Dynamical Systems: Course Works

This is a place where I store my notes and/or exercise attempts when studying MA4233 Dynamical Systems during the second semester of Academic Year 2024/2025, instructed by Dr. Daren Wei.

The main reference book used is A First Course in Dynamics, with a Panorama of Recent Developments by Boris Hasselblatt.

Chapter 1: Introduction

The famous Logistic equation is

$$f(x)=kx(1-x), \quad x \in [0,1].$$

1. Verify that if $k \leqslant 1$, $f^n(x) \to 0$ as $n \to +\infty$ for all $x \in [0,1]$.
2. Starting with $x = 0.01$, iterate $2x(1-x)$ until you get a clear pattern.
3. Starting with $x = 0.01$, iterate $3x(1-x)$ until you get a clear pattern.
4. Starting with $x = 0.01$, iterate $3.45x(1-x)$ and $3.5x(1-x)$ until you get a clear pattern.
5. Starting with $x = 0.5$, iterate $3.83x(1-x)$ until you get a clear pattern.

Please find the Sage implementations of the workings of parts 2. to 5. of this question here.

Let $f(x,y) = \left(\frac{x+y}{2},\frac{2xy}{x+y}\right)$, check that if you start with $(1,4)$, the iterations will converge to $(2,2)$.

note

The context of this exercise is Heron's method of obtaining the square root of an integer $z$ that is not a perfect square, which is by taking a rectangle with sides $x$ and $y$ such that $xy=z$, and $x$ is an educated guess of the square root. Iteratively, one then replaces the sides $x$ and $y$ by $(x+y)/2$ (the arithmetic mean) and $2xy/(x+y)$ (harmonic mean, so that their product yields the same number $z$) respectively. As the number of iteration increases, one then sees that the two sides tend to the same value, which gives a better approximation of $\sqrt{z}$.

A Sage computation of this procedure can be found here.

If $x_n=n^2p/q$, where $p,q \in \mathbb{N}$ and $p,q$ are coprime, prove that the last digits of $x_n$ is periodic.

(Here, the 'last digit' refers to the digit before the decimal point, where applicable.)

A Sage computation that illustrates this assertion can be found here.

A direct argument allows us to see that the period is in fact at most $10q$, because

$$x_{n+10q}-x_n=(n+10q)^2p/q-n^2p/q=10(2np+10pq)$$

is an integer multiple of $10$, so the last digit before the decimal point of $x_{n+10q}$ and $x_n$ are equal.