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- ItemGlobal bifurcation analysis of limit cycles for a generalized van der Pol system(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Schneider, Klaus R.; Grin, AlexanderWe present a new approach for the global bifurcation analysis of limit cycles for a generalized van der Pol system. It is based on the existence of a Dulac-Cherkas function and on applying two topologically equivalent systems: one of them is a rotated vector field, the other one is a singularly perturbed system.
- ItemGlobal bifurcation analysis of a class of planar systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Grin, Alexander; Schneider, Klaus R.We consider planar autonomous systems dx/dt =P(x,y,λ), dy/dt =Q(x,y,λ) depending on a scalar parameter λ. We present conditions on the functions P and Q which imply that there is a parameter value λ0 such that for &lambda > λ0 this system has a unique limit cycle which is hyperbolic and stable. Dulac-Cherkas functions, rotated vector fields and singularly perturbed systems play an important role in the proof.
- ItemGlobal algebraic Poincaré--Bendixson annulus for van der Pol systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Grin, Alexander; Schneider, Klaus R.By means of planar polynomial systems topologically equivalent to the van der Pol system we demonstrate an approach to construct algebraic transversal ovals forming a parameter depending Poincaré-Bendixson annulus which contains a unique limit cycle for the full parameter domain. The inner boundary consists of the zero-level set of a special Dulac-Cherkas function which implies the uniqueness of the limit cycle. For the construction of the outer boundary we present a corresponding procedure