Given your background and interest in modeling, I would highly recommend The Cambridge Handbook of Computational Psychology. The book provides an overview for several of the prominent modeling paradigms in cogsci, including dynamical systems, as well as many concrete examples--albeit most using other computational paradigms.
Dynamical systems, to my knowledge, are most used in the field of motor control. Though to be honest, dyn systems is not 'my' paradigm, and most of my knowledge comes from the aforementioned book.
In particular, the chapter "Dynamical Systems Approaches to Cognition" by
G. Schoner gives this example that has always stuck in my mind:
A highly illustrative example comes from the orientation behaviors of
the common house ﬂy (Reichardt & Poggio, 1976; Poggio & Reichardt,
1976). Flies orient toward moving objects, which they chase as part of
their mating behavior. Detailed analysis revealed that the circuitry
underlying this behavior forms a simple controller: a motion detection
system fed by luminance changes on the ﬂy’s facet eye drives the ﬂight
motor, generating an amount of torque that is a function of where on
the sensory surface motion was detected. If the speck of motion is
detected on the right, a torque to the right is generated. If the
speck is detected on the left, a torque to the left is generated. The
level of torque passes through zero when the speck is right ahead. The
torque changes the ﬂight direction of the ﬂy, which in turn changes
the location on the facet eye at which the moving stimulus is
detected. Given the aerodynamics of ﬂies, the torque and its on-line
updating generate an orientation behavior, in which the insect orients
its ﬂight into the direction in which a moving stimulus is detected.