Chaos in the Anisotropic Kepler Problem

(1)

Corso di Laurea Magistrale in Matematica

Tesi di Laurea Magistrale

Chaos in the Anisotropic Kepler

Problem

Candidato: Relatore:

Roberto Paoli

Prof. Claudio Bonanno

Anno Accademico 2017/2018 Sessione di Laurea 3 Maggio 2019

(2)

(3)

Abstract

The Anisotropic Kepler Problem (AKP), first introduced by M.C. Gutzwiller to describe the motion of an electron with anisotropic mass-tensor moving in a Coulomb field, is a Hamiltonian System which posseses chaotic dynamics. Following two different approaches, one by R.L. Devaney and the other by G. Contopoulos, we show the existence of chaotic behaviour in the AKP for all the cases in which there is anistropy, measured by the parameter µ. For

µ= 1 we have the Classical Kepler problem. In the first approach we use

the Conley-Moser conditions which provide a generalization of the Smale horseshoe. We start by making an accurate study of the flow of the problem inside the Hamiltonian energy level sets, by using many techniques from the theory of dynamical systems such as the regularization of the singularities and the use of Poincaré maps. Particular attention is given to the bi-collision orbits, which begin and end with a collision. We then isolate a collection of "windows" for the flow when µ > 9

8 and we define a Poincaré map of a two

dimensional square in phase space, considering only orbits which meet those windows in a prescribed order. We define horizontal and vertical strips as in the Smale horseshoe and we discuss a way to label these strips, so that we can associate a bi-infinite sequence of integers to each admissible orbit and we discuss the behaviour of such orbits in the configuration space. Following Smale and Moser we then find an invariant set on which the dynamics is topologically conjugate to a shift on a suitable set of sequences. The second approach uses the concept of Lyapunov characteristic number as a way to measure the sensitive dependence on initial conditions (and then chaos) of a given system. This number is zero in the proximity of a stable periodic orbit and it is a positive number in a chaotic domain. We describe some tests and their results by Contopoulos, which show that the AKP is chaotic for all

µ > 1. We conclude by exposing some periodic orbits and the Matlab

(4)

(5)

Introduction

The Anisotropic Kepler Problem (AKP) is a model to describe the motion of an electron with anisotropic mass-tensor in a Coulomb field and it is used for studying the donor states in silicon. It is also one of the earliest simple systems where chaos was rigorously proved. In its planar formulation it can be interpreted as the usual Kepler problem of gravitational attraction, where one of the two axes is weighted. This means that orbits tend to oscillate more and more about the heavy axis as the anisotropy increase.

The problem has been studied extensively by M. C. Gutzwiller, who first introduced it in 1967, and by other authors such as R. L. Devaney, R. Broucke and G. Contopoulos. At the time of its introduction, the problem has been proven to be chaotic mainly by means of symbolic dynamics, which requires a rigorous study of the flow of the problem, both in configuration and in phase space. In later years, however, as computers became more and more powerful, numerical computations have become more accessible and reliable, resulting in accurate studies of the problem from a numerical point of view.

In this work we show two different approaches, one by Devaney [6], which involves symbolic dynamics on an infinite alphabet, and another one by Contopoulos [4], which mainly uses Lyapunov characteristic numbers as a way to study the presence of chaos in a given system. This thesis is structured as follows.

In Chapter 1 we start by recalling some definitions and by exposing some preliminary results concerning chaos in dynamical systems. We de-scribe briefly the Smale horseshoe, the prototype of a chaotic map. We then introduce the Conley-Moser conditions as a way to describe maps which have a chaotic behaviour similar to the Smale horseshoe. Moreover, we outline some facts about Poincaré maps which we will use to reduce the anisotropic Kepler problem to a map.

In Chapter 2 we start exposing the approach by Devaney. We provide the Hamiltonian formulation of the problem and we introduce the

(6)

parame-ter µ ≥ 1 which measures the anisotropy. If µ = 1 the AKP reduces to the classical Kepler problem. We define the zero velocity curve Z for negative energy values. To study rigorously the flow of the AKP we make advantage of a smooth change of variables introduced by McGhee in his studies about triple collision in the three body problem. Following standard regulariza-tion techniques we blow up the singularity to a 2-torus, which we shall call

Collision manifold. To achieve some familiarity with this manifold we use

it to study the Kepler problem. To describe the invariant manifolds of some circles of equilibria that we find in the collision manifold we introduce the concept of normally hyperbolic submanifolds. A particular attention is given to bi-collision orbits, which begin and end with a collision.

In Chapter 3 we study the problem for µ > 1. We investigate the stability of the equilibria on the collision manifold and the topology of their corresponding invariant manifolds. A key result is that, for µ > 9

8 some

of the characteristic exponents of the sinks and sources are complex, and the corresponding invariant manifolds spiral around the equilibrium point. We will then restrict ourselves only to those values of µ. We study collision orbits and we examine the flow of the problem near Z. We then isolate a sets of sections transversal to the flow and we define some suitable Poincaré maps between them. We expose the resulting map from a square to itself and we prove that it satisfies the Conley-Moser conditions. We discuss a possible indexing on the square to associate a sequence of symbols to an orbit and its significance in configuration space, and to prove that the AKP is chaotic for µ >9

8.

Chapter 4 is dedicated to the numerical approach. We introduce the variational equations and the Lyapunov characteristic number (LCN) and some of its properties. We then provide the results of the numerical tests by Contopoulos which uses the LCN to prove that the AKP is chaotic for all

µ >1. We conclude by showing some numerically computed orbits for the

AKP, both periodic and non-periodic, and a Matlab code to draw some (unstable) periodic orbits in configuration space.

(7)

Preliminary results

In this chapter we collect all the definitions and general results necessary to formulate and study the Anisotropic Kepler Problem (which we will refer to as AKP in the following). Most of the work here is due to the excellent books by Wiggins, [12], and Moser, [10], which are the main references for the chapter.

1.1 Preliminary remarks and definitions

Most of our work will concern autonomous differential equations of the form ˙x = f(x, µ) x ∈ Rn

, µ ∈ R (1.1)

where f is a generic smooth function, possibly nonlinear. Here the dot means

d

dt, i.e. the differentiation with regard to the independent variable t, the x

are the dependent variables and we view the variable µ as a parameter. We will refer to the space of the dependent variables of (1.1) as the phase space of (1.1). We will also refer to f(x, µ) as a vector field on Rn_{. We remark that}

in many applications, such as the ones in one our work, the structure of the phase space could be radically different from Rn_{. One usually encounters}

cylindrical, spherical, or toroidal phase spaces.

Terminology and notation

There are several different terms synonymous with the term solution of (1.1) and also a lot of different equivalent notations.

(10)

I ⊂ R, into Rn, which we represent as follows

X: I → Rn, (1.2)

t 7→ X(t) (1.3)

such that X(t) satisfies (1.1), i.e.,

˙X(t) = f(X(t), µ).

We could geometrically interpet the solution of a curve in the phase space, and f gives the tangent vector at each point of the curve, hence the use of the term vector field.

