be replace by k + 1. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

84 Invariant objects: fixed points, periodic orbits, tori,... Numerical approximation of invariant manifolds Numerical approximation of invariant manifolds This same method can be applied to obtain unstable invariant manifolds of periodic orbits with one unstable direction by using Poincaré sections. A more difficult question is to obtain higher dimensional invariant manifolds. The key difficulty is that if we have k unstable directions with related eigenvectors v j, j = 1,..., k and eigenvalues λ j, j = 1,..., k, λ j > 1, starting at a point k p + t j v j, j=1 with k j=1 t j 2 small near the fixed point p, we shall escape essentially in the direction of the dominant eigenvalue. That is, if λ 1 > λ 2 λ 3... for almost any k ple of values (t 1,..., t k ) we obtain points very close to the ones generated by iteration starting at p + t 1 v 1. A general rule to overcome this behaviour consists on the use of high order expansions allowing to start the computations at a relatively big distance of p. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

85 Invariant objects: fixed points, periodic orbits, tori,... Numerical approximation of invariant manifolds Local approximation of the stable manifold of a quasiperiodic halo orbit We are going to see how to compute the linear approximation of the stable manifold of a quasiperiodic halo orbit (qpo). We know that if an orbit is periodic, the eigenvectors of the variational matrix computed over one period of the orbit, that is the monodromy matrix, give the directions of the tangent spaces to the invariant manifolds at the starting point. Now our orbits are not periodic but quasiperiodic, with small deviations from the halo periodic orbit of the restricted three body problem. So, the eigenvector associated to the eigenvalue with smallest absolute value of the variational matrix computed, over a revolution approximates quite well the stable direction when the orbit crosses y = 0. We want to use the stable manifold of quasiperiodic orbits computed for a given time span, covering several revolutions. If we want to use this manifold for the transfer, we shall be interested in the stable manifold associated to the initial revolutions. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

86 Invariant objects: fixed points, periodic orbits, tori,... Numerical approximation of invariant manifolds Local approximation of the stable manifold of a quasiperiodic halo orbit Instead of computing the stable direction associated to a selected revolution, the variational matrix Ā associated to the whole quasiperiodic orbit is taken as a monodromy matrix. We recall that Ā is a 7 7 matrix because time is added in the equations of motioni, in order to get an autonomous system,and its pattern is: g 1 g 2 Ā = g 3 A, g 4 g 5 g 6 where A is the part of Ā associated to positions and velocities. We note that if v is an eigenvector of A, then (0, v) is an eigenvector of Ā. We shall compute the stable direction for the 6 6 matrix A. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

87 Invariant objects: fixed points, periodic orbits, tori,... Numerical approximation of invariant manifolds Local approximation of the stable manifold of a quasiperiodic halo orbit If A i is the 6 6 matrix related to positions and velocities of a certain variational matrix Āi associated to the i-th revolution of a numerical qpo made of N revolutions, we have: A = A N A N 1... A 1. Because of possible overflows and rounding errors, due to the fact that the eigenvalue of each A i associated to the unstable manifold is of the order of 1700, the above product must not be computed. With the power method applied to A 1 = A 1 1 A A 1 N, we can compute the eigenvector v 1 of the matrix A associated to the stable manifold. Then the eigendirections v j related to the beginning of each j-th revolution are computed by means of: where v v j = A 1 j v j+1 A 1, j = N,..., 2, j v j+1 = v. The contraction (eigenvalue in the case of a periodic Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

88 Invariant objects: fixed points, periodic orbits, tori,... Numerical approximation of invariant manifolds Local approximation of the stable manifold of a quasiperiodic halo orbit When we have the direction of the stable manifold at the beginning of a revolution, we can obtain the stable direction at any intermediate point in the revolution transporting this vector by means of the 6 6 differential matrix of the flow corresponding to the coordinates of position and velocity. That is, if v j (t 0 ) = v j is the initial vector at time t 0 we have where A j (t) = Id. v j (t) = A j (t)v j (t 0 ), Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

