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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?

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

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  • $\begingroup$ 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? $\endgroup$ Jan 6, 2021 at 1:48
  • $\begingroup$ Yes, exactly. Any solution to $g(x)=x$ is a fixed point. $\endgroup$
    – sasquires
    Jan 6, 2021 at 1:51
  • $\begingroup$ Great, thanks. Now I can get past chapter 1. :) $\endgroup$ Jan 6, 2021 at 1:52

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