Limit cycles in planar piecewise smooth systems having non-regular switches, time scales or rotated properties
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Data
2022-09-30
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Universidade Federal de Goiás
Resumo
In this thesis, periodic trajectories in planar discontinuous piecewise linear systems with
a nonregular switching line are studied. We provide sharp upper bounds of one or two
limit cycles for certain classes of the model considered. We also establish the stability and
hyperbolicity of these limit cycles. In addition, we provide examples reaching one and two
limit cycles for these classes. We perform the global analysis of a representative model
through bifurcation theory to analyze the birth of limit cycles, sliding periodic trajectories,
and tangential ones. We also provide some results addressing the coexistence of periodic
trajectories. We studied Fast-Slow systems with nonregular switching line with a new
approach. This study allows proving that a specific sliding periodic trajectory is in fact
a homoclinic trajectory. This homoclinic trajectory arises from a bifurcation of sliding
limit cycles that are not topologically equivalents. We propose the theory of piecewise
rotated vector fields with the goal of understanding how the trajectories of two families
of rotated vector fields behave as the same parameter is varied. In this context, we prove
the non-intersection theorem for closed periodic trajectories for piecewise rotated vector
fields.
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Citação
ALVES, A. M. Limit cycles in planar piecewise smooth systems having non-regular switches, time scales or rotated properties. 2022. 112 f. Tese (Doutorado em Matemática) - Universidade Federal de Goiás, Goiânia, 2022.