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- ItemOn the construction of a class of generalized Kukles systems having at most one limit cycle(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Schneider, Klaus R.; Grin, AlexanderConsider the class of planar systems fracdxdt=y,quadfracdydt=−x+musumj=03hj(x,mu)yj depending on the real parameter $mu$. We are concerned with the inverse problem: How to construct the functions $h_j$ such that the system has not more than a given number of limit cycles for $mu$ belonging to some (global) interval. Our approach to treat this problem is based on the construction of suitable Dulac-Cherkas functions $Psi(x,y,mu)$ and exploiting the fact that in a simply connected region the number of limit cycles is not greater than the number of ovals contained in the set defined by $Psi(x,y,mu)=0.$
- ItemOn the construction of bifurcation curves related to limit cycles of multiplicity three for planar vector fields(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Cherkas, Leonid; Grin, Alexander; Schneider, Klaus R.For plane vector fields depending on three parameters we describe an algorithm to construct a curve in the parameter space such that to each point of this curve there belongs a vector field possessing a limit cycle of multiplicity three. One point of this curve is related to the bifurcation of a limit cycle of multiplicity three from an equilibrium point. The underlying procedure is a continuation method.
- ItemA new approach to study limit cycles on a cylinder(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Cherkas, Leonid; Grin, Alexander; Schneider, Klaus R.We present a new approach to study limit cycles of planar systems of autonomous differential equations with a cylindrical phase space $Z$. It is based on an extension of the Dulac function which we call Dulac-Cherkas function $Psi$. The level set $W:=vf,y) in Z: Psi(vf,y)=0$ plays a key role in this approach, its topological structure influences existence, location and number of limit cycles. We present two procedures to construct Dulac-Cherkas functions. For the general case we describe a numerical approach based on the reduction to a linear programming problem and which is implemented by means of the computer algebra system Mathematica. For the class of generalized Liénard systems we present an analytical approach associated with solving linear differential equations and algebraic equations
- ItemOn the construction of Dulac-Cherkas functions for generalized Liénard systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Cherkas, Leonid; Grin, Alexander; Schneider, Klaus R.Dulac-Cherkas functions can be used to derive an upper bound for the number of limit cycles of planar autonomous differential systems, at the same time they provide information about their stability. In this paper we present a method to construct such functions for generalized Liénard systems by means of linear differential equations. If the degree m of the polynomial is not greater than 3, then the described algorithm works generically. By means of an example we show that this approach can be applied also to polynomials with degree m larger than 3.
- ItemStudy of the bifurcation of a multiple limit cycle of the second kind by means of a Dulac-Cherkas function: A case study(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Schneider, Klaus R.; Grin, AlexanderWe consider a generalized pendulum equation depending on the scalar parameter having for = 0 a limit cycle Gamma of the second kind and of multiplicity three. We study the bifurcation behavior of Gamma for -1 ≤ ≤ (√5 + 3)/2 by means of a Dulac-Cherkas function.
- ItemConstruction of generalized pendulum equations with prescribed maximum number of limit cycles of the second kind(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Schneider, Klaus R.; Grin, AlexanderConsider a class of planar autonomous differential systems with cylindric phase space which represent generalized pendulum equations. We describe a method to construct such systems with prescribed maximum number of limit cycles which are not contractible to a point (limit cycles of the second kind). The underlying idea consists in employing Dulac-Cherkas functions. We also show how this approach can be used to control the bifurcation of multiple limit cycles.
- 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
- ItemAndronov-Hopf bifurcation of higher codimensions in a Liènard system(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Schneider, Klaus; Grin, AlexanderConsider a polynominal Liènard system depending on three parameters itshape a, b, c and with the following properties: (i) The origin is the unique equilibrium for all parameters. (ii) Ifitshape a crosses zero, then the origin changes its stability, and a limit cycle bifurcates from the euqilibrium. We inverstigate analytically this bifurcation in dependence on the parameters itshape b and itshape c and establish the existence of families of limit cycles of multiplicity one, two and three bifurcating from the origin.