# Fixed points which are not locally attractive can have distant basins of attraction?

I have conducted simulations of a discrete dynamical system. I know the fixed points of the system and which of them are attractive. According to the simulations, I obtain that certain fixed points that are not locally attractive can have basins of attraction away from the point.

I would like to know whether it is possible or it is just rounding errors.

Thank you!

Sorry, I think I can rise the question in a more precise way:

Let $x \in \mathbb{R}^n$ be an unstable fixed point of the discrete dynamical system $\mathcal{G}$. There may be a set $A \subset \mathbb{R}^n$ with non-zero Lebesgue measure such that $\forall y \in A, lim_{n\rightarrow \infty} \mathcal{G}^n(y) = x$ ?

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Under one interpretation of your terms, here's an example. The flow defined by $x' = x^2$ has a non-locally-attracting fixed point at 0, but any open set of negative numbers is attracted to it. But perhaps you mean something else? –  Martin M. W. Jul 23 '11 at 12:13
No, my question is related to your example. –  Navcar Jul 25 '11 at 11:11
Do you know any reference in which this specific issue is discussed? –  Navcar Jul 25 '11 at 11:23
You might look up "semi-stable equilibrium". Another interesting example is \eqalign{x' &= x^2 - y^2\cr y' &= 2 x y\cr} where the trajectories off the $x$ axis are circles tangent to the $x$ axis at the origin. The basin of attraction of the origin is the complement of the positive $x$ axis. –  Robert Israel Jul 25 '11 at 20:12

It's possible for stupid reasons:

Consider the dynamical system that is $T(x)=2x$ for $x\in[0,1/2]$, $1-100(x-1/2)$ for $x\in [0.5,0.51]$ and $0$ for $x\in[0.51,1]$.

Now 0 is locally repelling, but the basin contains open intervals.

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Thank you. I am using dynamical systems as a tool for other topics. I need to learn more about this specific issue i.e. about non-locally-attracting fixed points with basin of attraction. Could you recommend me any reference? Thanks again. –  Navcar Jul 25 '11 at 11:24
Your set $A$ might be a parabolic component of the Fatou set of $\mathcal{G}$. In that case $x$ would be an indifferent fixed point of $\mathcal{G}$ on the boundary of $A$ and for all $y$ in $A$, $\mathcal{G}^n(y)\to x$.