Sometimes we will define solutions of (1.1) by means of the integral flow, a function Φ(t, x0; t0) : I × Rn→ Rn which verifies the following:

• d

dtΦ(t, x0; t0) = f(Φ(t, x0; t0), µ)

• Φ(t0, x0; t0) = x0

so that Φ(t, x0; t0) represent the position of the point x0in the phase space at

time t under the effect of vector field (1.1). This two notation are equivalent, but the latter let us associate to a point x0 in the phase space the solution

passing through that point at time t = t0. If the value of t0 is obvious,

i.e. usually we set t0 = 0, we can drop the dependence on t0 and we write

Φ(t, x0) or Φt(x0). The graph of a solution Φ(t, x0; t0) over t will be called integral curve.

It is a well known fact that the solution through a point is unique under some smoothness assumption on the vector field. Let x0 be a point in the

phase space of (1.1). By the orbit through x0 we mean the set of points in

the phase space that lie on the unique trajectory passing through x0.

1.1.1 Equilibrium solutions and stability

We now state some facts about equilibrium solutions and their stability necessary for our study of the AKP. The curious reader can find more on [12]. We let the parameter µ fixed in the following and we drop it in our notation.

Definition 1.1.1. An equilibrium solution or fixed point of (1.1) is a point x∗_{∈ R}n such that

(11)

Other terms for equilibrium solution are critical point or stationary point. Once we find a solution, not necessarily an equilibrium one, of (1.1), it is natural to try to determine if this solution is stable. But what stable means?

Roughly speaking, we say that a solution x(t) is stable if solutions starting "close" to x(t) at a given time remain close to x(t) for all later times. It is

asymptotically stable if nearby solutions actually converge to x(t) as t → ∞.

We formalize these ideas in the following definitions.

Definition 1.1.2. (Lyapunov Stability) x(t) is said to be stable (or Lya-punov stable) if, given > 0, there exists a δ = δ() > 0 such that, for any

other solution, y(t) of (1.1) satisfying kx(t0) − y(t0)k < δ (where k · k is a norm on Rn_{, then kx(t) − y(t)k < for t > t}

0, t0 ∈ R.

Definition 1.1.3. (Asymptotic Stability) x(t) is said to be asymptot-ically stable if it is Lyapunov stable and for any other solution y(t) of

(1.1), there exists a constant b > 0 such that, if kx(t0) − y(t0)k < b, then

limt→∞kx(t) − y(t)k = 0.

To determine the stability of an equilibrium solution we linearize the vector field near the fixed point x∗_{, if it is not already a linear vector field.}

The linear system associated to (1.1) is given by

˙y = Df(x∗_)y _(1.4)

where y = x−x∗_{. The following theorem states a well known fact. Its proof}

can be found in [12].

Theorem 1.1.1. Suppose all of the eigenvalues of Df(x∗) have negative

real parts. Then the equilibrium solution x = x∗ _{of the nonlinear vector} field (1.1) is asymptotically stable.

If the eigenvalues of the associated linear vector field have nonzero real parts, then the orbit structure near an equilibrium solution of the nonlinear vector field is essentially the same as that of the linear vector field. This result is commonly known as Harman-Grobman Theorem. Such equilibrium solutions are given a special name.

Definition 1.1.4. Let x∗ be a fixed point of (1.1). Then it is called a hyperbolic fixed point if none of the eigenvalues of Df(x∗_{) have zero real} part.

In the special case n = 2 we have the following catalogation of hyperbolic fixed points based of their eigenvalues, λ and µ.

(12)

• If Reλ and Reµ are positive then x∗ _{is a source.}

• If Reλ and Reµ are negative then x∗ _{is a sink.}

• If Reλ and Reµ have different sign, then x∗ _{is a saddle point.}

Moreover, in the case of sinks and sources, if Imλ and Imµ are not zero, then the orbits tend to spiral to the sinks and away from the sources. The result is classical and somehow elementary and we will not give more details.

1.1.2 Invariant Manifolds

The concept of invariant manifold will be extremely important in the study of the flow of the AKP. The subject has been studied extensively and we refer the reader to [7] for more details.

Definition 1.1.5. Let f(x) a vector field on Rn. Let S ⊂ Rn be a set, then

S is said to be invariant under the vector field ˙x = f(x) if for any x0 ∈ S we have Φt_(x

0) ∈ S for all t ∈ R.

If we restrict ourselves to non-negative times (i.e. t ≥ 0) we refer to S as a positively invariant set and, for negative times, as a negatively invariant

set. An invariant set S with the structure of a Cr _{differentiable manifolds}

(r ≥ 1) is said to be a Cr _{invariant manifold. Similarly, a positively (resp.}

negatively) invariant set S is said to be a Cr _{positvely (resp. negatively)}

invariant manifold if it has the structure of a Cr _{differentiable manifolds}

(r ≥ 1).

If the vector field which defines the sistem is linear on Rn_{, i.e. ˙y =}

Ay where A is a constant n × n matrix, one can easily define the stable, unstable and center manifolds of the equilibrium solution y = 0, as the

eigenspace relative to the eigenvalues with negative, positive and zero real part respectively. Moreover it is easy to prove that solutions starting in Es

approach y = 0 asymptotically as t → +∞ while solutions starting in Es

approach y = 0 asymptotically as t → −∞.

The following theorem generalizes the existence of invariant stable, un-stable and center manifolds to the case of nonlinear vector fields by means of the linearized vector field we introduced before.

Theorem 1.1.2. Suppose ˙x = f(x) is a Cr vector field with r ≥ 2 and

let ¯x be an equilibrium solution. Then the fixed point possesses a Cr _local

stable manifold Ws

(13)

local center manifold Wc

loc(¯x), all intersecting at ¯x. These manifolds are all

tangent to the respective invariant manifolds of the linearized vector field

˙y = Df(¯x)y at ¯x, i.e. y = 0, they have the same respective dimension

and they are locally representable as graphs. Moreover Ws

loc(¯x) and Wlocu (¯x)

have the same asymptotic properties of Es _{and E}u_{, respectively. Namely}

solutions of ˙x = f(x) Ws

loc(¯x) (resp. Wlocu (¯x)) approach ¯x at an exponential

rate asymptotically as t → +∞ (resp. t → −∞).

For a proof of Theorem 1.1.2 we refer the reader to [12].

1.1.3 Some results on maps

In our study we will sometimes encounter dynamical system defined by maps. In particular we are interested in maps of the form

x 7→ g(x; µ) x ∈ Rn_{, µ ∈ R} (1.5) where µ is considered as a parameter. The goal is to study what happens to a point x0 under iterates of g, i.e. the study of gn(x0µ) for fixed µ. In this

sense we might interpret maps as discrete dynamical system where the time variable is a positive integer. If g is invertible then we can also consider negative integers. The orbit of x0 under g is then given by a bi-infinite

sequence of points

{. . . , g−n(x₀; µ), . . . , g−1(x₀; µ), g(x₀; µ), . . . , gn_(x

0; µ) . . . }

We state below the definition of equilibrium solution for maps.

Definition 1.1.6. An equilibrium solution for a map (1.5) is a fixed point

for the map g, i.e. g(x∗_{; µ) = x}∗_.

