Unbounded Orbits in Almost Circular Outer Billiards: a Numerical Exploration

U.U.D.M. Project Report 2013:4

Unbounded orbits in almost circular outer billiards: A numerical exploration

Nikos Kyriakopoulos

Examensarbete i matematik, 30 hp Handledare och examinator: Denis Gaidashev Januari 2013

Department of Mathematics Uppsala University

Abstract

In the present thesis we explore the behavior of orbits of the outer billiard map around almost circular tables. We first give some basic definitions and theorems about symplectic twist maps and the Kolmogorov-Arnol’d-Moser theory. After that we define the outer billiard map which is closely connected to the theory. Then we introduce the system we work on: The almost circular outer billiard, which is the circular billiard with part of it substituted with a flatter function of the form f (x) = a + bx2k and can be thought of as a perturbation of the circular one. The purpose of the thesis is to explore numerically if there can be unbounded orbits for this system, i.e. orbits that eventually escape to infinity. This would mean there are no invariant curves in the system. We present the results of the exploration in the last part.

i ii Contents

Abstract i

1 Introduction 1 1.1 Monotone twist maps ...... 1 1.1.1 Area preservation and symplecticity ...... 2 1.1.2 The twist condition ...... 4 1.2 Generating functions ...... 4 1.3 KAM theory ...... 4 1.3.1 Diophantine numbers ...... 5 1.3.2 The Kolmogorov - Arnol’d - Moser theorem ...... 5

2 The outer billiard map 7 2.1 Example: The circular outer billiard map ...... 9

3 Almost circular outer billiards 11 3.1 Method ...... 16 3.2 Results ...... 17 3.3 Conclusions - Final remarks ...... 19

A C++ code 25

iii iv Chapter 1

Introduction

iThe annulus is defined as A = 1 × [a, b] (1.1) where 1 is the circle /. The cylinder is

C = 1 ×

It is often less ambiguous to work with lifts of maps of A. These are maps of the strip { } A := (x, y) ∈ 2|a ≤ y ≤ b (1.2) where x, thought of as the angular variable, ranges over . The covering map κ : A 7→ A takes (x, y) to (x mod 1, y) and a lift of a map f of the annulus is a map F of the strip, which satisfies

κ ◦ F = f ◦ κ (1.3)

This implies that F(x + 1, y) = F(x, y + (n, 0)) for an integer n which, by continuity, only depends on f and is called the degree of f . In the case that f is an orientation preserving diffeomorphism of the annulus, its degree is 1 and F(x + 1, y) = F(x, y) + (1, 0) for any lift F of f . Denoting T the translation T(x, y) = (x + 1, y), (1.3) gives:

F ◦ T = T ◦ F (1.4)

Any map F which satisfies (1.4) is the lift of a map f with degree 1. We say that f is induced by F.

1.1 Monotone twist maps

Let F(x, y) = (X(x, y), Y(x, y))ii be a diffeomorphism of the strip A = × [a, b] which satisfies the following properties:

1. F preserves the boundaries of A: Y(x, a) = a, Y(x, b) = b

2. F satisfies a certain twist condition

iThe presentation in the majority of this chapter follows [Golé, 2001] iiF :(x, y) 7→ (X(x, y), Y(x, y))

1 3. F preserves orientation and area

4. F satisfies the condition F ◦ T = T ◦ F

Then F induces a map on the annulus A which is called an area and orientation preserving, monotone twist map of the annulus. The properties are analyzed in detail below.

1.1.1 Area preservation and symplecticity

An area preserving map F is a map for which

µ(A) = µ(F(A))

(where µ denotes a measure in n) for every set A ⊂ n, i.e. F maps any set A to a set with equal measure. For an area preserving map F(x, y) = (X(x, y), Y(x, y)) in a 2-dimensional space we have    ∂X(x,y) ∂X(x,y)  =  ∂x ∂y  = ± det J det  ∂Y(x,y) ∂Y(x,y)  1 ∂x ∂y

If det J = 1 the map is called orientation preserving.

Flux

We define the function S (x, y): A → : ∫ (x,y) S (x, y) = YdX − ydx (1.5) (x0,y0)

S is well defined in A (independent of the path of integration) and YdX − ydx = dS . Then the flux of the map is defined by Φ = S (F(x, y)) − S (x, y) (1.6) The flux of an area preserving map F of 2 satisfying (1.4) is defined by

Φ(F) = S (T(x, y)) − S (x, y) (1.7)

Geometrically, this can be interpreted as the oriented area between an embedded circle (y = y0) wrapped once around the cylinder C and its image under the map F (fig 1.1). If F has zero flux, then S ◦ T = S and thus S induces a function s on A such that

∗ f (ydx) − ydx = ds (1.8)

∗ Taking the exterior derivative on both sides of (1.8) we get d( f (ydx) − ydx) = ds2 and thus

∗ f (dx ∧ dy) = dx ∧ dy (1.9)

A map satisfying this equation is called symplectic because it preserves the symplectic form dx∧dy

2 Figure 1.1: The flux is the oriented area (green area is positive, red is negative) between an embedded circle (dashed line) and its image (thick line).

