How things move
A space (in the mathematical sense, for a continuous space, one often uses a Manifold, or a Topological space), with a Function (a.k.a. a map) that describes how a point in the space evolves (in "time").
Measure-theoretical dynamical system
Continuous dynamical system are dynamical systems where the space is continuous. It is often represented as a system of 1st order O.D.Es. Linear dynamical systems (O.D.E.s linear) are easy to analyze, and can be analyzed by looking at the eigenvalues of the Jacobian.
Discrete dynamical system are those where the space is discrete. They are often represented as systems of difference equations (see Nonlinear maps).
Measure-theoretical dynamical system
The richest class of dynamical systems are Nonlinear systems
A dynamical system, whether continuous or discrete, can be partitioned (coarse-grained), so that its dynamics can be studied as Symbolic dynamics. If the system is a Probabilistic dynamical system, then the coarse-graining gives rise to a stochastic process
Dynamical systems generally describe deterministic processes. Probabilistic processes are described as Stochastic processes. However, these can sometimes be described as deterministic dynamics of probability distributions, or as a probability measure over a deterministic process (i.e. a Probabilistic dynamical system).
Dynamical system software: https://dsweb.siam.org/Software
See Wiki page for good intro and different kinds
Dynamical systems on complex space (particularly discrete ones): Complex dynamics
Nonlinear Dynamics 1: Geometry of Chaos by Predrag Cvitanović (ChaosBook course)
https://en.wikipedia.org/wiki/Floquet_theory