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Assume that an autonomous differential equation $$ \frac{dx}{dt}=F(x), \quad x\in R^n,\quad F:R^n\to R^n, \quad n>2$$ has a homoclinic orbit $x=x_{h}(t).$ Does it ever imply that the equation has periodic orbits?

I guess for $n=2$ this is the case. Can something be said about higher dimensions?

JaneEyre
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1 Answers1

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It's not necessarily true in case $n = 2$: see this answer for example. In case $n \geqslant 3$ there is a Shilnikov theorem which describes the structure of neighbourhood of homoclinic loop to saddle-focus. Basically it states that the return map to some transversal near the saddle-focus has a Smale horseshoe which corresponds to saddle-limit cycles in the neighbourhood of homoclinic loop. Shilnikov also considered the case of homoclinic loop to saddle limit-cycle, see the paper On a Poincaré-Birkhoff Problem. As far as I understand in this case trajectories near the homoclinic loop admit symbolic description which is also derived from return map. In both cases the presence of homoclinic orbit instantly means complicated structure of phase space and plenty of limit cycles.

Evgeny
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  • thanks a lot for the useful references!

    I have realized that I am dealing with a Hamiltonian system which corresponds to the case (3) in "Topological type of saddle-foci" Shilnikov theorem with $\sigma_2=0.$

    So basically, the system has a homoclinic loop to bi-focus $x_0$ that has two pairs of leading eigenvalues $\lambda_{1,2}=\rho \pm i \omega$ and $\lambda_{3,4}=-\rho \pm i \omega.$

    Do you know if there are studies for this case?

     – JaneEyre Sep 20 '17 at 12:50
  • Hm, this time it's harder to pull reference off my head :) I think that this study should have been done by somebody. I'll try to find and add reference her, but in case you want to try to search it yourself pay attention to the articles of Shilnikov, Turaev, Tresser, Sparrow, Glendinning, Coullett, Arneodo, Bo Deng. I think it's highly likely to find your case in their works or works of their collaborators. – Evgeny Sep 20 '17 at 15:03
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    Also this one seems to be discussing your case, please take a look at it. – Evgeny Sep 20 '17 at 16:26
  • does Theorem A imply that there are at least countably many periodic orbits in the neighborhood of the homoclinic orbit? – JaneEyre Sep 21 '17 at 09:53
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    Yes, that's right. Because of correspondence between trajectories on hyperbolic set and symbolic dynamics of Bernoulli shift there is at least as many periodic orbits as periodic orbits in Bernoulli shift. If I'm not mistaken, periodic orbits of Bernoulli shifts is a countable union of finite-size sets so it's countable too. – Evgeny Sep 21 '17 at 10:36