1
$\begingroup$

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?

$\endgroup$

1 Answer 1

1
$\begingroup$

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

$\endgroup$
3
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.