I have a system of 3 ODEs that yields the attached graph when solved numerically. I was wondering: how can I calculate the local minimum of the green curve in this graph? Is this possible?
There is not enough information provided to answer the question. Do you have the functional form or a list of points or, ... Take a look at FindMinimum and this.
You can use the WhenEvent handling in NDSolve to detect when a derivative becomes zero, and thereby locate stationary points.