To investigate the stability of a fixed point whe proceed as we did for vector fields. We linearize the map g near the fixed point ¯x and we study the eigenvalues of the linearized map, i.e. a n × n matrix. In this situation the point x∗ _{is asymptotically stable if all of the eigenvalues have moduli strictly}

less than one.

It is possible to define invariant sets and stable and unstable manifolds for maps as we did for vecor fields, but we will state here only the definition of invariant set, the only one which will be of interest in the following. Definition 1.1.7. Let S ⊂ Rn be a set, then S is said to be invariant under

the map x 7→ g(x) if for any x0∈ S we have gn(x0) ∈ S for all n.

In the next section we will define Poincaré maps which, in some way, can be seen as the natural way of obtaining a discrete dynamical system (i.e. a map) from a continuos one (i.e. the flow of a vector field).

(14)

1.2 Poincaré maps

Citing Wiggins,"The idea of reducing the study of a continuous time system

(flows) to the study of an associated discrete time system (map) is due to Poincaré, who first utilized it in his studies of the three body problem in celestial mechanics. Nowadays virtually any discrete time system that is associated with an ordinary differential equation is referred to as a Poincaré map."

This technique offers several advantages, as the elimination of at least one of the variables of the problem, resulting in the study of a lower di-mensional problem. For example the study of the stability of an orbit could become the study of the stability of a fixed point, which requires only the knowledge of the eigenvalues of the linearized map about the fixed point.

The construction of a Poincaré map requires some preliminary knowledge of the flow. We construct a Poincaré map near a periodic orbit and then we generalize the situation to our needs.

Consider the following ordinary differential equation ˙x = f(x), x ∈ Rn

where f : U → Rn _{is C}r _{on some open set U ⊂ R}n_{. Suppose that this}

vector field has a periodic solution of period T . Namely we can find a point x0 so that Φ(t + T, x0) = Φ(t, x0) for all t ∈ R. Let Σ be a n − 1

dimensional "surface" transverse to the flow passing through the point x0.

Here by "transverse" we mean that the scalar product between f(x) and

n(x) is not zero, where n(x) is the normal to Σ at x.

We consider common knowledge that if f(x) is Cr _{then the flow Φ is also}

Cr. Thus we can find an open set V ⊂ Σ such that the points in V return to

Σ in a time close to T . The Poincaré map is the map which associates points in V to their points of first return to Σ, and we denote it by P . Summing up, we have

P : V → Σ

x 7→Φ(τ(x), x) (1.6)

where τ(x) is the time of first return to Σ. It is trivial that τ(x0) = T and P(x0) = x0. Therefore a fixed point of P corresponds to a periodic orbit

˙x = f(x), and a periodic point of P (i.e. a point x ∈ V such that Pk_{(x) = x}

correspond to a periodic orbit that pierces Σ k times before closing.

The existence of a periodic orbit is here just a way to guarantee that points near x0 return near x0 in roughly the same time. In our study of the

(15)

Figure 1.1: The way the Smale horseshoe map f acts on the horiziontal rectangles.

AKP we will gain extensive knowledge on the flow in the phase space and we will be able to choose the section in ways suitable to our purpouse. We will then use the term Poincaré map to define maps which follows the flow of a given vector space, even if the map is from a transversal section to a

different one.

1.3 Chaotic maps

In this section we show some example of choatic maps. The meaning of the adjective chaotic will be clarified lately. We first show some facts about the Smale horseshoe map and then we generalize this results to arbitrary maps which satisfy two assumptions, which will be called Conley-Moser conditions. Most of the work here is based on [10] and [12].

1.3.1 The Smale horseshoe

The Smale horseshoe map is possibly the simplest map which admits a chotic invariant set. It takes its name from the way the map acts on its domain, see Figure (1.1).

Consider the following map

f : D −→ R2

D= {(x, y) ∈ R2|0 ≤ x ≤ 1, 0 ≤ y ≤ 1}

which contracts the x-direction, expand the y-direction and folds D around, laying it back on itself as in Figure (1.1).

(16)

We will assume that f acts affinely on the horizontal rectangles H0 = (x, y) ∈ R2_|_{0 ≤ x ≤ 1, 0 ≤ y ≤} 1 µ and H1 = (x, y) ∈ R2_|_{0 ≤ x ≤ 1, 1 −} 1 µ ≤ y ≤1

sending them to the vertical rectangles

f(H0) ≡ V0 = n (x, y) ∈ R2_|_{0 ≤ x ≤ λ, 0 ≤ y ≤ 1}o and f(H1) ≡ V1= n (x, y) ∈ R2_|_{1 − λ ≤ x ≤ 1, 0 ≤ y ≤ 1}o .

The explicit restrictions of the map f to the rectangles H0 and H1 are

given below H0 : x y ! 7→ λ 0 0 µ ! x y ! , H1 : x y ! 7→ −λ 0 0 −µ ! x y ! + 1 µ ! with 0 < λ < 1

2 and µ > 2. The minus sign in the matrix corresponding to

restriction on H1 means that, in addition to being contracted and expanded,

the rectangles is rotated by π.

The inverse map acts in the opposite way, expanding in the x-direction and contracting in the y-direction, as shown in Figure (1.2). The explicit restrictions of the map f−1 _{to the rectangles V}

0 and V1 is easy to compute

and we leave it to the reader.

The following Lemma is crucial to define the chaotic properties of the horse-shoe map. In the following by horizontal and vertical rectangles we mean sets whose boundary are parallel to the ones of the square, similarly to Hi

and Vi, as in Figure (1.3).

Lemma 1.3.1.

• Suppose V is a vertical rectangle; then f(V ) ∩ D consists of precisely two vertical rectangles, one in V0 and one in V1, with their widhts each being equal to a factor of λ times the width of V .

• Suppose H is a horizontal rectangle; then f−1_{(H) ∩ D consists of} pre-cisely two horizontal rectangles, one in H0 and one in H1, with their widhts each being equal to a factor of 1

(17)

Figure 1.2: The way the inverse of the Smale horseshoe map f−1 _{acts on}

the vertical rectangles.

(18)

We remark that the qualitative feautures of Lemma 1.3.1 are independent of the analytical form for f given. They are indeed more geometrical in nature and their implications will be important in generalizing our result to arbitrary maps. Moreover even if it requires only the knowledge of the be-haviour of f and f−1_{, its implications allow us to understand the behaviour}

of all the iterates fn_.

We will now sketch the construction of the invariant set which consists of points which remains in D for all possible iterates of f. We call this set Λ and we have Λ = ∞ \ n=−∞ fn(D) We start by consideringT+∞

n=0fn(D), obtained as the limit of the sets

Tk

n=0fn(D)

as k → ∞. The process forT0

n=−∞fn(D) will be similar.

We introduce the following notation to keep track of the iterates of f. We let S = {0, 1} be and index set and we let the number si denote any of

the two elements of S for all possible integer values for i. The reason for this choice will be evident in the following.

