 $\tanh(z) \approx z$ for small $z$.  For $w
        \le 1$ the system has only a single fixed point.  For $w > 1$ the system
        has three fixed points : two stable and one unstable.</p>

        <figure> <img width="80%" src="figures/pend_autapse.svg"/>
        </figure>

        <p>These equations are not arbitrary - they are actually a model for one
        of the simplest neural networks, and one of the simplest models of
        persistent memory<elib>Seung00</elib>.  In the equation $x$ models th