u"r Y 2 q rR u a q&rp Y (&$ 5"r " 5 2 "u& 5
gcb4hFh)'bF6l"gbIcs4%)i(kx64"ji
'Pv'Fh)6g)4(42i5g!xgbbFfbeYhxiu42G)d!Gb)4(Yb'c%xgy)GG0
r& B B&rp (& p 5" qu Y"rrR a 2 f a 5 p ( 7 r" q Rr q 2 ("Y 2r f "r& u 0&uu" r R
"r u "& B (& rR f 5& fR au "u& a&" (w u B 5
MA 471-671
ASSIGNMENT 7 SOLUTIONS
Assignment: Page 202: Nos. 1,13,14,15,18
1. Without using the computer, predict the structure of the attractor generated by the
iterated function system with contraction factor beta and fixed point p_i.
a) beta=1/3; p_0=(
MA 471-671
Solutions to Homework Assignment #8
Page 243: Nos. 1, 2, 3, 4, 6, 7, 9
1. Describe the filled Julia set for F(z)=z^3.
Like the quadratic map Q_0, the filled Julia set for F(z)=z^3 is the closed unit disk.
Observe that
F(z)=z^3
F^2(z)=z^9
.
.
F^
MA 471-671
Solutions to Homework Assignment #9
1. 1 Prove that Q_c has a periodic point of prime period 2 at each root of the equation
z^2+z+c+1=0.
Notice first that Q^2_c(z)=(z^2+c)^2 + c = z^4+2z^2c+c^2+c. Hence, we need to solve
Q^2_c(z)=z, that is, fi
MA 471-671
ASSIGNMENT 6 SOLUTIONS
Assignment: Page 151: Nos. 1,2,3,4,8 Page 161: Nos. 1-8
Page 151
1. Can a continuous functon on R have a periodic point of period 48 and not of of period
56? Why?
Yes. Observe that 56=(2^3)*7 precedes 48=(2^4)*3 in the Sa
MA 471-671
ASSIGNMENT 5 SOLUTIONS
Assignment: Page 131: Nos. 1-4, 6-13, 15, 16, 21, 22, 24.
For each of the following sets , decide whether or not the set is dense in [0,1].
1. S1 is the set of all real numbers in [0,1] except those of the form 1/2^n for
MA 471-671
ASSIGNMENT 1 SOLUTIONS
Assignment: Page 26: Nos. 1,3,5,6,11,12,13
1. Let F(x)=x^2. Compute the first five points on the oribit of 1/2.
F^1(1/2)=F(1/2)=1/4
F^2(1/2)=F(1/4)=1/16
F^3(1/2)=F(1/16)=1/256
F^4(1/2)=F(1/256)=1/65536
F^5(1/2)=F(1/65536)
MA 471-671
ASSIGNMENT 2 SOLUTIONS
Assignment: Page 34: Nos. 1 A,C,F; 4 A,B,D; 5. Page 50: Nos. 1 A,b,f,j; 2 B,C; 4 A,C,E.
Page 34
1. Use graphical analysis to describe the fate of all orbits for each of the following
functions. Use different colors for or
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Solutions to Homework Assignment #3
Page 67: Nos. 1b,c,e,i; 3, 4, 5, 8, 9, 10, 11, 12, 13, 14
1. Each of the following functions undergoes a bifurcation of fixed points at the given parameter
value. In each case, use algebraic or graphical meth
MA 471-671
Solutions to Homework Assignment #4
Page 80: Nos. 9-15; Page 111: Nos. 1, 3, 6, 10, 11, 12, 18 b,e,g,i
Page 80:
The following exercises deal with the function
T(x)=
cfw_ 3x for x <= 1/2
cfw_ 3x-3 for x > 1/2
10) Find the fixed points for T. Wha