Noel G. Lloyd

Algorithmic Derivation Of Centre Conditions

The conditions for a critical point of a polynomial differential system to be a centre are of particular significance because of the frequency with which they are required in applications. We demonstrate how computer algebra can be effectively employed in the search for necessary and sufficient conditions for critical points of such systems to be centres. We survey recent developments and illustrate our approach by means of examples.

POLYNOMIAL SYSTEMS: A LOWER BOUND FOR THE HILBERT NUMBERS

Let Hn be the maximum possible number of limit cycles of systems ẋ = P(x, y), ẏ = Q(x, y), where P and Q are polynomials of degree at most n. We are concerned with the rate of growth of Hn as n increases: it is known that Hn ≽ kn2 for some constant k. In this paper we show that Hn grows at least as rapidly as n2 log n.

REDUCE and the bifurcation of limit cycles

A technique is described which has been used extensively to investigate the bifurcation of limit cycles in polynomial differential systems. Its implementation requires a Computer Algebra System, in this case REDUCE. Concentration is on the computational aspects of the work, and a brief resume is given of some of the results which have been obtained.

A cubic Kolmogorov system with six limit cycles

Abstract We consider a class of cubic Kolmogorov systems. We show in particular that a maximum of six small amplitude limit cycles can bifurcate from a critical point in the first quadrant, and we discuss the number of invariant lines.

Symmetry in Planar Dynamical Systems

In the study of dynamical systems the conditions for a critical point to be a centre are often sought. The sufficiency of such conditions is probed using various techniques; here we exploit the possible symmetry of a given system. We describe an application of Grobner bases in the search for a bilinear transformation of the system to one which is symmetric in a line.

Kukles revisited: Advances in computing techniques

We revisit the Kukles system to show how advances in computer hardware and software have improved symbolic calculations. We confirm directly that there are no non-persistent centres for the Kukles system. We also prove an earlier conjecture regarding the isochronous centres for the extended Kukles system. We introduce a technique whereby modular resultants can be used to investigate the properties of resultants that cannot be readily calculated using the existing software.

Computing integrability conditions for a cubic differential system

Abstract The integrability conditions for certain systems of differential equations χ = P(χ, y), y = Q(χ, y) , where P and Q are cubic polynomials, are obtained using two distinct computational approaches. Such conditions are required in the investigation of Hilberts 16 th problem and in the development of algorithms for the automatic solution of differential equations.

Space Saving Calculation of Symbolic Resultants

Abstract.We describe an approach to the computation of symbolic resultants in which factors are removed during the course of the calculation, so reducing the stack size required for intermediate expressions and the storage space needed. We apply the technique to three well-established methods for calculating resultants. We demonstrate the advantages of our approach when the resultants are large and show that some otherwise intractable problems can be resolved. In certain cases a significant reduction in the cpu time required to calculate the resultant is also evident.