We look at two example functions that have a built in time dimension.

As a warm up exercise we look at a “logistic map”, that investigates the simple function:

It plots what happens to x as we vary the parameter r. This function can be used to model population growth.

The diagram of the stable states starts with a simple curve that then starts bifurcating until it generates a complex veil-like pattern.

We can consider this a two-dimensional function with one space and one time dimension, both along the Cartesian axes.

### The Mandelbrot Universe

We can also consider the Mandelbrot set as a 2-D object with a space and a time axis, but not along the Cartesian axes. We can look at the Mandelbrot set as a collection of contour lines.

This way the 1-dimensional Mandelbrot map is a perfect circle at time t=0 and then at time t=1 is a bit deformed and continues to become more chaotic and complicated.

There are an infinity of different ‘contour’ lines, but at each finite time t there is a finite space which is more convoluted than the previous one at t-1. We call them contour lines because they are concentric and never intersect.

This is interesting because we have with the Mandelbrot set a 2-dimensional mathematical object with well defined time and space dimensions, where space is a continuous quantity while time is quantised into discrete time steps.

Now we are ready to look at multiple time dimensions.