a dynamical system consists of a space, or manifold, where the motion of the system takes place, and a rule of motion, or vector field. The starting point is called the initial state, and the path of motion the trajectory. The end point of a trajectory is the system's attractor.
Cooking receipes, computer programs, and dynamical systems are all isomorphic. (having the same structure.) They consist of inputs and vectors. "do this with that."(for a definition of isomorphism see Rudolf Carnap, Introduction to Symbolic Logic.) Because of this isomorphism and the undecidability of formal systems, first proved by Kurt Gödel and later extended by Gregory Chaitin, almost every interesting property of a dynamical system is undecidable. The reason for this is that dynamical systems are so versatile they can model the computational process itself. Shouldn't it be the other way around?
Stuart Kauffman often characterizes the three dynamical regimes as solid or frozen (the ordered regime), as gas (the chaotic regime), and as liquid (the complex regime). (see phase boundary ) Dynamical Systems can be characterized as conservative or dissipative, depending on whether their phase volume stays constant or contracts. Brian Goodwin proposes that the term natural selection could simply be replaced with the word dynamic stabilization, the emergence of stable states in dynamic systems.
see Ian Stewart, "the Dynamics of Impossible Devices", Nonlinear Science Today 1 (1991)
dynamics vs. kinematics: kinematics is geometrical decription of movement whereas dynamics is causal.