Silva, Paulo Ricardo da [UNESP]Lopes, Bruno Domiciano [UNESP]2016-04-062016-04-062016-03-18http://hdl.handle.net/11449/137797In this thesis we deal with non-smooth dynamical systems expressed by piecewise first order implicit differential equations of the form \[\dot{x}=1,\quad \left (\dot{y}\right)^2=\left\{\begin{array}{lll} g_1(x,y) \quad \mbox{if}\quad \varphi(x,y)\geq0, \\ g_2(x,y) \quad \mbox{if}\quad\varphi(x,y)\leq0, \end{array}\right. \] where $g_1,g_2,\varphi:U\rightarrow\R$ are smooth functions and $U\subseteq\R^2$ is an open set. The main concern is to study sliding modes of such systems around some typical singularities. The novelty of our approach is that some singular perturbation problems of the form \[\dot{x}= f(x,y,\e) ,\quad (\e\dot{ y})^2=g ( x,y,\e) \] arise when the Sotomayor--Teixeira regularization is applied with $(x, y) \in U$, $\e\geq0$, and $f, g$ smooth in all variables. For the cubic polynomial differential systems in $\R^2$ with centers we study the maximum number of limit cycles that bifurcate from some families of planar polynomial differential systems of degree 3 with rational first integrals of degree 2 when they are perturbed inside the classes of all cubic polynomial differential systems. We obtain an explicit polynomial whose simple positive real roots provide the limit cycles which bifurcate from the periodic orbits of any weight--homogeneous polynomial differential systems having centers with (weight--degree, (weight--exponent)) (3,(1,1)), (2,(1,2)) e (3,(1,3)) when it is perturbed inside the class of all polynomial differential systems of degree n, 3 and 5 respectivelyNesta tese trabalhamos com sistemas din\^{a}micos n\~{a}o-suaves expressos por equa\c{c}\~{o}es diferenciais impl\'{i}citas descont\'{i}nuas de primeira ordem da forma \[\dot{x}=1,\quad \left (\dot{y}\right)^2=\left\{\begin{array}{lll} g_1(x,y) \quad \mbox{se}\quad \varphi(x,y)\geq0, \\ g_2(x,y) \quad \mbox{se}\quad\varphi(x,y)\leq0, \end{array}\right \] onde $g_1,g_2,\varphi:U\rightarrow\R$ s\~{a}o fun\c{c}\~{o}es suaves e $U\subseteq\R^2$ \'{e} um conjunto aberto. O principal interesse \'{e} estudar a din\^{a}mica deslizante de tais sistemas em torno de algumas singularidades t\'{i}picas. A novidade da nossa abordagem \'{e} que alguns problemas de perturba\c{c}\~{a}o singular da forma \[\dot{x}= f(x,y,\e) ,\quad (\e\dot{ y})^2=g ( x,y,\e) \] surgem quando aplicamos a regulariza\c{c}\~{a}o Sotomayor--Teixeira com $(x, y) \in U$, $\e\geq0$, e $f, g$ s\~{a}o suaves em todas as vari\'{a}veis. Para os sistemas diferenciais polinomiais c˙bicos em $\R^2$ que possuem centros, estudamos o número máximo de ciclos limites que podem bifurcar de algumas famílias de sistemas diferenciais planares polinomiais de grau 3, com integrais primeiras racionais de grau 2, quando eles são perturbados dentro da classe de todos os sistemas polinomiais diferenciais cúbicos. Obtemos um polinômio explícito cuja as raízes simples reais positivas fornecem os ciclos limites que bifurcam a partir das órbitas periódicas de qualquer sistemas diferenciais polinomiais homogêneos--ponderados que tem um centro com ( grau--ponderado, (expoente--ponderado)) (3,(1,1)), (2,(1,2)) e (3,(1,3)) quando é perturbado dentro de todas as classes de sistemas diferenciais polinomiais de grau n, 3 e 5 respectivamente.porLimit cyclesPlanar Vector FieldsIsochronous CentersAveraging methodCiclos LimiteCampo de Vetores PlanaresCentros IsócronosMétodo do averagingTese de doutoradoAcesso aberto00086844133004153071P060509558611681610000-0002-1430-5986