# When is a dynamical system discrete vs. continuous?

I have a basic question to ask:

Let's say I am reading a paper which gives a good model that consists of a set of ordinary differential equations, with first and second derivatives. Continuity is a requirement for differentiability, as we learned in an Analysis course.

Now, I simulate this model, writing code in Matlab, and calling a basic ode solver, ode45, which is a version of Euler's formula but with adaptive time-stepping. Then, the equations are being solved with a discrete method, namely, with discrete time-steps. There are no exact solution formulas being solved for.

Then, is this a discrete or continuous dynamical system?

## 1 Answer

You're using a numerical method to approximate the solution to the continuous dynamical system. If you've done this carefully, that approximate solution could be adequate for investigating the properties of the continuous dynamical system.

You could also, if you wanted to, analyze the numerical approximation as a discrete dynamical system.

The key word in your question is "this" in the question "Is this a discrete or continuous dynamical system?" What exactly does "this" refer to?

• Actually, are you sure of the terminology? For instance, according to Wikipedia, the ODEs would be called evolution equations / rules, with the solutions being called the dynamical system, with a time-dependent state space. You're calling both the evolution equations and the numerical solutions "dynamical systems". Just want to be sure - thanks. – user35586 Apr 9 '20 at 21:31
• If you want to call the solutions to an evolution equation, a dynamical system, then it is equally the case that the solutions to the system of difference equations used in approximating the solutions to the continuous time evolution equation are a dynamical system. It is not uncommon in my experience for the term dynamical system to refer both to the evolution equations with initial conditions and the solution of those equations. – Brian Borchers Apr 10 '20 at 1:56
• Ok, thanks for clarifying dawg. – user35586 Apr 10 '20 at 3:20