
And this course we are mostly using the computer to do simulations

numerically, that is by number crunching.

The major reason for that is that most problems can't be solved

by clean equations, unlike school book problems.

The real road is way too complex to do that.

A second reason is that particularly we have to base our simulations

on huge numbers of measurements be it climate's data, be it financial data, be it data from biology,

but the numerically approach also has a series of drawbacks.

In particular, it's very hard to prove general results using numerics.

Is there no way that a particle in some system escapes to infinity?

You can look at a single trajectory at a time, but you cannot look at an infinite number of trajectories

and numerical methods almost always introduce discretization errors

also known as truncation errors.

In particular, we have to work with finite steps in time.