Mudassir Lodi
Homework for Lecture 14 of Dr. Z.’s Dynamical Models in Biology class
Email the answers (either as .pdf file or .txt file) to ShaloshBEkhad@gmail.com by 8:00pm Monday, Oct. 25,, 2021. Subject: hw14 with an attachment hw14FirstLast.pdf and/or hw14FirstLast.txt
Also please indicate (EITHER way) whether it is OK to post

1.	(Mandatory for all students who did not get today’s “coming on time” attendance quiz completely right). Carefully read, https://sites.math.rutgers.edu/~zeilberg/Bio21/att14.pdf .
understand it. Then do the similar question for the following dynamical system
 ,
but this time do it for all the fixed points.
(x,y) = (x + y^4 - 1/16, x^2 + y – 1/9)
X = x + y^4 – 1/16              0 = y^4 – 1/16
Y = x^2 + y – 1/9                 0 = x^2 – 1/9
X = ½, -1/2
Y = 1/3, -1/3
[(1/2, 1/3), (-1/2, 1/3), (-1/2, 1/3), (-1/2, -1/3)]

2.	Draw the directed graph with 16 vetrices, labeled 1,...,16 of the Dynamical system
	x → x3 mod 17	.
Find all the periodic cycles (including those of size 1, aka as fixed points), and write down explicitly all the 16 trajectories for each of the possible starting points. For example, the one that starts with 1 is simply
[1,1]
The one that starts with 2 is
[2,8,2]
RevOpTr:=proc(n,k) local L,n1:
if not (type(n,integer) and n>=0 and n<10^k) then
RETURN(FAIL):
fi:

L:=[]:
n1:=n:

while not member(n1,L) do
 L:=[op(L),n1]:
 n1:=RevOp(n1,k):
od:

[op(L),n1]:

end:

3.	Let Tk(n) be the following finite Dynamical system that takes place in all k-digit positive integers.
•	Arrange the digits in decreasing order, call it L(n)
•	Arrange the digits in incrasing order, call it S(n)
•	Define Tk(n) = L(n) − S(n).
For example if k = 3 and n = 327, then L(327) = 732, S(n) = 237 and T3(327) = 732 − 237 = 495
(i)	Take 10 random 2-digit numbers, and , by hand, find the trajectories of T2(n), until the first repeat, and determine their ending cycle (that may or may not be of length 1, i.e. a fixed point).
(ii)	Take 5 random 3-digit numbers, and , by hand, find the trajectories of T3(n), until the first repeat, and determine their ending cycle (that may or may not be of length 1, i.e. a fixed point).
(iii)	Take 3 random 4-digit numbers, and , by hand, find the trajectories of T4(n), until the first repeat, and determine their ending cycle (that may or may not be of length 1, i.e. a fixed point). 

4. Read and understand the procedures RevOp(n,k) and RevOpTr(n,k) in the Maple code https://sites.math.rutgers.edu/~zeilberg/Bio21/M14.txt	,
and convince yourself that it implements Tk(n). Use it to find all the periodic orbits (aka cycles) of T3(n) and T4(n). Are they all fixed points?

They are all fixed points. 

5.	Consider the dynamical system, defined on the infinite set of positive integers
	 if	n	is	is	odd	.
Take five random integers and find their trajectories. What do they end up with?

RevOpTr:=proc(n,k) local L,n1:
if not (type(n,integer) and n>=0 and n<10^k) then
RETURN(FAIL):
fi:

L:=[]:
n1:=n:

while not member(n1,L) do
 L:=[op(L),n1]:
 n1:=RevOp(n1,k):
od:

[op(L),n1]:

end: