MTG 5412 Keesling Assignment #1 1/11/08 due 1/23/08 Do all problems. The format of the homework should be a report on each problem done. The report should include (1) a statement of the problem, (2) a succinct description of the method of solution, (3) th

MTG 5412/MAT 4930 Dynamical Systems and Chaos
Keesling Study Guide 2/15/13
Test on 2/22/13
The test will consist of five problems based on the following questions. Each question
will be worth 20 points.
1.
2.
Let f : [ a, b ] R be continuous. Suppose that

MTG 5412/MAT 4930 Dynamical Systems
Keesling
Test 1
2/22/13
Name
Do all problems and show all work. Each problem is worth 20 points.
1.
Let f : [ a, b ] R be continuous. Suppose that f ([ a, b ]) [ a, b ] . Show that f
has a fixed point in [ a, b ] .
2.
S

MTG 5412 Dynamical Systems and Chaos
Quiz 1
1.
2.
Let f : [ a, b ] R be continuous. Suppose that f ([ a, b ]) [ a, b ] . Show that f
has a fixed point in [ a, b ] .
Suppose that f : [ a, b ] R is continuous and that f ([ a, b ]) [ c, d ] . Show
that there

MTG 5412 Dynamical Systems and Chaos
Quiz 2
1.
Suppose that f : R R is continuous and suppose that f has a periodic point
of period three. Show that f has a point of period seven. How many orbits of
period seven can you guarantee with this information?
2.

MTG 5412 Dynamical Systems and Chaos
Quiz 3
1.
Give an example of a period five orbit that does not imply a period three orbit.
2.
Let N be an integer greater than one. Let N = cfw_1, 2, N and let
i =1
: N N be defined by, ( ( xi ) = (

MTG 5412 Dynamical Systems and Chaos
Quiz 4
1.
Let f : R R be continuous and let x be a periodic orbit of period n . How
many orbit types are there for an orbit of period n ?
2.
Let f : R R be a differentiable function. Suppose that x is a fi

MTG 5412 Dynamical Systems and Chaos
Quiz 5
Name
1.
Let f : R R be continuous and let x be periodic of period three. How
many points of period five must there be for f ?

MTG 5412 Dynamical Systems and Chaos
Quiz 6
Name
1.
Let f : [0,1] [0,1] be defined by x (1 x ) . Show that for 0 < 1
zero is an attracting fixed point.
2.
Let f : [0,1] [0,1] be defined by x (1 x ) . Show that for 1 < < 3

MTG 5412 Dynamical Systems and Chaos
Quiz 7
1.
Let h( x ) : R R be given. Define the Schwarzian derivative of h( x ) to be
h( x ) 3 h( x )
Sh( x ) =
h( x ) 2 h( x )
2
Let f : [0,1] [0,1] be defined by x (1 x ) . Show that Sf ( x

MTG 5412 Dynamical Systems and Chaos
Quiz 8
1.
Let f : R R and g : R R . Suppose that Sf < 0 and Sg < 0 . Show that
S(g f ) < 0 .
2.
Suppose that f : R R and that Sf < 0 . Show that f '( x ) cannot have a
positive local minimum or a nega

MTG 5412 Dynamical Systems and Chaos
Quiz 9
1.
Let f : R R and that f has finitely many critical points. Suppose that
Sf < 0 . Then f has finitely many periodic points of period m for any
integer m.
2.
Suppose that Sf < 0

MTG 5412 Dynamical Systems and Chaos
Quiz 10
1.
Suppose that c is a critical point for f. Show that if c is periodic, then that
periodic orbit is attracting.
2.
Let : N N be the shift map on N objects. Let A be an N N matrix.
a11 a

MTG 5412 Dynamical Systems and Chaos
Quiz 11
Name
1.
Suppose that c is a critical point for f. Show that if c is periodic of period
three, then that periodic orbit is attracting.
2.
Show that the quadratic family, f ( x ) = x (1

MTG 5412 Dynamical Systems and Chaos
Quiz 12
1.
Let f = x (1 x ) and suppose that the critical point c is on an orbit of
period five. Suppose that the orbit of c = .5 is c = .500 , f (c ) = .935 ,
f 2 (c ) = .229 , f 3 (c)

