Math Insight

An introduction to discrete dynamical systems

Dynamical systems are about the evolution of some quantities over time. This evolution can occur smoothly over time or in discrete time steps. Here, we introduce dynamical systems where the state of the system evolves in discrete time steps, i.e., discrete dynamical systems.

When we model a system as a discrete dynamical system, we imagine that we take a snapshot of the system at a sequence of times. The snapshots could occur once a year, once every millisecond, or even irregularly, such as once every time a new government is elected.

When we take these snapshots, the idea is that we are recording whatever variable determine the state of the system: our chosen state variables that evolve through the state space (see dynamical systems idea). To complete the description of the dynamical system, we need to specify a rule that determines, given an initial snapshot, what the resulting sequence of future snapshots must be.

Here, we introduce these basic concepts of a dynamical system through an example involving the evolution of a population of moose.

Dynamics of moose population

Video 1

Introduction to discrete dynamical systems, part 1.

More information about video.

Question about video 1

With one exception, each of the following are valid descriptions of $m_{t+1}-m_{t}$. Which one is an invalid description?

Video 2

Introduction to discrete dynamical systems, part 2.

More information about video.

Question about video 2

If one starts with a moose population of $m_0=1134$ individuals, and the population grows by 5% per year (i.e., $m_{t+1}-m_t = 0.05 m_t$), what is the size of the moose population after after one year?

(no rounding)

Summary from moose videos

If a moose population starts with 1000 individuals and grows by 8% per year, we can model the population with a discrete dynamical system \begin{align*} m_{t+1}-m_t &= 0.08 m_t\\ m_0 &= 1000, \end{align*} where the state variable $m_t$ is the number moose in year $t$. The discrete dynamical systems gives a rule for going from a snapshot of the moose population to another snapshot of the moose population one year later. One can use the dynamical system to calculate the population $m_1$ after one year, $m_2$ after two years, etc.

You can view more examples of iterating discrete dynamical systems. You can also read how to solve linear dynamical systems like the moose model so that you can directly calculate the value of the moose population at any time, without having to iterate the system for each time point.