### Introduction to Dynamical Systems

In Dynamical Systems our main goal is to understand behavior of states in a system, given a rule for how the state evolves. The states are our variables, in fact we even call them *state variables*. Anything that one could represent with a number could be considered a state. Some examples of state variables may include the population of a colony, the density of a chemical in a solution, the amount of money in a bank account, the position of a particle, temperature or anything that can be represented by a number or set of numbers. In each of these examples the state of the system may be represented by a number. Given the state variable a dynamical system needs a rule which defines the dynamics or how the system changes. Determining the appropriate rule for which the state variable evolves or changes is where difficulty lies. This is where the modeling comes into play. In this course we will model population growth in which case the state variable will be the population. The rule under which the population evolves is not obvious and we will investigate behavior for different models.

Formal study of dynamical systems involves studying mathematical models which have been handed down from creative experts in fields such as Physics, Chemistry, Biology or Economics. Some systems may be derived from basic principles and tested to show experimental accuracy strong enough to create the technologies we use everyday. Newton's Laws for example are incredibly accurate under known situations and formulate a well defined dynamical system. For example, problems from Classical Mechanics may be written as a dynamical system where the position and velocity of a particle or even that of a continuous medium are the state variable and Newton's Laws used as the updating rule. In this case we are left with a dynamical system written as a system of differential equations. These physical models constitute a side of dynamical systems which may be used as a quantitative tool to analyze the environment around us.

On the other hand some dynamical systems may involve more simplifications and approximations and thus do not carry with them the same numerical accuracy or prediction of exact values. However, such systems allow for more profound statements about general behavior of given physical phenomena. This qualitative analysis will be a primary focus for this class. For example, understanding chemical oscillations does not require an exact knowledge of chemical densities at every point in space at every time. This would come from an exact solution of a dynamical system whose state variable represents chemical density. But, exact solutions for such complex systems are often too difficult to solve or too complicated to understand. Instead the system of differential equations which model say the Belousov-Zhabotinky reaction may still shed interesting information by utilizing simple dynamical system techniques without the need of an exact solution.

#### Constant rate of growth

The simplest type of dynamical system describes the evolution of a state variable which changes at a constant rate. As an example you may consider a persons age. Everyone ages at the same constant rate. Although our age is continuous our birthday is a discrete event. On our birthday our age makes a jump from one integer value to the next. If we measured our age as any positive number the change would not be noticeable, but it is a bit awkward to say things like "I am approximately 26.997 years old." Instead we say "tomorrow I will be 27 years old."

Another point worth addressing is the use of relevant units. A baby who is 1 month old is rarely referred to as $\frac{1}{12}$ of one year old. Something that we do naturally is express quantities in units so that the numerical value is clear. For some reason this practice is lost when doing mathematics. However, when modeling a problem dimensional analysis or just thinking about relevant units can be a useful tool.

###### Discrete Birthdays

Expressing someones age as a discrete dynamical system we need two things, a state variable and a rule for which the state variable changes after each time interval. We may denote the persons age in years between birthday $n$ and $n+1$ as $a_n$. Hence, someones age between their 20th birthday and 21st birthday is $a_{20}$. This defines the state variable which in this case is a representation of a persons age. We can write the rule for the evolution of the state variable as \[ a_{n+1}=a_n + 1. \] In a discrete dynamical system the rule which takes the previous value of the state variable and gives the next value is called an updating function. The right hand side of the above equation is the updating function for a person aging at a constant rate.

So far we have defined what it is that is changing, the state variable, and how it changes, the updating function. In order to complete the dynamical system we need an initial condition, which is the value the state variable begins at. The solution to a dynamical system depends on the systems initial condition or starting point. In the case of modeling a persons age we may simply choose \[ a_0=0. \] In other words an infant is zero until they reach their first birthday and turn 1 year old.

Now that we have a complete dynamical system which may be written as \[ \left\{ \begin{array}{r c l} a_{n+1} & = & a_n +1 \\ a_0 & = & 0\\ \end{array} \right. \] we would like to write down a solution to the system. First, we have to be clear as to what we mean by a solution to a discrete dynamical system. Since have the initial condition $a_0=0$ and the updating rule $a_{n+1} = a_n +1$ we can find $a_1$ by taking $n=0$ \[ a_1=a_0 + 1 = 0 + 1 =1. \] Each concurrent step is just as easy, $a_2=a_1 + 1 = 1 + 1 =2$ and $a_3=a_2 + 1= 2 + 1 = 3$ and as you may have already figured out we can deduce that $a_n= n$. In a pure math class one would prove this by induction, but for the benefit of this course it suffices to find a pattern and verify it. It is simple to see for this system that $a_0=0$ and $a_{n+1} = a_n +1 = n + 1$ and so the formula is valid. Of course it is no surprise that a person is $n$ years old for the year following their nth birthday. But, this is just so we can illustrate this simple concept in the framework of a dynamical system. It is important to note here that the solution to this dynamical system is the simple formula $a_n=n$. We can think of the solution to a discrete dynamical system as a list, the only thing is that we can never write down the entire list because it never ends. In this case we quickly computed the first few entires of the list 0,1,2,3. But, the actual solution consists of every non-negative integer 0,1,2,3,4,5,$\ldots$ So to actually write down the entire solution we write it as a formula for the nth entry of the list.

###### Continuous age

As you approach your next birthday the discrete age that is assigned to you in society becomes more and more inaccurate because in reality your age is a positive real number not a non-negative integer. Since your age increases at a constant rate and you are zero from the start of measurement, your age may be seen as a line through the origin. Representing your age as a continuous dynamical system will first require you to learn a little calculus. Although all we need to do is say that the rate of change of age with respect to time is constant. This is expressed simply by saying that the derivative of age with respect to time is constant. If $a$ is your age and $t$ is time this means that \[ \frac{da}{dt} = c. \] The distinction here about notation must be made that both $a$ and $\frac{da}{dt}$ are functions of time, $t$, so we should really write $a(t)$ and $\frac{da}{dt}(t)$ but this is a little messy so we often drop the $(t)$. It will always be assumed that the state variable of a continuous dynamical system is a function.

It is a simple exercise to determine the value of this constant $c$. All we need is a little dimensional analysis. If we are measuring age in years and time in years then it is clear a persons age changes by $+1$ year in 1 year time and so \[ c=\frac{1 \text{ year}}{1 \text{ year}} = 1. \]

Similar to the discrete system above we need a starting point from which the dynamics begin. Again we call this an initial condition, but for the continuous function $a(t)$ it is simply the function value at a specific point in this case when $t=0$. We can now be a little more explicit and write down a continuous dynamical system which represents a persons age. \[ \left\{ \begin{array}{r c l} \frac{da}{dt} & = & 1 \\ a(0) & = & 0\\ \end{array} \right. \] This continuous dynamical system is a very simple differential equation whose solution is a line through the origin with slope 1. We will spend a large portion of this course studying differential equations.