MTG 5412 Dynamical Systems and Chaos
Quiz 15
Name
1.
Consider the automorphism of the torus, fM : T 2 T 2 , where
M : R 2 R 2 is the linear transformation of the plane given by
3
10
2 1
. Consider the point p = 1 [ 0,1) [ 0,1

MTG 5412 Keesling 1/30/08 Due 2/1/08 1. Let F : S ! S be the Smale Horseshoe map. Show that the attractor
A = ! F n (S) is compact and connected.
n =1
!
2.
Let F : S ! S be the Smale Horseshoe map. Show that there is a Cantor set cross an interval C ! I "

MTG 5412 Keesling 2/1/08 Due 2/4/08 Let fM :T 2 ! T 2 be the toral automorphism associated with the matrix !3 4$ 2 M =# & . Determine the stable and unstable manifolds of the point 1 !T . "2 3%
1.
2.
Let f :T 2 ! T 2 be a hyperbolic toral automorphism. Le

MTG 5412 Keesling 2/4/08 Due 2/6/08 1. Prove the Baire Category Theorem: Let X be a complete metric space and
!
suppose that cfw_U i i =1 is a countable collection of dense open sets in X. Show that
!U
i =1
!
i
is dense in X.
2.
Let X be a complete separa

MTG 5412 Keesling Problem 4 2/6/08 Due 2/8/08
1.
Let f :T 2 ! T 2 be a hyperbolic toral automorphism. Show that there is a point x !T 2 whose orbit is dense under f.
2.
Solve the following linear system of differential equations using the exponential of a

MTG 5412 Keesling Problem 5 2/11/08 Due 2/13/08 1. Let f : X ! X be a homeomorphism on X. Define the mapping torus of f in the following way. Let T f = X ! I (x,1) ! ( f (x), 0) . Show that there is a flow F : R ! T f " T f so that f is equivalent to F at

MTG 5412 Keesling Problem 6 2/13/08 Due 2/15/08 1. Let X be a compact metric space. Let f : X ! Y be a function that is onto. Suppose that g : X ! Z is a continuous map onto a metric space Z such that for each x ! X , f !1 f (x) = g !1g(x) . Then the quot

MTG 5412 Keesling Problem 7 2/15/08 Due 2/18/08 1. Use Picard Iteration to find the solution of the following differential equation.
dx = t2x dt
x(0) = 1
2.
State and prove the Contraction Mapping Theorem.

MTG 5412 Keesling Problem 8 2/18/08 Due 2/20/08 1. Consider the following differential equation.
dx " x $ =# dt $ 0 %
x!0 x<0
x(0) = 0
Show that there are an infinite number of distinct solutions with this initial condition.
2.
Determine the flow for a li

MTG 5412 Keesling Problem 9 2/20/08 Due 2/25/08 1. Solve the following system of differential equations numerically. This is the Lorenz system of equations.
dx = ! (y " x) dt dy = x( # " z) " y dt dz = xy " $ z dt
! = 28 " = 10 8 #= 3
2.
Solve the followi

MTG 5412 Dynamical Systems and Chaos Quiz 1 Let f :[a,b] R be continuous. Suppose that f ([a,b]) [a,b] . Show that f has a fixed point in [a,b] .
1.
2.
Suppose that f :[a,b] R is continuous and that f ([a,b]) [c,d] . Show that there is an interval [g,h] [

MTG 5412 Dynamical Systems and Chaos Quiz 2 1. Suppose that f : R R is continuous and suppose that f has a periodic point of period three. Show that f has a point of period seven. How many orbits of period seven can you guarantee with this information?
2.

MTG 5412 Dynamical Systems and Chaos Quiz 3 1. Give an example of a period five orbit that does not imply a period three orbit.
2.
Let N be an integer greater than one. Let N = cfw_1,2,., N and let
: N N be defined by, ( (xi ) = (xi+1 ) . The map is cal

MTG 5412 Dynamical Systems and Chaos Quiz 4 1. Let f : R R be continuous and let x be a periodic orbit of period n . How many orbit types are there for an orbit of period n ?
2.
Let f : R R be a differentiable function. Suppose that x is a fixed point for