A CUBIC SYSTEM WITH 8 SMALL-AMPLITUDE LIMIT-CYCLES

In E.M. James and N.G. Lloyd's paper A Cubic System with Eight Small-A mplitude Limit Cycles [1], a set of conditions is given that ensures t he origin to be a fine focus of order eight and eight limit cycles to bifurcate from the origin by perturbing parameters. We find that one o f the conditions, a9 = sigmaa7, where 666/97 < sigma* < 103/15, can b e weakened as a9 = sigma.a7 or a9 = sigma1a7, where 283/125 < sigma1 < 284/125. In [1], deriving above conditions is reduced to finding the real solutions of a system of some algebraic equations and inequalitie s. When verifying these conditions by solving this system in a differe nt ordering, we find another real solution to the system, which is lea ding to above improvement of the conditions.