I have this question: What is a separatrix of a equilibrium point of a continuous dynamical system and why it is flow-invariant? Thanks

Hello and thanks for the answer. I explain better. I'm following a first undergraduate course in dynamical systems. My professor gave me this definition of separatrix curve :

"Let be $\dot X=F(X)$ a planar dynamical system and let be $\hat X$ an equilibrium point. A differentiable curve $g:I\rightarrow\mathbb R^2$ is a stable separatrix for $\hat X$ if:

1) For every $P\in Im(g)$ the solution with initial condition $P$ exists for every $t\in[0,+\infty)$ and has $\hat X$ for limit as $t$ tends to $ +\infty $ .

2)For every $P\in Im(g)$ exists a neighborhood $U$ of $P$ such that for every $Q\in U-Im(g)$ the solution with initial condition $Q$ does not have $\hat X$ for limit as $t$ tends to $+ \infty$ . "

This is the definition...

Well, it seems to not work...

For example if i consider the system: $$\begin{cases} \dot x=-x \\ \dot y=0 \end{cases} $$ the basin of attraction of $(0,0)$ is the $x$ - axis and it should be even the (image of the) stable separatrix. Now, according to the definition, even the curve $g:(0,1)\rightarrow \mathbb R^2 $ , with $g(s)=(s,0)$ is a stable separatrix for $(0,0)$ . Now i also know that the image of the separatrix should be positively invariant...but g is not! Where i get wrong?

  • $\begingroup$ Can you give more mathematical context to this? It seems too terse to invite a lengthy explanation. Are you taking a math course in dynamical systems? $\endgroup$ – hardmath Jan 31 '14 at 17:18
  • $\begingroup$ For me, $x$-axis seems to be surely flow-invariant. Note that left half of $x$-axis and right half of $x$-axis are two distinct separatrices. $\endgroup$ – Evgeny Feb 1 '14 at 6:31
  • $\begingroup$ Thanks for the answer!That the x-axis is invariant is clear, what is not clear for me is: for my definition, the x-axis, the left half and the right- half of the x-axis are separatrices...so...how many separatrices there are? Where i get wrong? My definition is correct? Thanks! $\endgroup$ – user125242 Feb 5 '14 at 18:09

With this definition, the separatrices are exactly the differentiable curves $g:I\to\mathbb R^2$ such that $g(I)\subseteq\mathbb R\times\{0\}$. Thus, $g(I)$ can be any interval of the $x$-axis. There are more restrictive definitions of this notion, though.


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