A 2n-dimensional manifold M2n is symplectic if it carries a natural symplectic structure, which in canonical (Darboux) coordinates can be written as ( ) Ω = 0n In −In 0n where 0n is the n-dimensional zero matrix and In is the n-dimensional identity matrix (if iab are the elements of In, then iab = δab, a, b = 1, 2,..., n). A map F is called symplectic if it preserves the symplectic structure, i.e. if the corresponding Jacobian matrix, J, is symplectic. JT ΩJ = Ω (1.10) the condition (1.10) is called the symplectic condition. For 2-dimensional maps, we see that ( )( )( ) ( ) j j 0 1 j j 0 1 JT ΩJ = Ω ⇔ 11 21 11 12 = j12 j22 −1 0 j21 j22 −1 0 or

j11 j22 − j12 j21 = 1 ⇔ det J = 1 where jab are the elements of J. We see that the symplectic condition implies area and orientation preservation for 2-dimensional maps.

3 1.1.2 The twist condition

A map F(X(x, y), Y(x, y)) satisfies the twist condition if the function X(x0, y) is monotone for any given x0. The map then is called a twist map. More specifically, the quantity

∂X(x0, y) , 0 (1.11) ∂y everywhere, so it is either always positive or always negative. If it is positive, then X(x0, y) is strictly increasing and F is called a positive twist map. If it is negative, F is a negative twist map. Geometrically, the twist condition means that there is a diffeomorphism between a fiber (x0, y) and its image under the map F, i.e. the image of a fiber forms a graph over the x-axis.

1.2 Generating functions

The lift F of a twist map of the annulus comes with a function S such that YdX − ydx = dS and S (x + 1, y) = S (x, y). On the other hand, the twist condition on F gives us a function ψ which we view as a change of coordinates ψ :(x, y) 7→ (x, X). In the (x, X) coordinates the equation YdX − ydx = dS implies immediately that the functions −y(x, X) and Y(x, X) are the partial derivatives of S ∂ = − S y ∂ x (1.12) ∂S Y = ∂X S is called the generating function( ) of F in the sense that we can retrieve F from S at least implicitly: ψ−1 , 7→ , − ∂S ψ is given by (x X) x ∂x hence is implicitly given by S . Thus, F can be defined by ( ) ∂S F :(x, y) 7→ X(ψ((x, y)), (ψ(x, y)) (1.13) ∂X

1.3 KAM theory iIf we consider functions defined in m × n, it is very convenient to use expansions which are Taylor expansions in the real variables and Fourier expansions in the angle: ∑ j f (θ, I) = f j,kI exp (2πik· θ) (1.14) j∈n,k∈m For these functions, it is convenient to define norms

∥ f ∥σ = sup | f (θ, I)| (1.15) |I|≤e2πσ,Im(θ)≤σ Under this supremum norm, the spaces constitute a Banach algebra, that is,

∥ f g∥σ ≤ ∥ f ∥σ∥g∥σ (1.16)

−1 −1 Therefore, if ∥ f ∥σ < 1, then ∥(1 + f ) ∥σ ≤ (1 − ∥ f ∥σ) .

iThe presentation in this section follows [De la Llave, 1999]

4 1.3.1 Diophantine numbers

A number ω is called Diophantine of type (K, ν) for K > 0 and ν ≥ 1, if:

p − −ν ω − > K|q| 1 (1.17) q

p ∈ D ,ν for all ∪q . We will denote by K the set of numbers that satisfy (1.17). We denote by D = D ν K>0 K,ν. For maps in higher dimensions, there is a condition that appears in KAM theory: − ν |ω· k − ℓ| 1 ≤ C|k| ∀(k, ℓ) ∈ (n − {0} × ) (1.18)

1.3.2 The Kolmogorov - Arnol’d - Moser theorem

Theorem 1.1. Consider the symplectic manifold M = n × n endowed with the canonical symplectic form. Consider the map F0 : M → M given by:

F0(I, ϕ) = (I, ϕ + ∆(I)) (1.19) where ∆ : n → n is an analytic function, ∂ ∆ j(I) = Φ(I) (1.20) ∂I j

Assume that ω ∈ n satisfies (1.18) and

ω = ∆(I0) (1.21) for some I0 and that ∂ ∆ ≥ κ > det i(I) 0 (1.22) ∂I j in a neighborhood of I0. Let F : M → M be an analytic, exact symplectic map. If ∥F − F0∥σ is sufficiently small, then the map F admits a quasiperiodic orbit of frequency ω. This orbit is dense in an analytic torus which (if ∥F − F0∥σ is sufficiently small) is arbitrarily close in the n analytic topology to the torus {I0} × which is filled densely by the orbit of frequency ω of F0.

5 6 Chapter 2

The outer billiard map

The outer (or dual) billiard map is defined as follows: 2 Let A ∈ be a bounded, convex set and take a point x0 < A. The point x1 is taken such that i the segment x0 x1 is tangent to A at its midpoint and A lies to the right of the vector x0x1. If this ii construction is defined , then the transformation T : x0 7→ x1 is called the outer billiard map (fig. 2.1).

Figure 2.1: There are two tangent lines from the point x0. We choose the one that fits the above definition (green).

i The map can also be defined with A lying to the left of the vector x0x1. The first definition is the time reversal of the second and vice versa. The results are identical. iiThe map is not defined when the boundary of A, γ, contains straight lines. In that case there are points of 2 (the extensions of the straight line segments of γ) for which there are infinitely many points of tangency. Even in that case, the set of the points for which the map is not defined is a set of measure zero.

