I was reading about horseshoes and [heterclinic bifurcation](https://matheuscmss.wordpress.com/2012/09/10/homoclinicheteroclinic-bifurcations-thin-horseshoes/) but my knowledge of dynamical systems is really old fashioned. 

as I understand the local stable manifold and the local unstable manifold intersect. I am looking for a differential equation or geometric construction that demonstrates this phenomenon

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

this seems highly improbable.  for example the flow defined by 

$$  \frac{df   }{ dt   } = (x^3 + ax   +b) f $$

could never exhibit such self-tangent orbits.  perhaps I am looking at the wrong place for example?