The earliest systems found to display some sort of memory were probably those with hysteresis.
Another important example are systems with time delay, e.g., described by delay differential equations, which can model a variety of different systems, e.g., "in physiological studies, in ecology, in lasers, etc.": Lakshmanan & Senthilkumar's Dynamics of Nonlinear Time-Delay Systems.
Networks also display memory effects, including hysteresis (see, e.g., Majdandzic et al Multiple tipping points and optimal repairing in interacting networks), and have been used to model the Information Processing Capacity of Dynamical Systems, as well as, of course, the brain.
I don't currently know of any book focused exclusively on memory mechanisms of dynamical systems. A search reveals Alonso-Sanz' Discrete Systems with Memory , which might be close to what you're looking for.