I was reading about horseshoes and heterclinic bifurcation 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

(actually the paper says "surface diffeomorphism" which I just took to mean "differential equation")

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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?

I also don't understand how the stable and unstable manifolds can intersect.

Grover is correct. The blog talks about a $C^2$ diffeomorphism $f:M\to M$ it has a horseshoe and a periodic point. And the local (un)stable manifolds have to do with the periodic point $p \in K$. These two manifolds have quadratic tangency at some other point $q \in M - K$ (which I guess means these curves agree to second $o(x^2)$ rather than just linearly.

I had not even begun to read the rest of the blog about horseshoe dynamics. The blog does not name any map in particular. Just looking for a specific $f$ that might work (if that's even possible). There's no reason to believe an algebraic example should work here but there must have been some classical instance of this.


It cannot happen in a continuous time 2D system, simply due to uniqueness of ODE property. At least (2+1)-D is needed, i.e. this phenomenon can be seen in 2D maps derived from taking time-T sections of a time-dependent 2D ODE system.

A common example is forced Duffing system.

A good resource is this document by Wiggins: http://docenti.lett.unisi.it/files/4/2/13/1/Wiggins_CHAOS.pdf

Also take a look at Wiggin's "Global bifurcations and chaos".

Other keywords to search for : "Melnikov method", "conley-moser conditions".


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