O. Ragos

ZEAL: A mathematical software package for computing zeros of analytic functions

We present a reliable and portable software package for computing zeros of analytic functions. The package is named ZEAL (ZEros of AnaLytic functions). Given a rectangular region W in the complex plane and a function f :W!C that is analytic inW and does not have zeros on the boundary ofW , ZEAL localizes and computes all the zeros off that lie inside W , together with their respective multiplicities. ZEAL is based on the theory of formal orthogonal polynomials. It proceeds by evaluating numerically certain integrals along the boundary of W involving the logarithmic derivative f 0 =f and by solving generalized eigenvalue problems. The multiplicities are computed by solving a linear system of equations that has Vandermonde structure. ZEAL is written in Fortran 90.

RFSFNS: A portable package for the numerical determination of the number and the calculation of roots of Bessel functions

A portable software package, named RFSFNS, is presented for the localization and computation of the simple real zeros of the Bessel functions of first and second kind, J,, (z), Y~ (z), respectively, and their derivatives, where v > 0 and z > 0. This package implements the topological degree theory for the localization portion and a modified bisection method for the computation one. It localizes, isolates and computes with certainty all the desired zeros of the above functions in a predetermined interval within any accuracy (subject to relative machine precision). It has been implemented and tested on different machines utilizing the above Bessel functions of various orders and several intervals of the argument.

Asymptotic and periodic motion around collinear equilibria in Chermnykh's problem

In this paper we study the asymptotic motion to the collinear equilibrium points of Chermnykhs problem. More specifically, we give three kinds of non-symmetric doubly-asymptotic solutions emanating from L 1 and L 3 . We also show that these solutions are closely connected to the families of periodic orbits generated around these equilibria.

On the existence of the “out of plane” equilibrium points in the photogravitational restricted three-body problem

The existence of equilibrium points laying out of the plane of the two primaries in the photogravitational restricted three-body problem is discussed. It is verified that such points do exist.

ZEBEC: A mathematical software package for computing simple zeros of Bessel functions of real order and complex argument

A reliable and portable software package, called ZEBEC (ZEros of BEssel functions Complex), is presented, which localizes and computes simple zeros of Bessel functions of the first, the second or the third kind, or their derivatives. The Bessel functions are of real order and complex argument. ZEBEC calculates with certainty the total number of zeros within a given box whose edges are parallel to the coordinate axes. Cauchys Theorem is used for this calculation. Then the program isolates each one of the zeros, utilizing the above-mentioned theorem, and finally computes them to a given desired accuracy using a generalized method of bisection.

Periodic motion around stable collinear equilibrium point in the photogravitational restricted problem of three bodies

In a binary system with both bodies being luminous, the inner collinear equilibrium pointL1 becomes stable for values of the mass ratio and radiation pressure parameters in a certain region. The kind of periodic motions aroundL1 is examined in this case. Second-order parametric expansions are given and the families of periodic orbits generated fromL1 are numerically determined for several sets of values of the parameters. Short- and long-period solutions are identified showing a similarity in the character of periodicity with that aroundL4. It is also found that the finite periodic solutions in the vicinity ofL1 are stable.

The topological degree theory for the localization and computation of complex zeros of bessel functions

We study the complex zeros of Bessel functions of real order of the first and second kind and their first derivatives. The notion of the topological degree is employed for the calculation of the exact number of these zeros within an open and bounded region of the complex plane, as well as for localization of these zeros. First, we prove that the value of the topological degree provides the total number of complex roots within this region. Subsequently, these roots are computed by a generalized bisection method. The method presented here computes complex zeros of Bessel functions, requiring only the algebraic signs of the real and imaginary part of these functions. It has been implemented and tested, and performance results are presented.

New kinds of asymmetric periodic orbits in the restricted three-body problem

The procedure of numerical ‘ascent’ from families of planar to three-dimensional periodic orbits and the subsequent ‘descent’ to the plane is proved efficient in determining new families of planar asymmetric periodic orbits in the restricted three-body problem. Two such families are computed and described for values of the mass parameter for which it has been found that they exist. Two new families of three-dimensional asymmetric periodic orbits are also presented in this paper.

On the application of optimization methods to the determination of members of families of periodic solutions

The techniques used for the numerical computation of families of periodic orbits of dynamical systems rely on predictor-corrector algorithms. These algorithms usually depend on the solution of systems of approximate equations constructed from the periodicity conditions of these orbits. In this contribution we transform the root finding procedure to an optimization one which is applied on an objective function based on the exact periodicity conditions. Thus, the determination of periodic solutions and families of such orbits can be accomplished through unconstrained optimization. In this paper we apply and compare some well-known minimization methods for the solution of this problem. The obtained results are promising.

ON THE COMPUTATION OF ALL THE EQUILIBRIUM POINTS IN HAMILTONIAN SYSTEMS WITH THREE DEGREES OF FREEDOM

In dynamical system theory the determination of the equilibrium points often requires the solution of systems of transcendental equations, whose exact number of solutions cannot be found analytically. In this paper, topological degree theory (especially the Kronecker-Picard integral) is implemented to obtain the exact number of these solutions, within a given region. These results are studied and applied to the accurate computation of the total number of equilibrium points of Hamiltonian systems with three degrees of freedom.