7 Theorem 2.1. The outer billiard map is area preserving. Proof. (see [Tabachnikov, 2005, ch. 9, p. 151]) Let’s take a set A with smooth boundary γ and ′ two infinitesimally close points X and X on γ. We take the line segments of length 2r which are ′ tangent to γ at their midpoints at X and X . The end points of the tangent line at point X are A ′ ′ ′ ′ and A and the ones of the tangent line at X are B and B . By this construction, A is the image ′ of A under the outer billiard map and B is the image of B. In other words, the outer billiard map ′ ′ takes AB to B A . We repeat the same construction with lines of length r − ϵ, where ϵ is an infinitesimal. The new ′ ′ ′ ′ ′ ′ points C, C , D and D lie on the respective segments XA, XA , X B and X B . We then have the ′ ′ ′ ′ map taking ABDC to A B D C (fig. 2.2).

A B' C D'

O ∆ ∆

X ' X

D C' B A'

′ ′ ′ ′ Figure 2.2: The outer billiard map maps the rectangle ABDC to the rectangle A B D C . We see that the area is preserved.

′ Since X and X are infinitesimally close to each other, we can assume that AX ≈ AO, where O is ′ ′ ′ the point of intersection of AA and BB . Similarly, BX ≈ BO. We then have for the area E of AOB: 1 1 E = OA × OB = (OA)(OB) sin δ AOB 2 2 where δ is the angle between the two vectors. Since δ is infinitesimal, we can make the approximation sin δ ≈ δ and have: 1 E = r2δ AOB 2 Similarly, for the area of COD, we get 1 1 E = (r − ϵ)2δ = r2δ − rϵδ COD 2 2 1 ϵ2δ where the term 2 is of order 3 and thus negligible. ′ ′ Finally, we have EABDC = EAOB − ECOD = rϵδ. Repeating the same steps for the triangles A OB ′ ′ and C OD , we have EA′B′D′C′ = rϵδ and we see that the area is preserved.

8 2.1 Example: The circular outer billiard map { √ } In this case, the billiard table A is the disk A = (x, y): x2 + y2 ≤ 1 . It is obvious by the definition of the map that every circle with center (0, 0) and radius greater than 1 is an invariant curve for the circular map. Depending on the value of the radius, the orbits can be periodic or quasiperiodic, but there is no displacement in the radial direction (fig. 2.3).

Figure 2.3: The segment x0 x1 is tangent to the circle at T , so it is perpendicular to the radius OT and, by construction, x0T = T x1, thus Ox0 = Ox1. Since this construction is repeated on each iteration, the points of the orbit always stay on the circle with radius R = Ox0

The map can be written explicitly as: R(r, θ) = r 1 Θ(r, θ) = θ − 2 arccos r for r > 1 and θ ≤ 0, mod(−2π) and we see that the Jacobian    1 0   J(r, θ) = − √2 1 r2 1− 1 r2 has determinant 1 for all (r, θ), thus the map is area and orientation preserving. Furthermore   ( )   ( ) 1 − √2   1 0 T  2 − 1  0 1   0 1 Ω =  r 1 2   2  = = Ω J J  r  − − √ 1 − 1 0 r2 1− 1 1 0 0 1 r2 thus the circular outer billiard map is symplectic. ∂X(x0,y) To check if the twist condition holds we calculate the derivative ∂y . In this case: ∂Θ(θ , r) 2 0 = − √ ∂ r 2 − 1 r 1 r2

9 which is negative for all r > 1, which means this is a negative twist map. The distance, r, of the initial condition from the center determines if the orbit will be periodic or quasiperiodic. If we think of the circular billiard map as a clockwise rotation of the position vector of the point (rn, θn) on each step, Θ(rn, θn)−θn is the angle by which the vector turns. The twist condition means that this rotation is faster the further away from r = 1 we are. If this angle is a divisor of −2ℓπ (the minus sign appears because we move clockwise, thus θ moves in the negative direction and ℓ is the number of “full circles” the orbit completes before closing), then the orbit will eventually come back to the same point after a number of iterations. This means, that for a k-periodic orbit we have: 1 1 ℓπ −2ℓπ = −2k arccos ⇔ = cos r r k k ∈ and we see that only for certain values of r can√ we have periodic orbits ( ℓ ). For example, k = if we choose ℓ 4, then r has to be equal to 2 (we can check it immediately with the initial i condition (x0, y0) = (1, 1), which maps to (1, −1) 7→ (−1, −1) 7→ (−1, 1) 7→ (1, 1) and we k = 5 see that after 4 iterations we end up where we started), while by choosing ℓ 2 , we have a 5-periodic orbit that completes 2 full circles before it closes (fig. 2.4).

x -3 -2 -1 1 2 3

-1

-2

-3

k = k = 5 Figure 2.4: Periodic orbits for ℓ 4 (blue) and ℓ 2 (red). The dashed lines are the respective invariant circles.

