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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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  • $\begingroup$ 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? $\endgroup$
    – Martin M. W.
    Commented Jul 23, 2011 at 12:13
  • $\begingroup$ No, my question is related to your example. $\endgroup$
    – Navcar
    Commented Jul 25, 2011 at 11:11
  • $\begingroup$ Do you know any reference in which this specific issue is discussed? $\endgroup$
    – Navcar
    Commented Jul 25, 2011 at 11:23
  • $\begingroup$ 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. $\endgroup$
    – Robert Israel
    Commented Jul 25, 2011 at 20:12

2 Answers 2

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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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  • $\begingroup$ 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. $\endgroup$
    – Navcar
    Commented Jul 25, 2011 at 11:24
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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 $A$, $\mathcal{G}^n(y)\to x$.

The book Iteration of Rational Functions, by Alan F. Beardon, might be useful to you. A significant portion of the book focuses on classifying the fixed points of certain discrete dynamical systems and on the properties of the basins of attraction of those fixed points. Be aware that Beardon focuses entirely on systems that arise from iterating rational functions on the Riemann sphere, which may or may not apply readily to your case.

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