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A Brief Introduction to Discrete Dynamical Systems

Introduction

Charles Stein

Charles Stein

Mathematician at heart and a gamer at leisure. Love theoretical computer science. Works on dsmodels, research in group theory, and perfecting Abzan Aristocrats deck in MTG. Top 3% in Hearthstone.


mathematics dynamical-systems introduction

A Brief Introduction to Discrete Dynamical Systems

Posted by Charles Stein on .
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mathematics dynamical-systems introduction

A Brief Introduction to Discrete Dynamical Systems

Posted by Charles Stein on .

Dynamical systems are a powerful mathematical tool first introduced in 1892 by H. J. Poincaré to model a three-body celestial system. Dynamical systems may seem scary in name and repertoire, but they become more approachable after walking through a few explanations and examples. These systems are interesting to mathematicians and researchers in similar fields as they are able to model nearly any quantifiable system through time such as pendulum motion, population growth (competition models), and game-theoretic games. Additionally, chaos theory is a topic born from dynamical systems, which we will get to later in this article.

The Black Box: Intuition Building

For younger students (think high-school and lower), teachers of mathematics usually neglect formal definitions and proofs in favor of simpler explanations to express difficult concepts. Amidst the myriad of workarounds used to introduce mathematics to newcomers, the idea of a "black box" is generally thrown around. A dynamical system can be thought of as a black box that feeds into itself. Let $x$ be a point in the real numbers notated as $\mathbb{R}$. Then let $f:\mathbb{R}\rightarrow\mathbb{R}$ be an arbitrary function. We denote the composition of $f(f(x))$ as $f\circ f(x)$ or $f^2(x)$. The orbit of $x$ is the "trajectory" of the point. Formally, it is the sequence

$x,f(x),f^2(x),f^3(x),\ldots$.

Readers familiar with recurrence relations may have an edge on other readers in understanding this idea. The $x$ can be thought of as a starting point, and the $f$ can be thought of as a recurrence relation. Like recurrence relations, these systems already exhibit interesting behavior. The above is a 1-dimensional dynamical system, but the function $f$ along with the point $x$ describe an entire range of motion throughout the real numbers. An $n$-dimensional system looks extremely similar. Let $x$ be a point defined by an $n$-tuple of real numbers, denoted $\mathbb{R}^n$. Then let $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be an arbitrary function. Then we have a similar, albeit higher-dimensional, dynamical system with an identically notated orbit.

Interpreting One-Dimensional Dynamical Systems

We will be looking at images called cobweb diagrams. An interface for generating cobweb diagrams can be found at desmos.com.

desmos-graph-3

This dynamical system is defined by the function

$f(x) = 1.32\cdot x\cdot(2-x)$.

The function is drawn in black and meets the $x$-axis at 0 and 2.
The starting point chosen is $0.5$ and is visualized as the black dot on the $x$-axis. To draw the cobweb diagram, we draw a vertical-line from the initial point to $f(0.5)=0.99$. From there, we draw a horizontal line to meet up with the identity function colored in red where the $x$- and $y$-axes both read $0.99$. Then we find $f^2(0.5)=f(0.99)=1.32$ and draw another vertical line from $0.99$ to $1.32$. Note that if our starting point was $0.99$, then we would have ended up in the same place with less applications of the function! We continue applying the function to our point until we get that $f^{10}(0.5)\approx1.2424$ and that $f^{11}(0.5)=f(1.2424)\approx1.2424$.

In dynamical systems, a fixed point refers to an orbit of a point $x$ which has "stagnated." Formally, for a function $f$ and point $x$, if $f(x)=x$ then $x$ is a fixed point. There are a few kinds of fixed points, which may be discussed in a future article. In a cobweb diagram, the points for which the function curve (black line defined by $f(x)$) crosses the identity curve (red line interpreted as $y=x$) are where the fixed points reside. This can be determined analytically, or estimated visually by the graph.

Chaos

Chaos theory embodies what is known as the "Butterfly Effect." A butterfly flaps its tiny wings in Tanzania, and a tiny gust is propelled across the Atlantic and develops into a hurricane in Florida. A tiny change in one system can massively influence another point in that system, possibly in unpredictable ways. Take that previous image we have and move the constant $1.32$ around a bit in your visualization. As the constant grows, you'll notice some strange behavior:

desmos-graph-cray

This image shows

$f(x)=1.96\cdot x\cdot(2-x)$.

Our intuition built in the previous section tells us that there are two fixed points, $0$ and $\sim 1.5$ which our orbit seems to approach; however, our system keeps frantically jumping from location to location, trying to settle at a fixed point. Changing any part of this function, even the tiniest bit, drastically alters the system. This is what is known as chaos. The butterfly flaps its wings and we have a whole new system with unpredictable and interesting behavior.

Conclusion

While we did not cover the uses of dynamical systems, we built a bit of intuition in them. We also began the journey into chaos theory, but any expansion on that must wait for another article. A future article may also include higher-dimensional dynamical systems. My research is in two-dimensional systems, but one-dimensional systems are beyond interesting, as well.

Charles Stein

Charles Stein

http://www.cs.trinity.edu/~sfogarty/dsmodels/index.html

Mathematician at heart and a gamer at leisure. Love theoretical computer science. Works on dsmodels, research in group theory, and perfecting Abzan Aristocrats deck in MTG. Top 3% in Hearthstone.

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