( √ ) ( √ ) ( √ ) ( √ ) ( √ ) i , π 7→ , 7π 7→ , 5π 7→ , 3π 7→ , π In polar coordinates: 2 4 2 4 2 4 2 4 2 4

10 Chapter 3

Almost circular outer billiards

Our aim in this thesis is to explore if there are escaping orbits (orbits which eventually reach any prescribed distance R > 1 from the center (0, 0)) around a table A with an almost circular boundary. By “almost circular” we mean the following: The boundary of A is the unit circle on which we have replaced a small part with another, “flatter” function. More specifically, the boundary γ consists of two curves:  √   1 − x2 if x ∈ [−1, −ϵ) ∪ (ϵ, 1] c : y =  1  2k a + bx if x ∈ [−ϵ, ϵ] (3.1) √ 2 c2 : y = − 1 − x if x ∈ (−1, 1) and γ = c1 ∪c2. 2ϵ is the size of the perturbation and k is the amount of flatness we introduce (fig. 3.1). The larger k is, the flatter the substituting function is (the absolute value of the derivative of 10 6 1 x is smaller than that of x near x = 0). a and b are calculated from ϵ and k so that c1 is a C curve.

ϵ 1 = − √ b − − ϵ2 2kϵ2k 1 √ 1 (3.2) a = 1 − ϵ2 − bϵ2k

y y y

1.0 1.0 1.0

0.5 0.5 0.5

x x x -1.0 -0.5 0.5 1.0 -1.0 -0.5 0.5 1.0 -1.0 -0.5 0.5 1.0

-0.5 -0.5 -0.5

-1.0 -1.0 -1.0

Figure 3.1: Different boundaries for different values of ϵ and k: (a) ϵ = 0.7, k = 2, (b) ϵ = 0.7, k = 4, (c) ϵ = 0.3, k = 4. The gray line is the unit circle.

11 Since the almost circular billiard map is identical to the circular one for most of its part (everywhere except a small area defined by ϵ), it can be thought of as a perturbation of the circular outer billiard map. √ √ The tangent lines of the table at the points (−ϵ, 1 − ϵ2) and (ϵ, 1 − ϵ2) define the area I (shaded region in fig. 3.2): {{ } { }} 1 − ϵx 1 + ϵx I = (x, y) ∈ (−∞, −ϵ] × : y < √ ∩ (x, y) ∈ (−∞, −ϵ] × : y > √ ∪ {{ 1}− ϵ2 }1 − ϵ2 1 − ϵx { } (x, y) ∈ (−ϵ, ϵ) × : y < √ ∩ (x, y) ∈ (−ϵ, ϵ) × : y > a + bx2k 1 − ϵ2 (3.3) For every point that lies outside this area, the map is defined by tangency on the unit circle, so it is the exact same map as the circular outer billiard. The distance r from the center (0, 0) remains unchanged and only the angle θ changes. For points that lie in I, though, the point of tangency 2k lies on the curve a + bx and we may have radial displacement, depending on yn.

y 2

I 1

x -2 -1 1 2

-1

-2

Figure 3.2: Outside the area I, the map follows the same rules as the circular outer billiard

Lemma 3.1. For points in I:

• If the point (xn, yn) is below the line y = a (if yn < a), the image (xn+1, yn+1) will be√ closer to , = 2 + 2 the center than (xn yn) (the distance rn+1 will be smaller than the distance rn xn yn).

• If the point (xn, yn) is above the line y = a, rn+1 will be greater than rn.

12 • If the point (xn, yn) is exactly on the line y = a, the point of tangency is the point (0, a), 2k which is the maximum of the curve a + bx , and thus the image (xn+1, yx+1) will be a reflection of (xn, yn) about the y-axis and in that case rn+1 = rn.

Proof. A point X = (xn, yn) of the system belongs to the invariant circle of the unperturbed system (the circular outer billiard) 2 + 2 = 2 Ci : x y rn (3.4)

Every point of this circle lies on the exact same distance rn from the origin. To see if the radial displacement is positive or negative (if the image is further away from the center or closer to it), we compare the distance of the image from the origin, rn+1, to the radius rn of the invariant circle (3.4) (dashed circle in fig. 3.3). For points in I, the point of tangency is a point XT = (xT , yT ) and the slope λ of the tangent line 2k LT is given by the derivative of a + bx at that point.

d ( ) λ = + 2k = 2k−1 (xT ) a bx 2kbxT (3.5) dx x=xT and the normal line at the tangency point, Ln (the almost vertical gray line in fig. 3.3), will have slope 1 1 n(x ) = − = − (3.6) T λ 2k−1 2kbxT

Since b < 0, the slope (3.6) is negative for xT < 0 and positive for xT > 0. For xT = 0 the normal line is the y-axis. ( √ ) , , = 2 + 2 Now, let’s compare this slope with that of the radius of a circle (0 0) rT xT yT which passes through (xT , yT ), i.e. the line LC that passes through XT and the origin (orange line in fig. 3.3). We have ( √ ) d x λ = 2 − 2 = − √ T T (xT ) rT x = dx x xT 2 − 2 rT xT and √ r2 − x2 1 T T yT nT (xT ) = − = = λT xT xT So, 2 2k 2 2k x + 2kbx yT 1 x + 2kbx yT n − n = − T T = T T (3.7) T 2k+1 2k+1 − 2kbxT xT 2kb The denominator of the second fraction is always positive (b < 0) and for the numerator, for small ϵ and taking (3.2) into account, we have:

ϵ2(1−k)y 2 − √ T 2k xT xT 1 − ϵ2 which is also positive, because |xT |, yT and ϵ all take values between 0 and 1. That means that for xT < 0 the line Ln is on the left of the corresponding radius LC of the circle CT and when xT > 0 it is on the right. When xT = 0 (and, of course, when xT = ±ϵ) the two lines coincide (fig. 3.3).

