单 位：西班牙巴塞罗那自治大学 

题 目：Explicit bifurcation diagrams with applications 

时 间：2013年10月11日上午 9:00 

地 点：理科楼-408 

摘 要：The key points for knowing the bifurcation diagrams for families of planar differential equations are the knowledge of the global behavior of the separatrices of their critical points and the control of the number of limit cycles that these equations can have. In this talk we present a method for studying the global behavior of the separatrices based on the construction of suitable piecewise algebraic curves without contact for the flow of the differential equation. These curves are obtained using local and global analytic information of the separatrices of the finite and infinite critical points of the vector field.  The study of the number of limit cycles is done by  applying the Bendixson-Dulac Theorem with suitable rational Dulac functions. We apply this approach to study an one parameter family that extends the celebrated van der Pol system and to the Bogdanov-Takens system. A similar method can also be used to determine the shape of the traveling waves appearing in the Fisher-Kolmogorov partial differential equation.