D ∩ f(D). By definition D ∩ f(D) consists of the two vertical strips V0 and V1 from before. We have

D ∩ f(D) = [

s−1∈S

Vs−1 (1.7)

D ∩ f(D) ∩ f2(D). We first note that D ∩ f(D) ∩ f2(D) = D ∩ f(D ∩ f(D)). By the previous observation we know that D ∩ f(D) consists of two

vertical rectangles and applying Lemma 1.3.1 we get that D ∩ f(D) ∩ f2_(D)

corresponds to four vertical rectangles, two each in V0 and V1. Now, using

(1.7) we have D ∩ f(D ∩ f(D)) = D ∩ f   [ s−2∈S Vs−2  . (1.8)

We emphasize the fact that, when substituting (1.7) in (1.8), we have changed the subscript s−1 to s−2. It is obvious that this doesn’t cause any

(19)

Figure 1.4: The sets D ∩ f(D) and D ∩ f(D) ∩ f2_(D)

of Lemma 1.3.1 and some set-theoretic manipulations, we get

D ∩ f(D) ∩ f2(D) = [ s−i∈S i=1,2 (f(Vs−2) ∩ Vs−1) ≡ [ s−i∈S i=1,2 Vs−1s−2 = {p ∈ D|p ∈ Vs−1, f −1_{(p) ∈ V} s−2, s−i ∈ S, i= 1, 2}. (1.9)

For example, the set V01 consists of all points p ∈ D such that p ∈ V0 and

the preimage of p is in V1.

It’s not difficult to see that, at the kth _{step, we have} D ∩ f(D) ∩ · · · ∩ fk(D) = [ s−i∈S i=1,2,...,k (f(Vs−2...s−k) ∩ Vs−1) ≡ [ s−i∈S i=1,2...,k Vs−1...s−k = {p ∈ D|f−i+1_{(p) ∈ V} s−i, s−i ∈ S, i= 1, . . . , k}. (1.10)

and that this set consists of 2k _{vertical rectangles, each of width λ}k_{. From}

the definition of f and the fact that V0 and V1 are disjoint, it easily follow

that the labelling of each vertical rectangle is unique at each step.

Now we let k → ∞. Since a decreasing intersection of compact sets is non-empty, it is clear that we obtain an infinite number of vertical rectangles of width zero, since limk→∞λk= 0, because 0 < λ < 1₂. We deduce that

∞ \ n=0 fn(D) = [ s−i∈S i=1,2,... (f(Vs−2...s−k. . .) ∩ Vs−1) ≡ [ s−i∈S i=1,2... Vs−1...s−k... = {p ∈ D|f−i+1_{(p) ∈ V} s−i, s−i ∈ S, i= 1, 2, . . . }. (1.11)

(20)

consists of an infinite number of vertical lines each labeled by an infinite sequence of 0’s and 1’s. These numbers represent the past iterates of the points in each line under the effect of the map f.

The inductive construction of Tn=0

−∞fn(D) is similar and we will not go

into the details, which can be found in [12]. We just note that

n=0 \ −∞ fn(D) = [ si∈S i=0,1,... (f(Hs1...sk. . .) ∩ Hs0) ≡ [ si∈S i=0,1,... Hs0...sk... = {p ∈ D|fi_{(p) ∈ H} si, si∈ S, i= 0, 1, . . . }. (1.12)

consists in this case of an infinite number of horizontal lines each labeled by an infinite sequence of 0’s and 1’s. These numbers represent the future

iterates of points in each line under the effect of the map f.

Thus, we have Λ = ∞ \ n=−∞ fn(D) = " ₀ \ n=−∞ fn(D) # ∩ "_∞ \ n=0 fn(D) # , (1.13)

which consists of an infinite set of points, since each vertical line intersects each horizontal line in a unique point. This implies that each point in Λ can be labeled uniquely by a bi-infinite sequence of 0’s and 1’s obtained by concatenating the sequences associated by the respective vertical and horizontal line which defines p itself.

For example, if p ∈ Vs−1...s−k...∩ Hs0...sk... we associate it to a unique

bi-infinite sequence as follows:

p 7→ . . . s−k. . . s−1s0. . . sk. . . .

We call this well defined map φ. Moreover, since f(Hsi) = Vsi, Vs−1...s−k...= {p ∈ D|f −i+1_{(p) ∈ V} s−i, i= 1, . . . } = {p ∈ D|f−i_{(p) ∈ H} s−i, i= 1, . . . } (1.14) and Hs0...sk...= {p ∈ D|f i_{(p) ∈ H} si, i= 0, . . . }. (1.15) we obtain that p= Vs−1...s−k...∩ Hs0...sk... = {p ∈ D|fi_{(p) ∈ H} si, i= 0, ±1, ±2 . . . } (1.16)

By (1.16), the sequence that we have associated to the point p contains information concerning the behavior of p under iterations of the map f, i.e.

(21)

the element sk of the sequence indicates that fk(p) ∈ Hsk. We further note

that the decimal point in the sequence associated to p separates the past iterates from the future iterates. This implies that the sequence associated to fk_{(p) is obtained by the one associated with p merely by shifting the}

decimal point k places to the left or to the right, according to the sign of

k. To make this clearer we introduce Symbolic Dynamics in the next

subsection.

Symbolic Dynamics

Let S = {0, 1} and Σ be the collection of all bi-infinite sequences of elements in S. If s ∈ Σ we have that

s= {. . . s−n. . . s−1.s0. . . sn. . . }, si ∈ S ∀i.

We introduce some structure on Σ by means of a metric as follows. Let s and ¯s ∈ Σ:

s= {. . . s−n. . . s−1.s0. . . sn. . . }

¯s = {. . . ¯s−n. . .¯s−1.¯s0. . .¯sn. . . }

and we define the distance d(s, ¯s) between s and ¯s as it follows

d(s, ¯s) = ∞ X i=−∞ δi 2|i| where δi=    0 if si = ¯si, 1 if si 6= ¯si.

Two sequences are then close if they agree on a long central block.

We now introduce a map from Σ to itself, which we call the shift map, defined as follows.

s= {. . . s−n. . . s−1.s0. . . sn. . . } 7→ σ(s) = {. . . s−n. . . s−1s0.s1. . . sn. . . }

or simply σ(s)i = si+1. σ is continuous and we have the following result

concerning the dynamics of σ on Σ, i.e. concerning the orbits of points in Σ under iteration by σ.

Theorem 1.3.1. The shift map σ acting on Σ has

• a countable inifnity of periodic orbits of arbitrarily high period; • an uncountable infinity of nonperiodic orbits;

• a dense orbit.

(22)

The Dynamics on the invariant set

We now wish to relate the dynamics of σ on Σ, about which we have a great deal of information given by Theorem (1.3.3), to the dynamics of the Smale horseshoe f on the invariant set Λ. As we noticed before, the sequence associated to f(p) can be deduced by the one associated to p by shifting the decimal point to the right. Using the shift map is then easy to see that the relation

σ ◦ φ(p) = φ ◦ f(p)

holds for every p ∈ Λ. If φ were invertible and continuous, the following relationship would hold for all p ∈ Λ:

f(p) = φ−1◦ σ ◦ φ(p) (1.17) In this case f and σ are said to be topologically conjugate.

