# Dynamical Systems

## What is a Dynamical System?

A dynamical system is any system, man-made, physical, or biological, that changes in time. Think of the Space Shuttle in orbit around the earth, an ecosystem with competing species, the nervous system of a simple organism, or the expanding universe... the list is endless!

To study dynamical systems mathematically, we represent them in terms of differential equations. The state of dynamical system at an instant of time is described by a point in an n-dimensional space called the state space (the dimension n depends on how complicated the systems is - for the double pendulum below, n=4). As time passes the point moves, sweeping out a curve in the state space, called a trajectory of the system (mathematically, this curve is a solution of the governing differential equations).

A very important aspect of dynamical systems is that they are nonlinear. This means they can exhibit unusual and surprising behaviour as described below.

## Chaos

### Double Pendulum

An example of a chaotic system is the double pendulum shown in the animation at the left. The behaviour of the pendulum is complex and never seems to repeat itself.

### Two Double Pendulums

In this animation, the two pendulum systems seem to start in the same place at the beginning of the animation. Why are they doing very different things after a while?

The two systems actually start with slightly different initial positions. But the small difference in their initial positions is amplified over time due to the chaotic behaviour of the system. This is called sensitive dependence on initial conditions.

## Bifurcation

The two videos below show a pendulum that is balanced in the upright position through the use of feedback control.

### Delay 0.06 sec

Remote video URL

In this video, the pendulum stays upright, even when it is given a tap.

### Delay 0.064 sec

Remote video URL

In this video, the pendulum starts to oscillate after it is given a tap.

### What is the difference?

In both videos, there is a time delay between when the computer measures the position of the pendulum and when it moves the cart to balance the pendulum. The time delay in the second video is 0.004 seconds longer than in the first video. This small change in the time delay causes a qualitative change in the behaviour of the system. This is a bifurcation.