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- ItemThe point charge oscillator: Qualitative and analytical investigations(Vilnius : Vilnius Gediminas Technical University, 2019) Schneider, Klaus R.We study the mathematical model of the point charge oscillator which has been derived by A. Belendez et al. [2]. First we determine the global phase portrait of this model in the Poincare disk. It consists of a family of closed orbits surrounding the unique finite equilibrium point and of a continuum of homoclinic orbits to the unique equilibrium point at infinity. Next we derive analytic expressions for the relationship between period (frequency) and amplitude. Further, we prove that the period increases monotone with the amplitude and derive an expression for its growth rate as the amplitude tends to infinity. Finally, we determine a relation between period and amplitude by means of the complete elliptic integral of the first kind K(k) and of the Jacobi elliptic function cn.
- 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
- ItemThe point charge oscillator: Qualitative and analytical investigations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Schneider, Klaus R.We determine the global phase portrait of a mathematical model describing the point charge oscillator. It shows that the family of closed orbits describing the point charge oscillations has two envelopes: an equilibrium point and a homoclinic orbit to an equilibrium point at infinity. We derive an expression for the growth rate of the primitive period Ta of the oscillation with the amplitude a as a tends to infinity. Finally, we determine an exact relation between period and amplitude by means of the Jacobi elliptic function cn.
- ItemLower and upper bounds for the number of limit cycles on a cylinder(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Schneider, Klaus R.; Grin, AlexanderWe consider autonomous systems with cylindrical phase space. Lower and upper bounds for the number of limit cycles surrounding the cylinder can be obtained by means of an appropriate Dulac-Cherkas function. We present different possibilities to improve these bounds including the case that the exact number of limit cycles can be determined. These approaches are based on the use of several Dulac-Cherkas functions or on applying some factorized Dulac function.
- 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 $Psi(x,y,mu)=0.$
- 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