If (1.17) holds, then we see that

fn(p) = (φ−1◦ σ ◦ φ)◦· · ·◦(φ−1◦ σ ◦ φ(p)) = φ−1◦ σn_{◦ φ}_{(p), n ≥ 0. (1.18)}

Equation (1.17) also implies that

f−1(p) = φ−1◦ σ−1◦ φ(p) (1.19) so that

f−n(p) = φ−1◦ σ−n◦ φ(p), n ≥ 0. (1.20) The previous relations imply that, if φ is invertible and continuous, the orbit of p ∈ Λ under f would correspond directly to the orbit of φ(p) under σ in Σ and the map f would have the same properties of the shift map on Σ. This is actually the case, thanks to the following theorem.

Theorem 1.3.2. The map φ : Λ → Σ defined before is a homeomorphism. A proof of this result can be found in [12].

We can now state a theorem regarding the dynamics of f on Λ which is a direct consequence of Theorem 1.3.3 and of the topological conjugacy of

f on Λ with σ on Σ.

Theorem 1.3.3. The smale horseshoe map f has

• a countable inifnity of periodic orbits of arbitrarily high period. These periodic orbits are all of saddle type;

(23)

• a dense orbit.

Let p ∈ Λ with corresponding symbol sequence

φ(p) = {. . . s−n. . . s−1s0. . . sn. . . }

We want to consider points close to p and how they behave under it-eration by f as compared with p. Let > 0 be given. We consider an

-neighborhood of p determined by the usual topology of the plane. Hence

there also exists an integer N = N() such that the corresponding neighbor-hood of φ(p) includes the set of sequences s0 _{= {. . . s}0

−n. . . s0−1s00. . . s0n. . . }

such that si = s0i for |i| ≤ N. Now suppose the N + 1 entry in the sequence

corresponding to φ(p) is 0, and the N + 1 entry in the sequence correspond-ing to some s is 1. Thus, after N iterations, no matter how small , the point p is in H0. The point, say p0, corresponding to s0 under φ−1 is in H1,

and they are at least a distance 1−2λ apart. Therefore, for any point p ∈ Λ, no matter how small a neighborhood of p we consider, there is at least one point in this neighborhood such that, after a finite number of iterations, p and this point are separated by some fixed distance.

A system displaying such behavior is said to exhibit sensitive dependence

on initial conditions. A dynamical system displaying sensitive dependence

on initial conditions on a closed invariant set (which consists of more than one orbit) will be called chaotic.

We emphasize that the particular form of the map f on D is not nec-essary for the system to expose a chaotic behaviour. In fact we just need a topological mapping F of the closed square D into the plane, such that

F(D) intersect D in two (connected) components, as in Figure (1.5). One

can then proceed as before to produce an invariant set Λ, so that F restricted to Λ is topologically equivalent to σ.

1.3.2 Conley-Moser conditions

We now wish to find conditions to relate a generical mapping on the square

Dto the shift σ. While in the previous section we considered a map so that D ∩ f(D) consisted of two connected components, we now consider maps so

that D ∩ f(D) consists of an arbitrary collection of connected components, possibly infinite. We expose a set of assumptions necessary for the map to admit a chaotic invariant set, called Conley-Moser conditions developed by Jurgen Moser in [10].

(24)

Figure 1.5: Generalization of the Smale horseshoe

Definition 1.3.1. • A vertical curve is the graph of a function x = v(y)

for which

0 ≤ v(y) ≤ 1, |v(y1) − v(y2)| ≤ µ|y1− y2| for y1, y2 ∈[0, 1]. (1.21) where µ ∈ (0, 1).

• A horizontal curve is the graph of a function y = h(x) for which

0 ≤ h(x) ≤ 1, |h(x1)−h(x2)| ≤ µ|x1−x2| for x1, x2 ∈[0, 1]. (1.22) where µ ∈ (0, 1).

We now want to use vertical and horizontal curves to define vertical and horizontal strips which will play the role of the connected components Vi

and Hi from the previous subsection.

Definition 1.3.2. • Given two nonintersecting vertical strips v1(y) < v2(y), y ∈ [0, 1] we define a vertical strip as

V = {(x, y) ∈ R2|x ∈[v1(y), v2(y)], y ∈ [0, 1]}. (1.23) • Given two nonintersecting horizontal strips h1(x) < h2(x), x ∈ [0, 1]

we define a vertical strip as

(25)

The width of horizontal and vertical strips is defined as d(V ) = max y∈[0,1] |v₂(y) − v₁(y)| (1.25) d(H) = max x∈[0,1]|h2(x) − h1(x)| (1.26)

The following two lemmas, whose proof can be found in ??, are necessary to the inductive process of constructing the invariant set, as we did in the previous subsection.

Lemma 1.3.2. • If V(1)⊃ V(2)_{⊃ . . . V}(k)_{⊃ . . .} _{is a sequence of nested} vertical strips with d(V(k)_{) → 0 as k → ∞ then}

∞

[

k=1

V(k) is a vertical curve.

• If H(1) _{⊃ H}(2)_{· · · ⊃ H}(k) _{⊃ . . .} _{is a sequence of nested horizontal} strips with d(H(k)_→_{0 as k → ∞ then}

∞

[

k=1

H(k) is a horizontal curve.

Lemma 1.3.3. A vertical curve and a horizontal curve (as in Definition

1.3.1) intersect in a unique point.

We come now to the formulation of the Conley-Moser conditions. We let

f : D −→ R2

.

Condition 1. Let S be the set

{1, 2, . . . , N}

if N < ∞ or the set of (positive) integers if N = ∞, and assume that Hi

and Vi for i ∈ S are given disjoint horizontal and vertical strips in D. The

map f takes Hi homeomorphically into Vi, i.e.

f(Hi) = Vi, ∀i ∈ S. (1.27)

Moreover it is required that the horizontal boundary of Hiare mapped onto

the vertical boundaries of Vi. Similarly the horizontal boundaries should

(26)

Condition 2. If H is a horizontal strip inS

i∈SHi then, for any i ∈ S

f−1(H) ∩ Hi = ˜Hi (1.28)

is a horizontal strip (in particular non-empty) and for some fixed ν ∈ [0, 1] we require

d( ˜Hi) ≤ νd(Hi). (1.29)

Similarly, if V is a vertical strip inS

i∈SVi, we require

f(V ) ∩ Vi= ˜Vi (1.30)

to be a non-empty vertical strip ∀i ∈ S, with

d( ˜Vi) ≤ νd(Vi). (1.31)

We note that Condition 2. is essentially equivalent to Lemma 1.3.1. The Conley-Moser conditions, then, describe the contraction and expansion necessary for the map to generate a complex mixing dynamics, as in the horsehoe map. In this case, however, we consider an arbitrary number of strips. The set S is used as index set precisely as before, and we assign to each orbit a bi-infinite sequence

s= {. . . s−k. . . s−1.s0. . . sk. . . }

whose entries now vary in the (possibly infinite) set S. We denote by Σ(S)

the set whose element are these sequences. The shift map is defined precisely as in the previous section. The following theorem relates the map f on D to the shift σ on Σ(S)_.

Theorem 1.3.4. If f is a map satisfying the Conley-Moser conditions with

respect to the horizontal and vertical strips Hi, Vi, i ∈ S,then it admits an

invariant set Λ, on which it is topologically conjugate to a full shift on N symbols, i.e. there is a homemorphism φ : Λ −→ Σ(S) _{so that}

f(p) = φ−1◦ σ ◦ φ(p) ∀p ∈ Λ.

