I have been having a hard time with the topic of discrete dynamical systems in my college linear algebra 2 course; I recall sketching phase portraits in a differential equations class, though it was several years ago and many of the intricacies are now lost on me.
Below is a problem from a past homework assignment that provides matrix A for the discrete dynamical system x(k+1)=Ax(k). The first two parts I am able to complete (finding the characteristic equation, determining the eigenvalues and eigenvectors, etc.), however, I'm stumped when asked to sketch the trajectory through a specific point.
I feel like I would be able to do this if I had a better understanding of how to determine trajectory "behavior." If the problem I provided is too long, here's a few straightforward questions I have:
- How do I determine if the origin is a saddle or node?
- How do I know if the origin is stable or unstable?
- Will the trajectories through specific points follow the same behavior as the trajectories of eigenvectors? (This one is the least ambiguous; I am assuming yes).
Anytime I try to search "sketching trajectories for discrete dynamical systems," I wind up more confused than I started. Any help or clarification is greatly appreciated; cheers!