Difference between system of differential equations and maps in the context of dynamical systems

Question

In Stephen Wiggins' "Introduction to Applied Nonlinear Dynamical Systems and Chaos," a distinction is made between systems of equations in the form $$\dot{x}=f(x,t)$$ and $$x \mapsto{} g(x)$$ Quite frankly, I don't understand what the latter expression means other than the fact that it is called a map. My understanding is that the first expression, for let's say $x \in \mathbb{R}^{2}$, is equivalent to the system $$ \dot{x}_1 = f_1(x_1, x_2, t) $$ $$ \dot{x}_2 = f_2(x_1, x_2, t) $$

There is a clear derivative (the $\dot{x}$) in the first expression but I don't see any derivatives in the second. So my question: what does the second method of describing a dynamical system, called a map, mean?

Answer 1

A dynamical system defined by a map uses discrete rather than continuous time steps. That is all. The dynamics are given by $$ x_{t+1} = g\left( x_t \right) $$