Chaos/Fixed points. I was reading a book by Strogatz and I encountered this.

Question

Now, I always thought a fixed point implied $f(x)=x$, so somebody tell me, what is he talking about here?

Thank you.

Answer 1

I suspect that you're thinking of the definition of fixed point that arises in discrete dynamics - i.e. the study of the iteration of a function $f$. For example, if $f(x)=x^2$, then zero is a fixed point, since $f(0)=0$. The "orbit" of $x=1/2$ tends towards the fixed point.

Now, you're studying a differential equation $x' = x^2-1$. A solution is a function $x(t)$ with the property that $x'(t) = x(t)^2 - 1$. It's easy to show that one solution is $x(t)=-1$. This is fixed in the sense that it doesn't change. Most other solutions are given by

$$x(t) = \frac{1-ce^{2t}}{1+ce^{2t}}$$

and many of these functions tend toward $-1$ as $t\rightarrow\infty$.