This article is from the Nonlinear Science FAQ, by James D. Meiss email@example.com with numerous contributions by others.
A dynamical system consists of an abstract phase space or state space, whose
coordinates describe the dynamical state at any instant; and a dynamical rule
which specifies the immediate future trend of all state variables, given only
the present values of those same state variables. Mathematically, a dynamical
system is described by an initial value problem.
Dynamical systems are "deterministic" if there is a unique consequent to every
state, and "stochastic" or "random" if there is more than one consequent
chosen from some probability distribution (the "perfect" coin toss has two
consequents with equal probability for each initial state). Most of nonlinear
science--and everything in this FAQ--deals with deterministic systems.
A dynamical system can have discrete or continuous time. The discrete case is
defined by a map, z_1 = f(z_0), that gives the state z_1 resulting from the
initial state z_0 at the next time value. The continuous case is defined by a
"flow", z(t) = \phi_t(z_0), which gives the state at time t, given that the
state was z_0 at time 0. A smooth flow can be differentiated w.r.t. time to
give a differential equation, dz/dt = F(z). In this case we call F(z) a
"vector field," it gives a vector pointing in the direction of the velocity at
every point in phase space.