In particular, if N < ∞, Λ is a closed Cantor set in D.

Proof. We sketch here the main steps required to prove this theorem, which

is similar to the one for the horseshoe map shown before. 1. Construct Λ

(27)

2. Define the map φ : Λ −→ Σ(S)

3. Show that φ is a homemorphism. 4. Show that φ ◦ f ≡ σ ◦ φ.

(28)

(29)

Chapter 2

Description of the problem

and first results

The Anisotropic Kepler Problem is one of the earliest nontrivial ex-amples of a map which exhibits a chaotic behaviour. This problem was introduced and developed by M.C. Gutzwiller in [8] to describe the motion of an electron with anisotropic mass tensor moving in a Coulomb field. It can be interpreted as a classical Kepler Problem where one of the axis is a

heavy axis, so that orbits tend to oscillate more about that axis than they

do about the other.

In the following chapters we will expose two methods to prove the ex-histence of chaos in the AKP. In this chapter we show a general formulation for the problems and we develop some of the tools, such as the collision

manifold, necessary to prove chaos by the first method, i.e. there is an

in-variant set in the phase space on which a suitably chosen Poincaré map is topologically conjugate to a shift on an infinite alphabet.

We will then use the new tools and techniques to study the ordinary Kepler problem, whose solution is well known.

2.1 General Outline

The planar anistropic Kepler problem, [9], is the first order system of ordinary differential equations defined by

     ˙q = Mp ˙p = − q |q|3 (2.1) The configuration space is the plane R2 _{\ {0}} _{with Cartesian coordinates}

(30)

q = (q1, q2) while the phase space is the tangent bundle T (R2\ {0}) with

coordinates p = (p1, p2) on the fibers. Equivalently we write

˙q ˙p !

= Xµ(q, p), (2.2)

where Xµ is a vector field on T (R2\ {0}) which satisfies

Xµ(q, p) =   M p − q |q|3  . (2.3)

M here is the 2 x 2 matrix

µ 0

0 1 !

, µ ≥1.

The parameter µ measures the anisotropy of the problem: when µ = 1 we have the usual Kepler problem, while when µ > 1 the problem is no longer spherically symmetric, though we still can find other symmetries in the phase space. We also note that Xµ has a singolarity in q = 0 and it is

analytic on R2_{\ {0} × R}2_{. In Figure () we show how the circular orbit for}

the Kepler problem, with energy fixed to the value H = −1

2, changes as the

parameter µ increase.

Moreover, Xµ correspond to a Hamiltonian system on T (R2 \ {0}) with

Hamiltonian E given by

E = K + V

with the usual central force potential energy

V(q) = − 1

|q| and kinetic energy defined as

K(p) = 1

2pTM p. We can then rewrite (2.1) as:

       ˙q = ∂E ∂p ˙p = −∂E ∂q (2.4)

(31)

Figure 2.1: The w ay the orbit with q = (1 ,0) and p = (0 ,1) with H = − 1 changes 2 for some values of µ .

(32)

Other authors, for example Broucke and Bai and Zheng, define the system by making the potential anisotropic. However it is easy to prove that the corresponding Hamiltonian system is linearly equivalent to (2.4).

The system being Hamiltonian, we know that it has E as an integral of motion. We denote the level set E−1_{(e) as Σ}

e and we will refer to them as

energy surfaces. We will consider only negative energy surfaces, because otherwise solutions in configuration space may escape, i.e. the orbit of a point may leave each compact set containing the origin.

In the next section we show that the projection of the energy surface on the configuration space is compact.

2.2 Change of variables

To study in greater detail the topology of the energy surfaces Σe we make

the following change of variables:    q= rs p= r−12u (2.5) where r is the Euclidean norm |q| and it’s a non negative real number, s is a point in the circle S1 _{and u is a vector in R}2_{. System (2.1) then assumes}

the form                ˙r = √1 r s T_{M u} ˙s = √1 r3 [Mu − (s T_{M u}_)s] ˙u = √1 r3 [ 1 2(sTM u)u − s] (2.6)

The right hand side of (2.6) is an analytic vector field on the open manifold

M = (0, ∞) × S1 × R2_{. By abuse of notation we shall again refer to it as} Xµ.

After a simple computation, the total energy relation on Σe can be written

as

re= 1

2uTM u −1. (2.7)

Choosing e to be negative and using (2.7) we have the following bound on

r:

0 ≤ r ≤ −1

(33)

i.e. for fixed values of e, the dynamics of the point mass are confined to the closed disk of radius −1

e in configuration space. From (2.7) also follows that

on the boundary of the disk we have u = 0. This curve, defined by

 



r = 1_e,

u= 0, (2.8)

is called the oval of zero velocity in Σe and we denote it by Z.

In the new variables, the set Σe for negative e is then

Σe = (r, s, u) ∈ M|r ∈0, −1 e , uTM u= 2(re + 1) (2.9) The following proposition gives us important information on the topology of negative energy surfaces:

Proposition 2.2.1. For e < 0, Σ_e is diffeomorphic to an open solid torus.

Proof. Let B the interior of the ellipse

utM u= 2.

For every fixed r ∈ (0, −1

e] and for arbitrary s the admissible momenta u

verify

uTM u= 2(re + 1),

i.e. they define ellipses which foliates B. Let Φ : S1_{× B →} _Σ

e be defined by: Φ(s, u) =1 e · 1 2uTM u −1 , s, u .

By the above arguments the inverse map Φ−1 _{is trival and Φ is the required}

diffeomorphism. We note that the core circle of the solid torus is mapped onto the oval of zero velocity. Moreover r = 0 correspond to the missing boundary of S1_{× B}_.

2.3 The Collision manifold

We wish to extend the vector field Xµ to the boundary r = 0 of (0, ∞) ×

S1× R2_{. To achieve this we made the following change of the time variable:} dt= r32dτ.

(34)

We write the vector field (2.6) in the new time scale and we obtain          ˙r = r(sT_{M u}₎ ˙s = Mu − (sT_{M u}_)s ˙u = 1 2(s T_{M u}_{)u − s} (2.10)

where, again by abuse of notation, the dot stands for the derivative with respect to τ.

The total energy relation is the same as in (2.7).

The change of time variable gives us some interesting properties: • the vector field is now analytical on the whole [0, ∞) × S1_{× R}2_;

• the submanifold of r = 0 is now invariant under the flow of the vector field.

We now restrict ourselves to a specific negative energy surface Σe. It

meets the boundary r = 0 of [0, ∞) × S1_{× R}2 _{along a submanifold Λ, which}

we shall call the collision manifold, defined by 1

2utM u= 1, r = 0, s ∈ S1 (2.11) because of the energy relation. Λ is trivially diffeomorphic to a 2-torus and the (extended) vector field on it assumes the form

   ˙s = Mu − (sT_{M u}_)s ˙u = 1 2(sTM u)u − s (2.12) This is a standard regularization technique. We emphasize the fact that the collision manifold is independent of the total energy. The change of time scale has also the effect of pasting ain invariant boundary on each of the Σe. In the new time scale orbits which began (ended) in collision (i.e.

q = 0 in the configuration space) with the orbit now tend asymptotically from (toward) Λ.

