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

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?

A dynamical system defined by a map uses discrete rather than continuous time steps. That is all. The dynamics are given by $$x_{t+1} = g\left( x_t \right)$$
• That...is much simpler than I would've thought. Thanks for the clarification! So a fixed point of a map would be when $g(x_t)=x_t$, right? Jan 6, 2021 at 1:48
• Yes, exactly. Any solution to $g(x)=x$ is a fixed point. Jan 6, 2021 at 1:51