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Heteroclinic orbits in slow-fast Hamiltonian systems with slow manifold bifurcations. (English) Zbl 1213.34072
The authors investigate a second order Hamiltonian system of the form ${x}_{\tau \tau }={g}_{x}\left(x,y\right)$ and $ϵ{y}_{\tau \tau }={g}_{y}\left(x,y\right)$ where $-g\left(x,y\right)$ can be regarded as the potential energy. Written as a four dimensional slow-fast system the 2-dimensional slow manifold undergoes a pitchfork bifurcation leading to a breakdown of normal hyperbolicity. The limiting problem has a heteroclinic connection between equilibria on the slow manifold. The authors prove, using a blow up technique, that the heteroclinic orbit persists despite the fact that not only normal hyperbolicity breaks down but as well the manifold property along the line of pitchfork bifurcations. Similar results have been obtained by [P. C. Fife, A phase plane analysis of a corner layer problem arising in the study of crystalline grain boundaries. unpublished preprint. University Utha] based on a shooting argument and the authors in [C. Sourdis and P. C. Fife, Adv. Differ. Equ. 12, No. 6, 623–668 (2007; Zbl 1157.34047)] by a contraction mapping principle. The problem is motivated by models of cristalline states where a heteroclinic orbit represents a moving interface between ordered and disordered cristalline states.
##### MSC:
 34E15 Asymptotic singular perturbations, general theory (ODE) 34C37 Homoclinic and heteroclinic solutions of ODE 34C23 Bifurcation (ODE) 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
##### References:
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