13 Lt 1.0 XT

X Ci

0.5

CT M

O -1.0 -0.5 0.5 1.0

-0.5

-1.0

LC L

Figure 3.3: The construction described above. X is a random point in I and XT is the point of tangency. The orange circle, CT is the circle centered at the origin with radius rT .

If we take the line yn L = x (3.8) xn which is the line that passes through (xn, yn) and the origin, then the normal Ln cuts L somewhere between xT and (0, 0) if xT < 0, on the origin if xT = 0 and between the origin and +∞ if xT > 0 (fig. 3.4). We denote M the point of intersection of Ln and L. This point has two properties:

• It lies on the line Ln. Any point on the normal Ln is equidistant to the points (xn, yn) and (xn+1, yn+1), as Ln is the perpendicular bisector of the segment that connects the point X and its image under the map and so, for any point Q ∈ Ln, a circle centered at Q with radius QX, will contain both X and its image.

• It lies on the line L. For any point P ∈ L, the circle C with center P and radius PX is tangent to the invariant circle of the unperturbed system, Ci, which contains X (the dashed circle in fig. 3.3) at the point X. Thus, if PX > rn, the circle C will lie outside the circle Ci

14 (except for the point X) and if PX < rn it will lie inside of Ci. If PX = rn the two circles will coincide (fig. 3.4).

X 1 P

-2 -1 1 2 3 4 O P

-1

-2 L

-3

Figure 3.4: Circles tangent to the invariant circle Ci (dashed circle) at the point X for P between O and X (red circle) and for P between O and +∞ (green circle).

Since the position of M depends on xT , the radius of the circle C centered at M will depend on xT .

M satisfies both these properties (it lies on both lines), so we see that the image of X will be a point on a circle which is either bigger or smaller than Ci, depending on if xT is larger or smaller than 0, which depends on the exact position of X in I. If yn > a then the circle C will be bigger than Ci and thus the image of X will lie on a distance grater than rn, while if yn < a, the distance rn+1 < rn.

Remark 3.2. If we split I into two parts, Iu and Il, above and below the line y = a respectively, we can easily see that the area of Iu is larger than that of Il. This means that we generally expect more frequent “outward” jumps than “inward” ones, as we assume that the orbits are equidistributed on the plane xy, which leads us to believe there must be some escaping orbits. In particular, if we take an arbitrary circle Cg with radius g, we can calculate its points of +ϵ −ϵ intersection with the lines y = √1 x and y = √1 x , which define the area I, and with the line 1−ϵ2 1−ϵ2 i y = a, which splits the area into the upper and lower regions Iu and Il. Using polar coordinates , the first two lines can be written as:

1 ru = √ cos θ 1 − ϵ2 + ϵ sin θ (3.9) 1 rl = √ cos θ 1 − ϵ2 − ϵ sin θ

iWith a 90 degrees counter-clockwise rotation, so that for θ = 0 we are on the y-axis: x = −r sin θ, y = r cos θ

15 and the line y = a becomes a r = (3.10) m cos θ

By equating ru, rl and rm with the radius g, we find the corresponding θ of the intersection points (fig. 3.5). The arc θu − θm is always longer than the arc θm − θl, which means that outward jumps are more probable than inward ones. The difference between the two arc lengths gets smaller the further away from the origin we go, which makes the rate of escape slower for greater distances, but never zeroi.

4 rm, Θm

rl, Θl ru, Θu H L

H L H L 3

Θ -3 -2 -1 0 1

Figure 3.5: Calculation of the points of intersection of an arbitrary circle (dashed line) and the lines that define the areas Iu and Il.

3.1 Method

We choose the values of ϵ and k and an initial point (x, y) and we simulate the orbit for a certain number of steps (i). On each iteration we need to calculate the next point according to the map (xn, yn) 7→ (xn+1, yn+1). 2 For a random point Xn = (xn, yn) ∈ lying outside the table, the point of tangency XT = (xT , yT ) with the billiard table is either√ on the circular part, where the function describing the boundary of the table is simply f (x) = ± 1 − x2 or on the perturbed piece, where f (x) = a + bx2k.

In both cases, the point XT can be found by solving the equation

− yn yT = d f − (3.11) xn xT dx x=xT iexcept for r → ∞

16 √ • When f (x) = ± 1 − x2, (3.11) yields

2 + 2 2 − + − 2 = (xn yn)xT 2xn xT 1 yn 0 (3.12)

with solution √ ± 2 + 2 − xn yn xn yn 1 x = (3.13) T 2 + 2 xn yn Since in this case we can find the solutions analytically, the only thing remaining is choosing the one of the two that is in accordance with the definition of the map (the one leaving the 2 table to the right), which we can determine by xn and yn, i.e. by the position of Xn in .

