Isochronicity conditions for some planar polynomial systems

We study the isochronicity of centers at O∈R² for systems , , where A,B∈R[x,y], which can be reduced to the Liénard type equation. Using the so-called C-algorithm we have found 27 new multiparameter isochronous centers.     

High-order non-reflecting boundary conditions for dispersive waves in polar coordinates using spectral elements

High-order non-reflecting boundary conditions are introduced to create a finite computational space and for the solution of dispersive waves using a spectral element formulation with high-order time integration. Numerical examples are used to demonstrate the synergy of using high-order spatial, time, and boundary discretization. We show that by balancing all numerical errors involved, high-order accuracy can be achieved for unbounded domain problems in polar coordinate systems.     

Isochronicity conditions for some planar polynomial systems II

We study the isochronicity of centers at O∈R² for systems where A,B∈R[x,y], which can be reduced to the Liénard type equation. When deg(A)?4 and deg(B)?4, using the so-called C-algorithm we found 36 new multiparameter families of isochronous centers. For a large class of isochronous centers we provide an explicit general formula for linearization. This paper is a direct continuation of a previous one with the same title [Islam Boussaada, A. Raouf Chouikha, Jean-Marie Strelcyn, Isochronicity conditions for some planar polynomial systems, Bull. Sci. Math. 135 (1) (2011) 89–112], but it can be read independently.     

B-spline finite-element method in polar coordinates

A numerical solution method for two-dimensional electromagnetic field problems is presented using the B-spline finite-element expression based on polar coordinates. The technique has two main advantages: (1) to avoid the truncation errors at some curved boundaries and (2) to improve the accuracy of singular boundary-value problems with a sharp corner. The B-spline finite-element formulation in polar coordinates is derived and its numerical applications are illustrated by an eddy current problem and several waveguide eigenvalue problems.     

Numerical solution of a nonlinear wave equation in polar coordinates

We study a finite element scheme and a difference scheme for the radial solutions of a nonlinear Klein-Gordon equation.     

Isochronicity for a class of reversible systems

This paper deals with the relation between isochronicity and first integral for a class of reversible systems: , , which associates to the first integral of the form H(x,y)=F(x)y²+G(x). Two necessary and sufficient conditions are given to characterize isochronicity for these systems. Moreover, we apply these results to show that there exists a class of polynomial reversible systems of degree n with isochronous center for any n.     

Equivalent conditions for hyperbolic coordinates

The existence of hyperbolic coordinates is equivalent to the pseudo orbits tracing property and expansiveness. Further equivalent conditions are also considered.     

Generalized polar coordinates on Lie groups and numerical integrators

Motivated by developments in numerical Lie group integrators, we introduce a family of local coordinates on Lie groups denoted generalized polar coordinates. Fast algorithms are derived for the computation of the coordinate maps, their tangent maps and the inverse tangent maps. In particular we discuss algorithms for all the classical matrix Lie groups and optimal complexity integrators for n-spheres.     

Strong isochronism of a center and a focus for systems with homogeneous nonlinearities

We suggest a new approach to studying the isochronism of the system

${{dx} \mathord{\left/ {\vphantom {{dx} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = - y + p_n (x,y),{{dy} \mathord{\left/ {\vphantom {{dy} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = x + q_n (x,y),$

where p _n and q _n are homogeneous polynomials of degree n. This approach is based on the normal form

${{dX} \mathord{\left/ {\vphantom {{dX} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = - Y + XS(X,Y),{{dY} \mathord{\left/ {\vphantom {{dY} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = X + YS(X,Y)$

and its analog in polar coordinates. We prove a theorem on sufficient conditions for the strong isochronism of a center and a focus for the reduced system and obtain examples of centers with strong isochronism of degrees n = 4, 5. The present paper is the first to give examples of foci with strong isochronism for the system in question.

     

Isochronicity of centers in a switching Bautin system

In this paper isochronicity of centers is discussed for a class of discontinuous differential system, simply called switching system. We give some sufficient conditions for the system to have a regular isochronous center at the origin and, on the other hand, construct a switching system with an irregular isochronous center at the origin. We give a computation method for periods of periodic orbits near the center and use the method to discuss a switching Bautin system for center conditions and isochronous center conditions. We further find all of those systems which have an irregular isochronous center.     

On boundary conditions for polar materials

Summary We discuss some restrictions on boundary conditions in continuum mechanics arising from the requirement that the basic balance laws, invariance conditions and entropy inequality must hold for the material outside of the bounding surface. We illustrate our assertions for polar materials, but the ideas we use are applicable to all continuum theories.