2.4 Brief study of the Kepler problem

In the following sections we will study in detail the dynamics on the collision manifold, but, at first, we will restrict ourselves to the case µ = 1, the classic Kepler problem of central force. We do this to show our approach on a familiar problem with known solution. In the next chapter we will then

(35)

move to the case µ > 1, with great attention to the properties of the flow which persist and to the ones which don’t.

2.4.1 The Collision manifold for the Kepler problem

Let µ = 1. Then the vector field becomes          ˙r = r(st_u₎ ˙s = u − (st_u_)s ˙u = 1 2(s t_u_{)u − s} (2.13) with total energy relation

re= 1

2|u|2−1 We use angle variables θ and ψ given by

 



s= (cos θ, sin θ)

u =p2(re + 1)(cos ψ, sin ψ) (2.14) so that the vector fields becomes

           ˙r = rp 2(re + 1) cos(ψ − θ) ˙θ =p 2(re + 1) sin(ψ − θ) ˙ψ = p 1 2(re + 1)sin(ψ − θ) (2.15)

We note that in the new variables, the system has a singularity at r = −1

e,

precisely on the oval of zero velocity Z. This is due to the fact that the direction of the momenta u is reversed after a solution meets the curve Z in the configuration space.

We now wish to study the restriction of the flow on the collision manifold. On Λ the vector field is given by:

     ˙θ =√2 sin(ψ − θ) ˙ψ = 1√ 2sin(ψ − θ) (2.16) The use of the new angle variables let us represent the flow easily on the plane, as the 2-torus is obtained simply by identifying the opposite side of the square in the following pictures. We represent the collision manifold (and the flow on it) on a square on the plane by identifiying the opposite

(36)

Figure 2.2: The flow on the collision manifold for µ = 1 and the circles of equilibria.

sides of the square. The axes correspond to the variables θ and φ. The sides of the square are parallel to the axes and correspond to the interval [0, 2π].

Because of (2.16), θ − 2ψ is an integral of motion. Moreover the system has two circles of equilibria, given by θ = ψ and θ = ψ + π.

In Figure (2.2) we have drawn some of the orbits of the problem together with the circles of equilibria. We note all these orbits tends asymptotically away from the circle in black and toward the circle in red. In the following we will refer to these circles as C1 and C2:

C1 = {ψ = θ} (2.17)

C2 = {ψ = θ + π} (2.18)

Let’s consider orbits which begin and/or end in collision. These orbits will be extremely important when we will study the AKP and introduce symbolic dynamics.

Definition 2.4.1. An orbit in the (Anisotropic) Kepler Problem which tends

asymptotically (forward or backward in time) to the collision manifold Λ is called a collision orbit. If an orbit tends to Λ in both time directions is called a bicollision orbit.

(37)

We wish to point out that orbits which stay on Λ at all times are not collision orbits.

2.4.2 Normally hyperbolic submanifolds

To go further on, we introduce the concept of normal hyperbolicity and a useful theorem which guarantees the existence of smooth stable and unstable manifolds for normally hyperbolic submanifolds.

Definition 2.4.2. Let Φtbe a smooth flow on the manifold M and C a

sub-manifold of M made entirely of equilibria for Φt. C is said to be normally

hyperbolic if the tangent bundle of M over C can be decomposed in three

sub-bundles T C, Es _{and E}u _{so that:}

• T C is the tangent bundle of C, • dΦt contracts Es exponentially,

• dΦt expands Eu exponentially.

Theorem 2.4.1. Let C be a normally hyperbolic submanifold of equilibrium

points for Φt. Then there exist smooth stable and unstable manifolds tangent

along C to Es_{⊕ T C} _{and E}u_{⊕ T C} _{respectively. Furthermore, both C and}

the stable and unstable manifolds are permanent under small perturbations of the flow.

Proof. For the proof of Theorem 2.4.1 we refer the reader to [7].

Citing Devaney, we emphasize that Theorem 2.4.1 does not say that a

submanifold of zeroes persists; indeed all zeroes may be destroyed by a small perturbation. However, there must be some invariant manifold nearby.

2.4.3 Collision orbits for the Kepler problem

We now wish to use the previous results to study the set of collision orbits for the Kepler problem. To do this we will apply Theorem 2.4.1 to the circle of equilibria for the Kepler problem. However we first need to prove the following:

Proposition 2.4.1. The circles C₁ and C₂ are normally hyperbolic circles

(38)

Proof. To prove this we first study the differential of X1 at points of C1. We have: DX1(0, θ, θ) =      √ 2 0 0 0 −√2 √2 0 −√1 2 1 √ 2     

which has eigenvalues

√ 2, √−1

2, 0, with corresponding eigenvectors

   1 0 0   ∈ E u_,    0 2 1   ∈ E s_,    0 1 1   ∈ T C.

so that we have the decomposition of T Σe in invariant subspaces required

by the definition.

Similarly, for C2 we have

DX1(0, θ, θ + π) =      −√2 0 0 0 √2 −√2 0 √1 2 − 1 √ 2     

which has eigenvalues

−√2, √1 2, 0, with corresponding eigenvectors

   1 0 0   ∈ E s_,    0 2 1   ∈ E u_,    0 1 1   ∈ T C.

and again we have found the required decomposition of T Σe.

We emphasize the fact that for both C1 and C2 the vector

   1 0 0   is "normal" to the collision manifold Λ, i.e. it points in the direction tangent to r, away from the collision manifold Λ.

(39)

Let’s consider C1. It admits two dimensional stable and unstable manifold

tangent respectively to Span         0 2 1   ,    0 1 1         and Span         1 0 0   ,    0 1 1         . Similarly, C2 admits two dimensional stable and unstable manifolds tangent

respectively to Span         1 0 0   ,    0 1 1         and Span         0 2 1   ,    0 1 1         .

It’s easy to see that the stable manifold of C1, which we shall refer to as Ws(C1), is entirely contained in Λ, and so is the unstable manifold of C2, Wu_(C

2). On the contrary, Wu(C1) and Ws(C2) are transversal to Λ.

We then have proven the following:

Proposition 2.4.2. The set of collision orbits with negative energy consists

(locally) of two smooth cilinder of orbits.

Actually, much more can be said about these two cilinders. Pollard (1966) has proven that these cilinders meet along the oval of zero velocity Z. This means that orbits which leaves Λ reach a point of zero velocity in Z and then fall back again toward Λ. In our terminology we can express this result by

Proposition 2.4.3. All collision orbits in the Kepler problem are bicollision

orbits.

(40)

(41)

Chapter 3

Chaotic dynamics in the

AKP for µ >

9 ₈

In this chapter we show that, choosing the parameter µ in an open and dense subset of (1, +∞), the AKP admits an invariant set in the phase space on which the dynamics is topologically conjugate to a Bernoulli shift on the closure of a suitably choosen subset of Σ, the collection of all bi-infinite sequence of positive and negative integers.

Our procedure will be similar to the one outlined in Chapter 2. At first we will make a change of variables to extend the vector field in an analytical way to the manifold and its boundary. Then we will focus on the dynamics on the collision manifold, and we will point out the differences between the classical problem and the anisotropic one.

