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