You must upload a two$-$page pdf where the first page is your computing report and the second page is your

scripts and code. The assignment is due at $11$:$00$pm$.$ I have set the due time in Canvas to $11$:$05$pm and if

Canvas indicates that you submitted late, you will be given $0$ on the assignment. Your computing report

must be exactly $1$ page. There will be a penalty given if your report is longer than one page. The same is

true for your scripts$/$code$.$

$$ Please read the Guidelines for Assignments first.

$$ Keep in mind that Canvas discussions are open forums.

$$ Acknowledge any collaborations and assistance from colleagues$/$TAs$/$instructor$.$

Computing Assignment $$ Newton$$s Method vs Fixed$-$Point methods of Discrete Dynamical Systems

Required submission: $1$ page PDF document and Matlab scripts uploaded to Canvas.

The below equation is a discrete dynamical system, that can be interpreted as a population growth for a

species with non$-$overlapping generations.

Nt$+1=$ rNt $(1$ Nt$)$

with Nt denoting the size of the population at time t$,$ and the variable r$,$ the intrinsic growth rate.

In general discrete dynamical systems can be written as such

Nt$+1=$ F$($Nt$),$

and equilibrium points are points that satisfy Nt $=$ F$($Nt$).$ That is$,$ suppose N$$

is an equilibrium point of

our dynamical system then, a population of size N$$

remains the same size for all the years following. This

is an important feature in understanding the behaviour of a dynamical system.

Here is your computational experiment:

$($a$)$ From our population growth equation find all the equilibrium points by setting up the equation

N$=$ F$($N$$

$)$ and solving for N$$

$.$ State for which values of r your equilibrium points have realistic

meaning.

$($b$)$ Using Newton$$s method, find the largest realistic root for r in $[0\mathrm{.}1,4].$ Note that you have to set up

your Newton$$s function G and G$$

$,$ and determine a good initial guess. $($You may use newton.m$,$

but make modifications to improve the stopping criteria.$)$ Graphically verify that Newton$$s method

found the correct root by comparing it to your analytical solution.

$($c$)$ You$$ve probably noticed that a discrete dynamical system looks exactly like a fixed$-$point method,

and that equilibrium points are exactly the fixed$-$point of the corresponding continuous equation.

Bravo! Use the fixed$-$point method now for r $=0\mathrm{.}5,$ r $=1\mathrm{.}5,$ r $=2,$ and for some r in $[3,3\mathrm{.}4].$

$($You may use fixedpt.m$,$ but make modifications to improve the stopping criteria.$)$ Explain what

you observe and why you observe this? Illustrate graphically the convergence or divergence that you

observe.

You should notice that the fixed$-$point has different convergence$/$divergence patterns. In fact, these are

interpretable as population dynamics, and relates to the concept of the existence of chaos in discrete

dynamical systems. For fun, use your code to see what happens as you continue to increase r$.$