89 Invariant objects: fixed points, periodic orbits, tori,... Numerical approximation of invariant manifolds Globalization of the stable manifold of the QPO Once the local approximation is available, the next thing to do is the globalization of the stable manifold. Given a displacement, D, in kilometers from the selected point in the qpo, in the right sense of the stable manifold (for instance the one which gives approaches to the Earth if we want to use the manifold for transfer), initial conditions X 0 ws in the linear approximation of the manifold are given by means of: X 0 ws = X qpo + D V ws, where X qpo is the selected point of the qpo and V ws is the scaled stable direction in the point X qpo. It must be noted that the magnitude D can not be too small, in absolute value, in order to prevent rounding errors and large integration periods of time when globalizing the manifold. However, it can not be too large because the linear approximation is good near the point X qpo. Values between 200 and 250 km are adequate. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

90 The restricted n and three body problems The n-body problem as a perturbation of Kepler s problem The restricted n and three body problems Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

91 The restricted n and three body problems The n-body problem The n-body problem as a perturbation of Kepler s problem The inertial Newton equations of motion of n punctual masses m 1,m 2,...,m n are n m j m k m k r k = G rjk 3 (r j r k ), k = 1,..., n, (2) where r jk = r j r k. j=1, j k P (m ) 2 2 r d j P (m ) 1 1 z r 1 r 2 r j ρ j P j (m j ) x y Inertial and relative coordinates Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

92 The restricted n and three body problems The n-body problem as a perturbation of Kepler s problem The n-body problem as a perturbation of Kepler s problem To get the equation of relative motion of m 2 with respect to m 1, we take k = 1, 2 in the preceding equation, to get d 2 r 1 dt 2 = G m n 2 r21 3 (r 2 r 1 ) + G j=3 d 2 r 2 dt 2 = G m n 1 r12 3 (r 1 r 2 ) + G j=3 m j rj1 3 (r j r 1 ), m j rj2 3 (r j r 2 ). (3) Subtracting these two equations we get the one of the relative motion ( ) d 2 r dt 2 + µ r n r 3 = G d j m j dj 3 + ρ j ρ 3, (4) j where r = r 2 r 1, ρ j = r j r 1, d j = r ρ j and µ = G(m 1 + m 2 ). If m 3 =... = m n = 0, then either (3) as (4) are the equations of the two body problem. j=3 Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

93 The restricted n and three body problems The restricted three body problem The restricted three body problem The inertial equations of the three body problem are r 1 = Gm 2 r 1 r 2 r 3 21 r 2 = Gm 1 r 2 r 1 r 3 12 r 3 = Gm 1 r 3 r 1 r 3 13 r 1 r 3 Gm 3 r31 3, r 2 r 3 Gm 3 r32 3, (5) r 3 r 2 Gm 2 r23 3, where r jk = r j r k. If one of the masses can be neglected, in front of the other two, the above equations can be simplified. Taking, for instance, m 3 = 0 we get r 1 r 2 r 1 = Gm 2 r21 3, r 2 r 1 r 2 = Gm 1 r12 3, (6) r 3 = Gm 1 r 3 r 1 r 3 13 r 3 r 2 Gm 2 r23 3. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

94 The restricted n and three body problems The restricted three body problem The restricted three body problem The first two equations of system (6) describe the Keplerian motion of m 1 and m 2, which are usually known as primaries and the third equation defines the restricted three body problem (RTBP). P (m =0) 3 3 z r 1 P (m ) 1 1 r 1 z r 3 r B x y r 3 r 2 r 2 P (m ) 2 2 x y Barycentric coordinates Since the equations are independent of any particular inertial frame, we can set the origin at any point, for instance at the barycentre of m 1 and m 2 (m 3 = 0). Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