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Zusammenfassung Es werden einige Einschränkungen für die Randbedingungen in der Kontinuumsmechanik behandelt, die von der Forderung herrühren, dass für die Materie ausserhalb der Grenzfläche die fundamentalen Bilanzgleichungen, die Invarianzbedingungen und die Entropieungleichung gelten sollen. Diese Behauptungen werden für polare Materialien erläutert, jedoch sind die hier eingeführten Begriffe auf Kontinuumstheorien aller Art anwendbar.

     

Some conditions for a center of autonomous and nonautonomous systems

We give some conditions which imply that the origin is a center for polynomial two- dimensional systems. A related cubic nonautonomous equation is also considered.     

The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems

In this work we study the centers of planar analytic vector fields which are limit of linear type centers. It is proved that all the nilpotent centers are limit of linear type centers and consequently the Poincaré-Liapunov method to find linear type centers can be also used to find the nilpotent centers. Moreover, we show that the degenerate centers which are limit of linear type centers are also detectable with the Poincaré-Liapunov method.     

The IIM in polar coordinates and its application to electro capacitance tomography problems

In a numerical study of the electric field inside an Electro Capacitance Tomography (ECT) device and other applications, often a Poisson equation with a discontinuous coefficient needs to be solved in polar coordinates. This paper is devoted to the Immersed Interface Method (IIM) in polar coordinates and the application to the solution of the electric potential inside an ECT device. The numerical algorithm is based on a finite difference discretization on a uniform polar coordinates grid. The finite difference scheme is modified at grid points near and on the interface across which the coefficient is discontinuous so that the natural jump conditions are satisfied. The algorithm and analysis here is one step forward in applying the IIM for 3D problems in axisymmetric situations or in the spherical coordinates. Numerical examples against exact solutions and the application to ECT problems are also presented.     

A note on Green element method discretization for Poisson’s equation in polar coordinates

In the work described herein, the Green element method (GEM) is applied to arrive at new sets of element matrices for the solution of Poisson’s equation in polar coordinates. A unique flexibility possessed by GEM allows integration within a typical element. This special GEM characteristic, in addition to providing exact values of element coefficient matrices, facilitates the handling of nodal singularities and variable coefficients that accompany such problems. Numerical solutions provided by GEM confirm reliability.     

Bifurcation and Isochronicity at Infinity in a Class of Cubic Polynomial Vector Fields

In this paper, we study the appearance of limit cycles from the equator and isochronicity of infinity in polynomial vector fields with no singular points at infinity. We give a recursive formula to compute the singular point quantities of a class of cubic polynomial systems, which is used to calculate the first seven singular point quantities. Further, we prove that such a cubic vector field can have maximal seven limit cycles in the neighborhood of infinity. We actually and construct a system that has seven limit cycles. The positions of these limit cycles can be given exactly without constructing the Poincare cycle fields. The technique employed in this work is essentially different from the previously widely used ones. Finally, the isochronous center conditions at infinity are given.     

一类五次多项式系统无穷远点等时中心条件与极限环分支

研究一类五次系统无穷远点的中心、拟等时中心条件与极限环分支问题.首先通过同胚变换将系统无穷远点转化成原点,然后求出该原点的前8个奇点量,从而导出无穷远点成为中心和最高阶细焦点的条件,在此基础上给出了五次多项式系统在无穷远点分支出8个极限环的实例.同时通过一种最新算法求出无穷远点为中心时的周期常数,得到了拟等时中心的必要条件,并利用一些有效途径一一证明了条件的充分性.     

Sufficient conditions for existence and uniqueness of a Chebyshev center of a nonempty bounded set in a geodesic space

We obtain sufficient conditions for existence and uniqueness of a Chebyshev center of a nonempty bounded set in a geodesic space. We also establish conditions under which the unique Chebyshev center of a nonempty bounded set in a geodesic space belongs to the closure of the convex hull of this set.     

An O(K2 + h4) finite difference method for one-space Burger's equation in polar coordinates

A two-level implicit difference method of O(k² + h⁴) for a class of singular initial boundary value problem, where α, β, γ, and ν are constants, is discussed using three spatial grid points. The method is shown to be unconditionally stable when applied to linearized equations. The fourth-order convergence for a fixed mesh ratio parameter is illustrated with the help of two examples. © 1996 John Wiley & Sons, Inc.