Equações diferenciais implícitas com descontinuidade - Lopes, Bruno Domiciano [UNESP] (2016)
In 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 respectively