• When f (x) = a + bx2k, (3.11) yields

− 2k − 2k−1 + − = (2k 1)bxT 2kbxn xT yn a 0 (3.14)

This equation is not analytically solvable for k ≥ 3. This leads us to use a numerical method in order to determine the point of tangency XT . We use the bisection method, as the speed − of convergence to the point even with this method is very fast (to the accuracy of 10 16) and it is the most straightforward one that presents no problems during the process.

After finding the point of tangency (xT , yT ) it is trivial to find the point Xn+1 = (xn+1, yn+1) by solving xn + xn+1 xT = 2 (3.15) y + y + y = n n 1 T 2 for xn+1 and yn+1. During the simulation we take the value r of the distance of (xn, yn) from the origin and the maximum value of r, rmax.

3.2 Results

From simulations we can get a general picture of how orbits look like on the plane (fig. 3.6, fig. 3.7). These particular pictures were made with a relatively large ϵ so that it is clearly shown that there exist islands of stability. As ϵ gets smaller these white “holes” in the picture still exist but are very difficult to see because they are very narrow. For the actual, 200 billion step simulations we used smaller ϵ values, between 0.0001 and 0.1. We ran the simulation (code in appendix A) for a large number of steps (200 billion iterations), for different values of ϵ and k and for different initial conditions. The evolution of an orbit contains both outward and inward jumps but we only care about the growth of the maximum distance from the origin, rmax (fig. 3.8). The first thing we observe is that rmax is growing with the number of steps i with the rate of this outward movement depending strongly on ϵ and less on k (fig. 3.9, 3.10, 3.11), unless, of course, we start at a point which belongs to a periodic or quasiperiodic orbit, or an area sufficiently close to such a point. In fig. 3.6 and fig. 3.7 the white areas are “stable” regions in the sense that if a point starts in them it will stay there (fig. 3.12). The black area is the chaotic region which we are

17 Figure 3.6: The first 2 million iterations of the orbit with initial condition (−1, 1) for k = 3 and different values of ϵ: (a) ϵ = 0.15, (b) ϵ = 0.25, (c) ϵ = 0.4.

Figure 3.7: The first 2 million iterations of the orbit with initial condition (−1, 1) for ϵ = 0.1 and different values of k: (a) k = 2, (b) k = 3, (c) k = 4.

interested in, as it is in this region that the orbits move inward or outward, by moving through the chaotic sea between these islands of stability. In this chaotic region we observe sensitivity to initial conditions, i.e. orbits with initial conditions very close to each other evolve in completely different and unpredictable ways (fig. 3.13). If our initial condition does not belong to such a bounded orbit (if it is in the chaotic region), then the orbit moves along the unperturbed system’s invariant circle until it enters region I and then jumps to another invariant circle of the unperturbed system. This new circle can have a bigger or smaller radius (depending on if the point was in Iu or Il before the jump) but, eventually -at least for the first 200 billion iterations- there was always a jump to an outer circle, although it was observed less frequently the further away we went from (0, 0), which means that the rate of moving outward does not only depend on the pair (ϵ, k) and the initial condition, but on the distance r too (fig. 3.14).

For some pairs (ϵ, k), there are values of rmax where this growth seems to stop (see for example fig. 3.15), something that could be evidence of an invariant curve of the system. This cannot be the case, though, as for different initial conditions, this particular value is overcome by the orbit, meaning that another orbit grows further away and we even have cases for which the same orbit has grown past the value on which it seems to have settled later on (fig. 3.16).

18 r rmax

100 100

80 80

60 60

40 40

20 20

i i 5.0 ´ 1010 1.0 ´ 1011 1.5 ´ 1011 2.0 ´ 1011 5.0 ´ 1010 1.0 ´ 1011 1.5 ´ 1011 2.0 ´ 1011 r rmax 25 25

20 20

15 15

10 10

5 5

i i 5.0 ´ 1010 1.0 ´ 1011 1.5 ´ 1011 2.0 ´ 1011 5.0 ´ 1010 1.0 ´ 1011 1.5 ´ 1011 2.0 ´ 1011

Figure 3.8: Evolution of the distance r and the maximum distance rmax of the orbits (a) with initial condition (−10, 1), ϵ = 0.03 and k = 4 and (b) with initial condition (−1, 1), ϵ = 0.005 and k = 3.

ln rmax ln rmax

H L 3.8H L 3.0 3.6

2.5 3.4

3.2

2.0 3.0

2.8 ln i 20 21 22 23 24 25 26

H L ln i 20 21 22 23 24 25 26

H L Figure 3.9: Logarithmic rmax-i plots for k = 3 and different ϵ: (a) ϵ = 0.005, (b) ϵ = 0.05

3.3 Conclusions - Final remarks

From the simulations, it looks like unbounded orbits do exist. In fact, it looks like every orbit that does not start in one of the trapping regions of the system escapes to infinity. This hints that there are no invariant curves of the system, as they would be uncrossable (moving outward or inward). Of course, we have not produced any theoretical results and we have not come up with a rigorous explanation of why that happens. There is always the chance that if we ran the simulation for longer times we would see the growth stopping. The orbits that seem to have settled at a distance have either come close to the boundary of a

19 r

1.4

1.2

1.0

0.8

0.6

0.4

0.2

i 5.0 ´ 1010 1.0 ´ 1011 1.5 ´ 1011 2.0 ´ 1011

Figure 3.10: Evolution of an orbit with ϵ = 0.0001, k = 3. We see the radial displacement is small due to the fact that the area I is very narrow but still existent and probably important for a greater number of iterations.

rmax rmax

3.0 3.0

2.5 2.5

2.0 2.0

1.5 1.5

i i 20 21 22 23 24 25 26 20 21 22 23 24 25 26

Figure 3.11: Logarithmic rmax-i plots for ϵ = 0.003 and different k: (a) k = 3, (b) k = 4

“trapping” region and will eventually move away (we see the same behavior for smaller numbers of steps in many r-i graphs, like, for example, fig. 3.17 at about 185 billion steps) or there could be a computational error that makes the orbit jump inside the trapping region. In any case, the conclusion is that these anomalies are not compatible with the existence of invariant curves of the system.

