A dynamical system is any process that moves or changes in time.
Dynamical systems occur in every branch of science.
The motions of the planets in celestial mechanics are a good example of a process of something that evolving in time. The weather is another system that changes dramatically over time. Similarly, the Stock Market, economic systems are good examples of very chaotic at times, dynamical systems. Finally, in chemistry, simple chemical reactions are examples of processes that evolve in time. If you look at some of the examples giving of dynamical systems, it is clear that some of them are predictable.