EXTERIOR BILLIARDS
Math118, O. Knill
ABSTRACT. We look here briey at the dynamical system called exterior billiard. Ane equivalent tables
lead to conjugated dynamical systems. One does not know, whether there is a table for which an orbit can
escape to in
THE LORENZ SYSTEM II
Math118, O. Knill
ABSTRACT. This is a continuation of the discussion about the Lorenz system and especially on the r dependence of the attractor.
OVERVIEW OVER BIFURCATIONS. We x the parameter = 10, b = 8/3.
For 0 < r < 1, the origin
2/2/2004 INTRODUCTION
Math118, O. Knill
ABSTRACT. We discuss the methodology and organization of the course.
The subject. Dynamical system theory has matured into an independent mathematical subject. It is linked to
many other areas of mathematics and has
FRACTALS
Math118, O. Knill
ABSTRACT. In order to dene a strange attractor, we have to look at the notion of a fractal, a set of
fractional dimension. The term fractal had been introduced by Benoit Mandelbrot in the late 70ies. We will
see more about fract
Calculating square
roots with
dynamical systems
Details to a result mentioned in the lecture
of February 4th 2005. The example can also found in
the book (introduction).
Math118r O.Knill
Theorem:
Dene
With the
initial
condition
The orbit
converges to
expo
2/9/2005: LYAPUNOV EXPONENT
Math118, O. Knill
ABSTRACT. We demonstrate that the logistic map f (x) = 4x(1x) is chaotic in the sense that the Lyapunov
exponent, a measure for sensitive dependence on initial conditions is positive.
LYAPUNOV EXPONENT. For an
2/7/05 ICE
Math118, O.Knill
Each of the graphs are dened by a function f . Iterate T (x) = f (x) using the cobweb construction,
starting near a xed point.
8
2
1.5
6
1
4
0.5
-1.5
-1
-0.5
0.5
1
1.5
1
1.5
2
2
-0.5
-1
0.5
1
1.5
2
2.5
3
-1.5
0.5
2
0.4
0.3
1
0.
2/11/2005: ENTROPY AND CHAOS
Math118, O. Knill
ABSTRACT. We look today at some notions of chaos. One denition is the positivity of a number called the
positive entropy, an other is the positivity of the Lyapunov exponent for every orbit which is not event
1.4 We dene the map f (x) = 4x + sin(x) mod 1 on the interval [0, 1].
Homework 1. Week
Math118, O.Knill
1.1 We consider the interval map f (x) = f4 (x) = 4x(1 x). For this particular
value, the logistic map is also called the Ulam map. We have met it in t
2/9/05 ICE
Math118, O.Knill
Find the Lyapunov exponent of a periodic orbit of period 2 of the map
f (x) = 3x mod 1
The graph below shows the second iterate of f .
1
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1
5/4/05 FINAL QUIZZ
Math118, O.Knill
1) Which of the following dynamical systems have a discrete time? We replace map or differential equation with system.
a) Henon system
b) Geodesic system
c) Cellular automata system
2) What is a semigroup?
a) A set G wi
4/25 QUIZ NR 11
Math118, O.Knill
Name:
1) True or False: there are always equilibrium solutions to the Newtonian n-body problem.
2) What is the minimal number of bodies for which one can prove that an escape to innity in
nite time is possible?
a) 2
b) 3
c
THE LORENZ SYSTEM
Math118, O. Knill
ABSTRACT. In this lecture, we have a closer look at the Lorenz system.
THE LORENZ SYSTEM. The dierential equations
x
y
z
= (y x)
= rx y xz
= xy bz .
are called the Lorenz system. There are three parameters. For = 10, r
3/7/05 QUIZZ NR 5
Math118, O.Knill
Name:
a) The graph of the function f (x) = sin(x) in the plane.
b) A lled triangle.
c) The set cfw_1, 1/2, 1/3, 1/4, 1/5, 1/6, .
d) The Cantor set.
5) Which of the following properties does a strange attractor K of a die
3/14/15926. QUIZ NR 6
Math118, O.Knill
Name:
5) Which of the following are open mathematical problems?
a) Every billiard in a triangle has a periodic orbit.
b) Every exterior billiard has the property that for (x, y) outside the table, T n (x, y) .
c) The
4/4 QUIZ NR 8
Math118, O.Knill
5) A xed point of a quadratic map f (z) is dened to be stable, if (only one answer applies):
a) f (z) < 1.
Name:
b) |f (z)| 1
c) |f (z)| = 0.
d) |f (z)| < 1.
1) With fc (z) = z 2 + c, which of the following statements are tr
4/11 QUIZ NR 9
Math118, O.Knill
Name:
8) When doing symbolic dynamics for the Arnold cat map T (x, y) = (2x + y, x + y) mod 1,
one uses a subshift of nite type over an alphabet with a minimal amount of letters. This
alphabet has
a) 2 elements.
b) 3 elemen
2/28/05 QUIZZ NR 4
Math118, O.Knill
a) An orbit in the plane which stays in a bounded region is either asymptotic to an equilibrium point or to a limit cycle.
b) An orbit in the plane which is not asymptotic to a limit cycle is attracted to an equilibrium
3/21 (Start of Spring!) QUIZ NR 7
Math118, O.Knill
Name:
6) True or false?
If d(x, y) = 1/10, then d(x), (y) = 1/10, where is the shift.
7) A lattice gas cellular automaton
a) conserves the total momentum of the particles
b) is used to simulate uids
1) Wh
4/18 QUIZ NR 10
Math118, O.Knill
5) The dynamical logarithm problem is the problem
a) to nd the time to reach from a point x to a neighborhood of a point y.
b) to nd the point y which is reached after time t when starting from x.
c) to nd the initial poin
2/2/04 WHAT ARE DYNAMICAL SYSTEMS? Math118, O.Knill
ABSTRACT. We discuss in this lecture, what dynamical systems are and where the subject is
located within mathematics.
A FIRST DEFINITION.
The theory of dynamical systems deals with the evolution of syste
2. Homework set
Math118, O.Knill
2.1 a) Realize the Henon map
T (x, y) = (y + 1 ax2 , bx)
as a second order dierence equation. A second order dierence equation
is a recursion of the form xn+1 = F (xn , xn1 ).
b) An orbit x0 , x1 , x2 , . of a dierence equ
2/7/2005: THE LOGISTIC MAP
Math118, O. Knill
ABSTRACT. Our rst dynamical system is the logistic map f (x) = cx(1 x), where 0 c 4 is a
parameter. It is an example of an interval map because it can be restricted to the interval [0, 1].
You can read about th
5. Homework set
Math118, O.Knill
5.1 Given a rectangle of length 1 and height b > 1. We play billiards in this table. For
which angles does a trajectory (which does not hit a corner) get arbitrarily close to
any point on the boundary of the table?
5.2 We