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 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 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
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 5, MATH 489, Due Date: 10/17/2014
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) Using information from
HW 3, MATH 489, Due Date: 11/21/2014
Problem 1:
Show that [0, 1] is a chaotic attractor for the 2x(mod1) map.
Problem 2:
Show that the logistic map ga (x) = ax(1 x), for a > 4, has a chaotic set,
which is not a chaotic attractor.
Problem 3:
Show that if x
HW 5, MATH 489, Due Date: 10/03/2014
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
b 1!: b I q. 3 -2
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116 3-way has MM w, _ -3 _.A;LaciL 5H4; 19} lo 25mg) W_._+_QAILDL_-al «a? m_-sgr_q}: L9: _M_
[email protected] - H611me mgmgl 44:. i :1: _:4_t_;_4.{._2$: i M= #
1%4- 4 [L a L) 3 3 R "=01
gym
90k XZ'JK =9
9'64) *3
Problem 1 (45 points): Consider the 1-dim map f (x) = 2x 2:2.
1. Find all xed points of f and determine their stability (classify them as
sinks or sources).
2. Are there any period-2 points of f? If yes, determine their stability. If
Problem 1 (40 points): Consider the piecewise linear map
32.1: 0
f($)={ §$§
.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.
(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: 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,
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
HW 4 (Extra Credit), MATH 489, Due Date: 12/05/2014
Problem 1: Draw the slope eld and phase portrait for x = x3 x. Sketch the
resulting family of solutions. Which initial conditions lead to bounded solutions?
Problem 2: Determine the possible phase plane
HW 3, MATH 489, Due Date: 09/26/2014
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
7
2
n
6
9
.
Problem 3: Let b = 0.3 in the Hnon map f (x) = (a x2 + by, x). (a) Find the
e
range of parameters a f
Math 446, HW # 3
(quiz on Fri, Feb 10)
(1) Using results in the Section on Differentiation Formulas, show that
(a) a polynomial
P (z) = a0 + a1 z + a2 z 2 + + an z n
(an 6= 0)
of degree n (n 1) is differentiable everywhere, with derivative
P (z) = a1 + 2a
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 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 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 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