Then we will study the topology of the stable and unstable manifolds in great details, so that we will be able to build suitable two-dimensional sections in the phase space. These section will be fundamental to introduce symbolics dynamics. To do this we will make extensive use of Poincaré maps and Conley-Moser conditions, outlined in Chapter 1.

3.1 Change of variables

We start by considering the variables (s, u) given by (2.5). To simplify the form of the equations, especially on the collision manifold, we make another change of variables.

(42)

         s= (cos θ, sin θ) u=p2(re + 1)(√1 µcos ψ, sin ψ) dτ =p 2(re + 1)dτ0 (3.1) The new variables assume values in [0, −1

e] × S

1_{× S}1_{. The set r = 0 is the} collision manifold Λ, while r = −1

e is the zero velocity manifold Z.

We would like to point out that in (3.1) we have made another change in the time variable, so that orbits are slowed down when approaching the zero velocity curve Z.

In the new variables, vector field (2.10) assumes the form:         

˙r = 2r(re + 1)(√µcos ψ cos θ + sin ψ sin θ)

˙θ = 2(re + 1)(sin ψ cos θ − √µ cos ψ sin θ) ˙ψ = √µ sin ψ cos θ − cos ψ sin θ

(3.2) which is analytical on [0, −1

e] × S

1_{× S}1_.

In the following we will study the dynamics of the AKP on both Λ and Z. The topology of the orbits on these manifolds is fundamental to understand how the orbits behave inbetween the two manifolds.

3.2 The Collision manifold in the AKP

On Λ, where r = 0, the vector field is given by:

 



˙θ = 2(sin ψ cos θ − √µ cos ψ sin θ)

˙ψ = √µ sin ψ cos θ − cos ψ sin θ. (3.3) We shall refer to the vector field defining system (3.3) as Xµ. In the next

section we study the equilibria of Xµ.

3.2.1 Equilibria

In the classical Kepler problem, as we saw in Chapter 2 , the vector field admits two normally hyperbolic circle of equilibria. We will show in the following that those circles are broken in the anistropic case. In fact only eight points of equilibrium persist for all values of µ, as we will show in the following proposition.

(43)

Proposition 3.2.1. On Λ, vector field Xµ admits exactly eight equilibrium

solutions, which are given in Table 3.1, together with their characteristic exponents.

Proof. We begin by noting that ˙θ = 0 if and only if both θ and ψ are

multiples of π or else cot θ = √µcot ψ. Moreover ˙ψ = 0 if and only if

again the angles are both multiples of π or else cot ψ = √µcot θ. Then,

simply noting that µ > 1 we get that cot ψ = 0 = cot θ. After examining all of the possibilities we obtain only eight points, as in Table 3.1. To compute the characteristic exponents of the equilibria we just need to study the differential of the vector field:

                                                                                                                       ∂˙r ∂r(r, θ, ψ) = 2(2re + 1)( √

µcos ψ cos θ + sin ψ sin θ)

∂˙r

∂θ(r, θ, ψ) = 2r(re + 1)(−

√

µcos ψ sin θ + sin ψ cos θ)

∂˙r

∂ψ(r, θ, ψ) = 2r(re + 1)(−

√

µsin ψ cos θ + cos ψ sin θ)

∂ ˙θ

∂r(r, θ, ψ) = 2e(

√

µcos ψ cos θ + sin ψ sin θ)

∂ ˙θ

∂θ(r, θ, ψ) = −2(re + 1)(sin ψ sin θ +

√

µcos ψ cos θ)

∂ ˙θ

∂ψ(r, θ, ψ) = 2(re + 1)(cos ψ cos θ +

√ µsin ψ sin θ) ∂ ˙ψ ∂r(r, θ, ψ) = 0 ∂ ˙ψ ∂θ(r, θ, ψ) = −( √

µsin ψ sin θ + cos ψ cos θ)

∂ ˙ψ

∂ψ(r, θ, ψ) =

√

µcos ψ cos θ + sin ψ sin θ

(3.4)

And evaluating on Λ, i.e. r = 0, we obtain:

DX(0, θ, ψ) = ν 0

T

0 A

(44)

where

• ν is a real number, • 0 = 0₀

! ,

• A is a 2 × 2 matrix which gives us the linearization of the restriction of the vector field to Λ.

A simple computation shows that ν is either ±2 or ±2√µ.

Equilibrium point

Characteristic Exponents _{Type on} Λ Dimension of on Λ off Λ Ws Wu (−π 2, − π 2) − 1 2 ± 1 2 √ 9 − 8µ 2 Sink 2 1 (0, 0) − √ µ 2 ± 1 2 √ 9µ − 8 2√µ Saddle 1 2 (π 2, π 2) − 1 2 ± 1 2 √ 9 − 8µ 2 Sink 2 1 (π, π) − √ µ 2 ± 1 2 √ 9µ − 8 2√µ Saddle 1 2 (−π 2, π 2) 1 2 ± 1 2 √ 9 − 8µ 2 Source 1 2 (0, π) √µ 2 ± 1 2 √ 9µ − 8 −2√µ Saddle 2 1 (π 2, − π 2) 1 2 ± 1 2 √ 9 − 8µ 2 Source 1 2 (π, 0) √µ 2 ± 1 2 √ 9µ − 8 −2√µ Saddle 2 1

Table 3.1: The eight equilibria on the collision manifolds and the dimension of the corresponding invariant manifolds.

In Figure (3.1) we show the eight equilibrium points which persist for all values of µ. The red asterisks mark the sources, the blue ones mark the sinks and the black ones mark the saddles. The following result is then trivial: Corollary 3.2.1. If µ > 9₈, the sinks and the sources on the collision

man-ifold have non-real characteristic exponents.

Corollary 3.2.1 implies that the orbits on the collision manifold tend to spiral towards the sinks or away from the sources for an open set of parameters.

(45)

Figure 3.1: The eight points of equilibrium for µ>1.

3.2.2 Topology of the stable and unstable manifolds on Λ

We now wish to study one final qualitative feature of the flow on the collision manifold, that is the ultimate behavior of the stable and unstable manifolds

of the saddle points.

We recall that the stable (resp. unstable) manifold of an equilibrium point consists of the points which tend asymptotically toward (resp. away

from) the equilibrium.

In the case of saddle points, the stable and unstable manifolds on Λ are analytic curves which consist of exactly two orbits each, tending toward or away from the equilibrium. We will refer to the stable (resp. unstable) manifold of an equilibrium point p by Ws_{(p) (resp. W}u_{(p)). Now, to proceed}

in our qualitative study, we need to introduce the notion of heteroclinic orbits and saddle connections.

Definition 3.2.1. Let p and q be two distinct hyperbolic equilibria for a

flow. An orbit γ is said to be heteroclinic if γ lies in Ws_{(p) ∩ W}u_{(q) or}

in Ws_{(q) ∩ W}u_{(p). If p = q, γ is said to be homoclinic. Finally, in the}

two dimensional case, when p and q are saddle points, γ is called a saddle

connection.

In the following sub-sections we will prove the following

Theorem 3.2.1. For an open and dense set of parameters in µ > 1, the

Chaos in the Anisotropic Kepler Problem