Título:
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The Melnikov method and subharmonic orbits in a piecewise smooth system
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Autor/a:
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Granados, Albert; Hogan, S. John; Martínez-Seara Alonso, M. Teresa
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Otros autores:
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Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I; Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions |
Abstract:
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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 ", using
an impact map, we rigorously prove that, for every n and m relatively prime and " > 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 " > 0 with
respect to the magnitude of this jump is large enough.
We also obtain similar conditions for the splitting of the heteroclinic connections. |
Materia(s):
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-Àrees temàtiques de la UPC::Matemàtiques i estadística -Differential equations -Hamiltonian systems -Equacions diferencials -Hamilton, Sistemes de -Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems -Classificació AMS::37 Dynamical systems and ergodic theory::37N Applications |
Derechos:
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Attribution-NonCommercial-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/es/ |
Tipo de documento:
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Artículo - Borrador Informe |
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