<p>I think
<span class="math-container">$$
f(x) = -\left(x-1\right)^{2}x\left(x+1\right)^{2}
$$</span>
works, perhaps there are simpler examples.</p>
<p><a href="https://i.sstatic.net/ZhtqV.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ZhtqV.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2024-03-20T05:58:54.063Z
2024-03-20T05:58:54.063Z
Giskard
<p>I am trying to find a (relatively) simple closed formula for a continuous dynamic system where a local asymptotic equilibrium <span class="math-container">$x^*$</span> is bracketed by two 'one-sided' attractors <span class="math-container">$x_l$</span> and <span class="math-container">$x_h$</span>. So I want to find a continuous <span class="math-container">$f()$</span> for a system
<span class="math-container">$$
\dot{x} = f(x)
$$</span>
where <span class="math-container">$x \in \mathbb{R}$</span>, such that from states <span class="math-container">$x &lt; x_l$</span> we converge to <span class="math-container">$x_l$</span>, from <span class="math-container">$x &gt; x_h$</span> we converge to <span class="math-container">$x_h$</span>, and for <span class="math-container">$x_l &lt; x &lt; x_h$</span> we converge to <span class="math-container">$x^*$</span>.</p>

<dynamical-systems>
One sided attractors in a one dimensional continuous time system
Giskard
# One sided attractors in a one dimensional continuous time system

(Asked by: username_0 on 2024-03-20)

I am trying to find a (relatively) simple closed formula for a continuous dynamic system where a local asymptotic equilibrium $x^\*$ is bracketed by two 'one-sided' attractors $x\_l$ and $x\_h$. So I want to find a continuous $f()$ for a system
$$
\dot{x} = f(x)
$$
where $x \in \mathbb{R}$, such that from states $x < x\_l$ we converge to $x\_l$, from $x > x\_h$ we converge to $x\_h$, and for $x\_l < x < x\_h$ we converge to $x^\*$.

**Tags:** dynamical-systems


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## 1 answers

### Answer by username_0 on 2024-03-20. Score: 1

I think
$$
f(x) = -\left(x-1\right)^{2}x\left(x+1\right)^{2}
$$
works, perhaps there are simpler examples.

[![enter image description here](https://i.sstatic.net/ZhtqV.png)](https://i.sstatic.net/ZhtqV.png)


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