تقرير
The Melnikov method and subharmonic orbits in a piecewise smooth system
| العنوان: | The Melnikov method and subharmonic orbits in a piecewise smooth system |
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| المؤلفون: | Granados, A., Hogan, S. J., Seara, T. M. |
| سنة النشر: | 2012 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Dynamical Systems |
| الوصف: | In this work we consider a two-dimensional piecewise smooth system, defined in two domains separated by the switching manifold $x=0$. We assume that there exists a piecewise-defined continuous Hamiltonian that is a first integral of the system. We also suppose that the system possesses an invisible fold-fold at the origin and two heteroclinic orbits connecting two hyperbolic critical points on either side of $x=0$. Finally, we assume that the region closed by these heteroclinic connections is fully covered by periodic orbits surrounding the origin, whose periods monotonically increase as they approach the heteroclinic connection. When considering a non-autonomous ($T$-periodic) Hamiltonian perturbation of amplitude $\varepsilon$, using an impact map, we rigorously prove that, for every $n$ and $m$ relatively prime and $\varepsilon>0$ small enough, there exists a $nT$-periodic orbit impacting $2m$ times with the switching manifold at every period if a modified subharmonic Melnikov function possesses a simple zero. We also prove that, if the orbits are discontinuous when they cross $x=0$, then all these orbits exist if the relative size of $\varepsilon>0$ with respect to the magnitude of this jump is large enough. We also obtain similar conditions for the splitting of the heteroclinic connections. |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/1201.5475 |
| رقم الأكسشن: | edsarx.1201.5475 |
| قاعدة البيانات: | arXiv |
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