Computing centre conditions for certain cubic systems

We present necessary and sufficient conditions for a critical point of certain two-dimensional cubic differential systems to be a centre. Extensive use of the computer algebra system REDUCE is involved. The search for necessary and sufficient conditions for a centre has long been of considerable interest in the theory of nonlinear differential equations. It has proved to be a difficult problem, and full conditions are known for very few classes of systems. Such conditions are also required in the investigation of Hilbert's sixteenth problem concerning the number of limit cycles of polynomial systems.     

Calculating the coefficients of a discrete elliptic equation from spectral data

A numerical analytical algorithm is developed for calculating the coefficients of a discrete two-dimensional elliptic equation from given eigenvalues and certain symmetry conditions imposed on the basis eigenfunctions.