Browsing by Author "Schneider, Klaus R."
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- ItemAsymptotics and stability of a periodic solution to a singularly perturbed parabolic problem in case of a double root of the degenerate equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Butuzov, Valentin F.; Nefedov, Nikolai N.; Recke, Lutz; Schneider, Klaus R.For a singularly perturbed parabolic problem with Dirichlet conditions we prove the existence of a solution periodic in time and with boundary layers at both ends of the space interval in the case that the degenerate equation has a double root. We construct the corresponding asymptotic expansion in the small parameter. It turns out that the algorithm of the construction of the boundary layer functions and the behavior of the solution in the boundary layers essentially differ from that ones in case of a simple root. We also investigate the stability of this solution and the corresponding region of attraction.
- 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.
- ItemExistence and asymptotic stability of a periodic solution with boundary layers of reaction-diffusion equations with singularly perturbed Neumann boundary conditions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Butuzov, Valentin F.; Nefedov, Nikolai N.; Recke, Lutz; Schneider, Klaus R.We consider singularly perturbed reaction-diffusion equations with singularly perturbed Neumann boundary conditions. We establish the existence of a time-periodic solution u(x; t; epsilon) with boundary layers and derive conditions for their asymptotic stability The boundary layer part of u(x; t; ") is of order one, which distinguishes our case from the case of regularly perturbed Neumann boundary conditions, where the boundary layer is of order epsilon.
- ItemExponential asymptotic stability via Krein-Rutman theorem for singularly perturbed parabolic periodic Dirichlet problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Nefedov, Nikolai N.; Recke, Lutz; Schneider, Klaus R.We consider singularly perturbed semilinear parabolic periodic problems and assume the existence of a family of solutions. We present an approach to establish the exponential asymptotic stability of these solutions by means of a special class of lower and upper solutions. The proof is based on a corollary of the Krein-Rutman theorem.
- 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
- 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 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.
- ItemInvariant manifolds for random dynamical systems with slow and fast variables(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Schmalfuss, Björn; Schneider, Klaus R.We consider random dynamical systems with slow and fast variables driven by two independent metric dynamical systems modelling stochastic noise. We establish the existence of a random inertial manifold eliminating the fast variables. If the scaling parameter tends to zero, the inertial manifold tends to another manifold which is called the slow manifold. We achieve our results by means of a fixed point technique based on a random graph transform. To apply this technique we need an asymptotic gap condition.
- 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.
- 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
- ItemNew approach to study the van der Pol equation for large damping(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Schneider, Klaus R.We present a new approach to establish the existence of a unique limit cycle for the van der Pol equation in case of large damping. It is connected with the bifurcation of a stable hyperbolic limit cycle from a closed curve composed of two heteroclinic orbits and of two segments of a straight line forming continua of equilibria. The proof is based on a linear time scaling (instead of the nonlinear Liénard transformation in previous approaches), on a Dulac-Cherkas function and the property of rotating vector fields.
- ItemOn a singularly perturbed initial value problem in case of a double root of the degenerate equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Butuzov, Valentin F.; Nefedov, Nikolai N.; Recke, Lutz; Schneider, Klaus R.We study the initial value problem of a singularly perturbed first order ordinary differential equation in case that the degenerate equation has a double root. We construct the formal asymptotic expansion of the solution such that the boundary layer functions decay exponentially. This requires a modification of the standard procedure. The asymptotic solution will be used to construct lower and upper solutions guaranteeing the existence of a unique solution and justifying its asymptotic expansion.
- ItemOn existence and asymptotic stability of periodic solutions with an interior layer of reaction-advection-diffusion equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Nefedov, Nikolai N.; Recke, Lutz; Schneider, Klaus R.We consider a singularly perturbed parabolic periodic boundary value problem for a reaction-advection-diffusion equation. We construct the interior layer type formal asymptotics and propose a modified procedure to get asymptotic lower and upper solutions. By using sufficiently precise lower and upper solutions, we prove the existence of a periodic solution with an interior layer and estimate the accuracy of its asymptotics. Moreover, we are able to establish the asymptotic stability of this solution with interior layer.
- ItemOn the approximation of the limit cycles function(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Cherkas, Leonid; Grin, Alexander; Schneider, Klaus R.We consider planar vector fields depending on a real parameter. It is assumed that this vector field has a family of limit cycles which can be described by means of the limit cycles function $l$. We prove a relationship between the multiplicity of a limit cycle of this family and the order of a zero of the limit cycles function. Moreover, we present a procedure to approximate $l(x)$, which is based on the Newton scheme applied to the Poincaré function and represents a continuation method. Finally, we demonstrate the effectiveness of the proposed procedure by means of a Liénard system. The obtained result supports a conjecture by Lins, de Melo and Pugh.
- 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.
- 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.
- 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.
- 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.
- 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.