95 The restricted n and three body problems The restricted three body problem The restricted three body problem One particular case of the RTBP is the circular restricted three body problem, in which m 1 and m 2 describe circular orbits around their barycentre. We will use a synodic reference system, with origin at the barycentre, in which the two primaries remain at rest on the x axis If ρ denotes the synodical position and r the corresponding inertial (sidereal) one ρ = R(t)r, where R(t) is the transformation (rotation) between both reference frames, defined by the constant angular velocity ω = (0, 0, n) T of the rotating frame with respect to the fixed one. Recall that d 2 r dt 2 = d 2 ρ dt 2 + 2ω dρ + ω (ω ρ), (7) dt Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

96 The restricted n and three body problems The restricted three body problem The restricted three body problem z m 3 = 0 P =(x,y,z) r 1 r 2 O L 1 m = 2 µ L 2 P =( µ 1, 0, 0) 2 y x m = 1 1 µ P =( µ, 0, 0) 1 The synodic reference frame Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

97 The restricted n and three body problems The restricted three body problem The restricted three body problem Denoting by ρ = (x, y, z) T the synodic position of m 3, its Coriolis and centripetal accelerations are ω dρ ẏ x = n ẋ, ω (ω ρ) = n 2 y, dt 0 0 and the synodic equations of the RTBP become ẍ 2nẏ n 2 x = Gm 1 (x x 1 ) r 3 13 ÿ + 2nẋ n 2 y y y = Gm 1 r13 3 Gm 2 r23 3, z z z = Gm 1 r13 3 Gm 2 r23 3. (x x 2 ) Gm 2 r23 3, where (x 1, 0, 0) and (x 2, 0, 0) denote the two fixed synodic positions of m 1 and m 2, respectively. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

98 The restricted n and three body problems The restricted three body problem The restricted three body problem It is useful to introduce a suitable set of units in such a way that: the distance between the primaries is equal to one, n = 1, the sum of the masses of the primaries is also one. So m 1 = 1 µ, m 2 = µ, with µ [0, 1]. With this choice G = 1 and we can set ρ 1 = (µ, 0, 0) T, ρ 2 = (µ 1, 0, 0) T, and get r 13 = (x µ) 2 + y 2 + z 2, r 23 = (x µ + 1) 2 + y 2 + z 2. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

99 The restricted n and three body problems The restricted three body problem The restricted three body problem Using the auxiliary function Ω = 1 2 (x 2 + y 2 ) + 1 µ r 1 + µ r 2, where r 1 = r 13 and r 2 = r 23, the RTBP equations become ẍ 2ẏ = Ω x, ÿ + 2ẋ = Ω y, (8) z = Ω z. System (8) has a first integral, the Jacobian integral, which can be obtained multiplying equations (8) by ẋ, ẏ and ż, adding the results and integrating; in this way we get ẋ 2 + ẏ 2 + ż 2 2Ω(x, y, z) = C J. (9) Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

100 The restricted n and three body problems The restricted three body problem Zero velocity surfaces The Jacobian integral of the RTBP can be used to obtain the velocity V = ẋ 2 + ẏ 2 + ż 2 of the third body at any arbitrary position. For a fixed value of C J one can have different values of V at the same position of the small particle. In the case when V = 0 the Jacobian integral becomes x 2 + y 2 + 2(1 µ) (x µ) 2 + y 2 + z 2 ) 1/2 + 2µ (x µ + 1) 2 + y 2 + z 2 ) 1/2 = C J, and gives the geometrical location of points at which the velocity of the infinitesimal body is zero. On one side of this surface the velocity will be real and on the other complex. Though, when the velocity is real, we can say nothing about the orbit but, at least, we can be sure that in that region motions are possible. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121

101 The restricted n and three body problems Zero velocity surfaces The restricted three body problem The shape of the zero velocity surfaces depends on the value of the Jacobi constant. Gómez - Masdemont (UB & IEEC, UPC & IEEC) Invariant Manifold Dynamics Glasgow, 3rd - 7th June / 121