HW 5, MATH 489, Due Date: 09/30/2016
Problem 1: For the map
4
1
30
3
,
is the origin a sink, source, or saddle?
Problem 2: Find
lim
n
9
2
2
8
27
n
6
9
.
Problem 3: Let b = 0.3 in the Henon map f (x) = (a x2 + by, x). (a) Find the
range of parameters a f
HW 7, MATH 489, Due Date: 10/31/2014
Problem 1: Consider the middle-half Cantor K(4)set formed by deleting the
middle half of each subinterval. Find its length and its box-counting dimension.
Problem 2: Find the box-counting dimension of the middle
1
5
Ca
HW 8, MATH 489, Due Date: 11/14/2014
Problem 1: Assume that the map f on Rm has constant Jacobian determinant,
say det Df (x) = J for each x Rm . Explain why the sum of all m Lyapunov
exponents is ln J.
4
Problem 2: Let f (x, y) = ( arctan x, y ). Find al
b 1!: b I q. 3 -2
LL22 4mhf'aQ1Jr 1: z w! _ 2 ' '
' 3 L o. .322 L " :3 "' n
116 3-way has MM w, _ -3 _.A;LaciL 5H4; 19} lo 25mg) W_._+_QAILDL_-al «a? m_-sgr_q}: L9: _M_
_§@¥ - H611me mgmgl 44:. i :1: _:4_t_;_4.{._2$: i M= #
1%4- 4 [L a L) 3 3 R "=01
gym
HW 9, MATH 489, Due Date: 11/21/2014
Problem 1: The ODE governing a linear oscillator, such as a spring moving
according to Hookes law is x + kx = 0. Convert it to a system of rst-order
equations and sketch the phase plane.
Problem 2: Consider the equatio
HW 10, MATH 489, Due Date: 12/05/2014
Problem 1: Consider the system
x
= x3 + xy
y
= y 3 x2
(a) Prove that (0, 0) is an asymptotically stable equilibrium. (b) Show that the
basin of attraction of the equilibrium (0, 0) is R2
Problem 2: Consider the system
Problem 1: Consider the function f shown in the gure below.
(a) (18 pts) Find a partition and draw its transition graph.
(b) (15 pts) Determine what periods the periodic orbits of f have?
(03 T09. Parhi'hn 15 A) B] C, D
0/3 Qan .n «HQ
rpm.3, F®=67 4m: D,
(Q
Problem 1: Let a E R and consider the 1-dim map f (x) = 9:3 am.
(a) (8 pts) Find its xed point(s). Do they exist for all a 6 R?
(b) (8 pts) Determine their stability.
(c) (14 pts) Where do the orbits go under iterations of f when a = 0.44?
Consider ini
Problem 1 (40 points): Consider the piecewise linear map
32.1: 0
f($)={ §$§
.1.
4
441 g
1
4g
4
IAIAIA
|/\|/\l/\
H
(I:
(I:
(I:
1. Sketch the graph of f.
2. Find a partition and a transition graph for f.
3. For what periods does f have a periodic orbit?
j.
90k XZ'JK =9
9'64) *3
Problem 1 (45 points): Consider the 1-dim map f (x) = 2x 2:2.
1. Find all xed points of f and determine their stability (classify them as
sinks or sources).
2. Are there any period-2 points of f? If yes, determine their stability. If
HW 6, MATH 489, Due Date: 10/24/2014
Problem 1:
(a) Sketch the graph of the map f (x) = 2x2 + 8x 5.
(b) Find a set of two subintervals that form a partition.
(c) Draw the transition graph of f . What are the possible periods for periodic
orbits?
Problem 2
HW 2, MATH 489, Due Date: 09/09/2016
Problem 1: Find the period-2 orbit of f (x) = 5x 5x2 . Is it a sink, a source
or neither?
Problem 2:
(a) Find the fixed points of f (x) =
(b) Solve the inequality
3xx3
.
2
|f (x) 0| > |x 0|.
This identifies points whos
HW 6, MATH 489, Due Date: 10/07/2016
Problem 1:
3
(a) Find the fixed point(s) of the map f (x, y) = ( x2 , 2y 15
8 x ).
(b) Determine its (their) stability.
(c) Show that the yaxis serves as the unstable manifold. Also, show that
the curve y = x3 serves a
HW 3, MATH 489, Due Date: 09/16/2016
Problem 1: Let 0 = p1 < p2 < p3 < p4 < p5 < p6 = 34 < p7 < p8 be the 8
fixed points of G3 , where G(x) = 4x(1 x). Group the 6 points that are not
fixed points of G into two orbits of 3 points each. Justify your answer
HW 1, MATH 489, Due Date: 09/02/2016
Problem 1: Give an example of a function h for which h0 (0) = 1 and x = 0 is
an attracting fixed point. Justify/Verify your answer.
Problem 2: Give an example of a function h for which h0 (0) = 1 and x = 0 is
a repelli
HW 4, MATH 489, Due Date: 09/23/2016
Problem 1: Show that for any two-dimensional map f , a source has sensitive
dependence on initial conditions.
Problem 2:
(a) If (x1 , y1 ) and x2 , y2 are the two fixed points of the Henon map f (x, y) =
(a x2 + by, x)
HW 7, MATH 489, Due Date: 10/21/2014
Problem 1: Let f (x) = rx(1 x), r > 2 + 5. Show that the Lyapunov
exponent of any orbit that remains in [0, 1], is greater than zero, if it exists.
Problem 2: Find the Lyapunov exponent shared by most bounded orbits of
HW 8, MATH 489, Due Date: 10/28/2014
Problem 1: Consider the map fa (x) = 2x(mod1) on [0, 1].
(a) Which points are periodic, eventually periodic, and asymptotically periodic?
(b) Which orbits of f are chaotic orbits?
Problem 2: (a) Find a conjugacy betwee
HW 2, MATH 489, Due Date: 09/12/2014
Problem 1: Find the period-2 orbit of f (x) = 5x 5x2 . is it a sink, a source
or neither?
Problem 2: Let 0 = p1 < p2 < p3 < p4 < p5 < p6 = 3 < p7 < p8 be the 8
4
xed points of G3 , where G(x) = 4x(1 x). Group the 6 poi
HW 1, MATH 489, Due Date: 10/17/2014
Problem 1: Dene
xn+1 =
xn + 2
.
xn + 1
Find L = limn xn for IC x0 0.
Describe the set of all negative x0 fr which the limit limn xn either
exists and is not equal to L or does not exist (e.g., x0 = 1)
Problem 2: Let
HW 2, MATH 489, Due Date: 10/31/2014
Problem 1:
(a) Decide on a partition for the map f (x) = 2x(mod 1) on [0, 1] and draw
its transition graph and schematic itineraries.
(b) Determine whether there are xed point and periodic points. If periodic
points ex