Resource recommendations for numerical methods involved in dynamical systems analysis

Question

I am interested in learning numerical methods that specifically have to do with analyzing dynamical systems. In particular:

• drawing phase plane diagrams
• drawing phase portraits
• analyzing bifurcations and drawing bifurcation diagrams

I have found the following sources so far:

Numerical Methods for Nonsmooth Dynamical Systems

Numerical Continuation Methods for Dynamical Systems

Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems

Numerical Methods for Bifurcations of Dynamical Equilibria

However, these are just results I have found from doing searches using keywords like "numerical dynamical systems bifurcation". I don't know the quality of these resources, and thus their usefulness.

Can you comment if you have experience with a resource I have listed, or another one I might not have listed? By the way, I would use a dynamical-systems tag, but I realized there isn't one!

Answer 1

In addition to the references you listed, you might like Geometric Numerical Integration by Hairer, Lubich and Wanner. They go into a lot of detail about "structure-preserving" methods for solving systems of ODE. For example, given a Hamiltonian system

$ \dot q = \frac{\partial H}{\partial p}, \quad \dot p = -\frac{\partial H}{\partial q}$,

the trajectories lie along curves of constant $H$ and you would like your numerical scheme to reflect this fact as much as possible. Unfortunately, the usual forward and backward Euler methods don't do this at all, to the point that they can give you the wrong impression about the long-term stability of a dynamical system. However, for symplectic methods, e.g. Stormer-Verlet, the numerical solution lies on an exact trajectory of a slightly different Hamiltonian $H + \delta H$.