20 Figure 3.12: Trapped orbit for k = 3 and ϵ = 0.15 (a) The first 3 million steps of the orbit with initial condition (0, 6.5) and (b) detail of the orbit. We see that even within the islands there is no order. There are smaller islands within the islands which are areas close to periodic and quasiperiodic orbits, but the majority of the orbits are chaotic, only contained, as opposed to the chaotic sea on the outside.

i 1 ´ 1010 2 ´ 1010 3 ´ 1010 4 ´ 1010 5 ´ 1010 0.00001,-2

-0.00001,-2 H L

H L Figure 3.13: Evolution of two orbits with initial conditions starting on the same invariant circle of the unperturbed system and very close to each other (k = 4, ϵ = 0.05).

21 rmax

i 5.0 ´ 1010 1.0 ´ 1011 1.5 ´ 1011 2.0 ´ 1011 -1,1 -10,1 H-15,L1 H L ln rmax H L 3.5 H L

3.0

2.5

2.0

ln i 20 21 22 23 24 25 26

H L -1,1 -10,1 H-15,L1 H L H L 11 Figure 3.14: Linear and logarithmic rmax-i plots for 2· 10 steps for ϵ = 0.005, k = 3 and different initial conditions. rmax is taken every 100 million steps.

rmax r

80 80

60 60

40 40

20 20

i i 5.0 ´ 1010 1.0 ´ 1011 1.5 ´ 1011 5.0 ´ 1010 1.0 ´ 1011 1.5 ´ 1011

Figure 3.15: rmax-i and r-i plots for k = 4 and ϵ = 0.007 with initial condition (−10, 1). The growth seems to stop but the distance from the origin has already become greater than the settling value earlier in the simulation.

22 r

i 5.0 ´ 1010 1.0 ´ 1011 1.5 ´ 1011 2.0 ´ 1011 -10,1

-50,1 H L

H L Figure 3.16: r-i plots for 2· 1011 steps for ϵ = 0.007, k = 4 and different initial conditions.

i 5.0 ´ 1010 1.0 ´ 1011 1.5 ´ 1011 2.0 ´ 1011

Figure 3.17: r-i plot for k = 3 and ϵ = 0.008 with initial condition (−1, 1).

23 24 Appendix A

C++ code

#include ”stdafx .h” #include #include #include #include #include #include using namespace std; void upper (double x,double y,double *xx,double *yy); void lower (double x,double y,double *xx,double *yy); void c1 (double x,double a,double b,double epsilon,double ye, double *y); void c2 (double x,double a,double b,double epsilon,double ye, double *y); void f (double x,double a,double b,double k,double *y); void bisection (double a,double b,double k,double x0, double y0,double l, double r, double *x,double *y); double g (double x,double a,double b, double k, double x0, double y0); int main() {double x0,y0,x1,y1,x2,y2,r,epsilon,a,b,ye,yc1,yc2,fy,rmax,theta,i , steps ; int k,c,cc, intv ,xyout;

cout<<”Input␣epsilon␣and␣k\n”; cin>>epsilon; cin>>k;

steps=2*pow(10,11.0); /* number of steps for the simulation*/ intv=pow(10,8.0); /*intv is the number of steps every which rmax output is produced*/ xyout=0; /*if xyout is 0 no x, y, r, theta output is produced, if not, it is produced every xyout steps*/

b=(−epsilon)/sqrt(1−pow(epsilon,2))*1/(2*k*pow(epsilon,2*k−1)); a=sqrt(1−pow(epsilon,2))−b*pow(epsilon,2*k);

ye=sqrt(1−pow(epsilon,2));

25 cout<<”Input␣x0␣and␣y0\n”; cin>>x0; cin>>y0; r=sqrt(pow(x0,2)+pow(y0,2)); theta=atan2(−x0,y0); rmax=sqrt(pow(x0,2)+pow(y0,2));

i=1.0; c=1; cc=1; ofstream file ; file .open(”xyrlist .dat”); file <

if (xyout==1) { file <

/*Deciding which part (x0,y0) is in and calculating the point of tangency x1 according to the corresponding function*/ if (x0>1) { lower(x0,y0,&x1,&y1); } else if (x0>epsilon&&x0<=1) { if (y0<0) { lower(x0,y0,&x1,&y1); } else upper(x0,y0,&x1,&y1);

