Résumé:
In this thesis, we rst gave essential information such as de nitions, lemmas, and theorems used in
our study. Second, we have presented the averaging theory for computing limit cycles for the systems,
focus-focus, focus-center, with the straight separation line y = 0; we proved that there are at most
1, 2, or 3 limit cycles, and the systems, center-center can not have limit cycles. Next, we studied the
limit cycles of planar piecewise di erential systems formed by quadratic systems and linear centers.
We proved that these piecewise systems, with the straight separation line y = 0; have a continuum
of periodic orbits and can have at most 2, 3, 4, 5, and 8 limit cycles. Finally, we have tackled the
number of periodic solutions of the piecewise di erential systems formed by the linear center and
isochronous cubic system separated by the straight-line x = 0 and y = ax + b by treating the two
cases as continuous and discontinuous. We proved that piecewise systems have three solutions with
a long formula.
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