Generalized Hopf Bifurcation for Planar Filippov Systems Continuous at the Origin

In this paper, we study the existence of periodic orbits bifurcating from stationary solutions of a planar dynamical system of Filippov type. This phenomenon is interpreted as a generalized Hopf bifurcation. In the case of smoothness, Hopf bifurcation is characterized by a pair of complex conjugate eigenvalues crossing through the imaginary axis. This method does not carry over to nonsmooth systems, due to the lack of linearization at the origin which is located on the line of discontinuity. In fact, generalized Hopf bifurcation is determined by interactions between the discontinuity of the system and the eigen-structures of all subsystems. With the help of geometrical observations for a corresponding piecewise linear system, we derive an analytical method to investigate the existence of periodic orbits that are obtained by searching for the fixed points of return maps.     

Complete Duality for Quasiconvex and Convex Set-Valued Functions

This paper provides a unique dual representation of set-valued lower semi-continuous quasiconvex and convex functions. The results are based on a duality result for increasing set-valued functions.     

Measures and Integrals in Conditional Set Theory

The aim of this article is to establish basic results in a conditional measure theory. The results are applied to prove that arbitrary kernels and conditional distributions are represented by measures in a conditional set theory. In particular, this extends the usual representation results for separable spaces.     

The synthesis and transformations of 9H-imidazo[1,2-b]pyrazolo[4,3-d]-pyridazine and 9H-pyrazolo[4,3-d]-s-triazolo[4,3-b]pyridazine derivatives

The nucleophilic and electrophilic substitutions of 6-substituted 9,9-dimethyl-9H-imidazo[1,2-b]pyrazolo- [4,3-d]pyridazines 2 , nucleophilic substitutions of 6-substituted 9,9-dimethyl-9H-pyrazolo[4,3-d]-s-triazolo- [4,3-b]pyridazines 7 and some other transformations to give compounds 3 and 8 , respectively, were studied. It was shown that both heterocyclic systems are stable under the conditions employed in these transformations.     

Multidimensional Markovian FBSDEs with super-quadratic growth

We give local and global existence and uniqueness results for multidimensional coupled FBSDEs for generators with arbitrary growth in the control variable. The local existence result is based on Malliavin calculus arguments for Markovian equations. Under additional monotonicity conditions on the generator we construct global solutions by a pasting technique along PDE solutions.