26 } else if (x0>0&&x0<=epsilon) { if (y0<0) { lower(x0,y0,&x1,&y1); } else if (y0>=yc2) { upper(x0,y0,&x1,&y1); } else bisection (a,b,k,x0,y0,x0,epsilon,&x1,&y1); } else if (x0>−epsilon&&x0<=0) { if (y0<0) { lower(x0,y0,&x1,&y1); } else if (y0>=yc2) { upper(x0,y0,&x1,&y1); } else if (y0=yc1) { bisection (a,b,k,x0,y0,−epsilon,epsilon,&x1,&y1); } else bisection (a,b,k,x0,y0,x0,epsilon,&x1,&y1); } else if (x0>−1&&x0<=−epsilon) { if (y0<0) { lower(x0,y0,&x1,&y1); } else if ((y0>=0&&y0=yc2)) { upper(x0,y0,&x1,&y1); } else bisection (a,b,k,x0,y0,−epsilon,epsilon,&x1,&y1); } else if (y0=yc2) { upper(x0,y0,&x1,&y1);

27 } else bisection (a,b,k,x0,y0,−epsilon,epsilon,&x1,&y1); x2=2*x1−x0; y2=2*y1−y0; r=sqrt(pow(x2,2)+pow(y2,2)); theta=atan2(−x2,y2);

/*exporting data to file */

if (xyout!=0) { if (cc==xyout) { file <

if (r>rmax) { rmax=r; file2 <

x0=x2; y0=y2;

if (c==intv) { cout<

i=i+1.0; c++; cc++; } file . close (); file2 . close (); file3 . close ();

28 return 0;}

void upper (double x,double y,double *xx,double *yy) /*Calculating the point of tangency and the image for the ”Upper” case*/ {double xx1,xx2; xx1=(x+y*sqrt(pow(x,2)+pow(y,2)−1))/(pow(x,2)+pow(y,2)); /*1st root for the tangency point’s x−component*/ xx2=(x−y*sqrt(pow(x,2)+pow(y,2)−1))/(pow(x,2)+pow(y,2)); /*2nd root for the tangency point’s x−component*/

if (y>0) /*choose root*/ { *xx=max(xx1,xx2); } else { *xx=min(xx1,xx2); }

/* if (*xx>=1||*xx<=−1) { *xx=1; *yy=0; } else { *xx=−1; *yy=0; } } else {

}*/ *yy=sqrt(abs(1−pow(*xx,2))); } void lower (double x,double y,double *xx,double *yy) /*Calculating the point of tangency and the image for the ”Lower” case*/ {double xx1,xx2; xx1=(x+y*sqrt(pow(x,2)+pow(y,2)−1))/(pow(x,2)+pow(y,2)); xx2=(x−y*sqrt(pow(x,2)+pow(y,2)−1))/(pow(x,2)+pow(y,2));

if (y>0) { *xx=max(xx1,xx2); } else {

29 *xx=min(xx1,xx2); } /* if (*xx<=−1||*xx>=1) { if (y>0) { *xx=1; *yy=0; } else { *xx=−1; *yy=0;

} } else {

} */ *yy=−sqrt(abs(1−pow(*xx,2))); } void c1 (double x,double a,double b,double epsilon,double ye,double *y)/*Calculating yc1*/ { *y=ye+(epsilon/ye)*(x+epsilon); } void c2 (double x,double a,double b,double epsilon,double ye,double *y)/*Calculating yc2*/ { *y=ye+(−epsilon/ye)*(x−epsilon); } void f (double x,double a,double b,double k,double *y) { *y=a+b*pow(x,2*k); } double g (double x,double a,double b, double k, double x0, double y0)/*equation for the point of tangency on f*/ {double y; y=(2*k−1)*b*pow(x,2*k)−2*k*b*x0*pow(x,2*k−1)−a+y0; return(y); } void bisection (double a,double b,double k,double x0, double y0,double l, double r, double *x,double *y) { double accuracy; accuracy=pow(10,−16.0);

if (g(l ,a,b,k,x0,y0)<0) { while(r−l>accuracy) { if (g(( l+r)/2,a,b,k,x0,y0)<0) {

30 l=(l+r)/2; } else r=(l+r)/2; } } else while(r−l>accuracy) { if (g(( l+r)/2,a,b,k,x0,y0)>0) { l=(l+r)/2; } else r=(l+r)/2; }

*x=(r+l)/2; *y=a+b*pow(*x,2*k); }

31 32 Bibliography

[Dolgopyat et al., 2009] Dolgopyat, D. and Fayad, B.: Unbounded orbits for semicircular outer billiard, Ann. Henri Poincaré 10 (2009), 357-375

[Ichtiaroglou et al., 2000] Ichtiaroglou, S., Meletlidou, E. and Wodnar, K.: A method for evaluating the stability of fixed points in perturbed symplectic maps, Chaos, Solitons and Fractals 11 (2000), 245-250

[Arnol’d, 1980] Arnol’d, V.I.: Mathematical Methods of Classical Mechanics, Springer (1980)

[De la Llave, 1999] De la Llave, R.: A tutorial on KAM theory, Smooth Ergodic Theory and its Applications (1999), 175-292

[Golé, 2001] Golé, C.: Symplectic twist maps: Global variational techniques, World Scientific Publishing Co. (2001)

[Katok, Hasselblatt, 1995] Katok, A., Hasselblatt B.: Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press (1995)

[Tabachnikov, 2005] Tabachnikov, S.: Geometry and Billiards, A.M.S. Mathematics Advanced Study Semesters (2005)