A Limit Cycle Of High-dimensional System Algorithmically Constructed

Recently, with the development of computer algebra systems and the discovery (or rediscovery) of algorithmic approaches to many of the basic computation, the techniques of polynomial manipulation have found significant applications. The Wu's well ordering principle and Wu's zero decomposed theorem are the basic theories commonly used and with a great deal of applications. A given set of polynomials can be decomposed to a series of triangle polynomials with the same zeros. In this paper, based on Wu's method and the estimation for maximal and minimal polynomials, we extend the real root isolation algorithm from univariant integral polynomial to multivariant integral polynomial systems. It gives the solutions of polynomial systems in the form of multi-dimensional boxes. The advantage of this algorithm is that the obtained solution can be used in future needed for its rational form.Based on the qualitative theoy of differential polynomial system, algorithm to calculate the focal values, the construction of small amplitude limit cycles, Hirsch's monotone theory and the center manifold theorem etc, we apply mrealroot algorithm to many problems, such as to obtain the real solutions of polynomial systems, to confirm the number of limit cycles in differential system and to construct the limit cycles. Based on this algorithm, we have checked the results of the construction of limit cycles in recent literature. We also consider the Lotka-Volterra system. We construct the examples of the classes 26, 28, 29 (under Zeeman's classification) in three dimensional Lotka-Volterra competitive system with two limit cycles and the class 27 with three limit cycles. In this way, we solve the open problem of Hofbauer and So partially, and give an negative answer to their conjecture about the maximal number of the limit cycles in such systems. At the same time, we construct the examples of a non-competitive and a prey-predator three dimensional Lotka-Volterra system with two limit cycles. And for the reason of the equivalence of the replicator dynamic system and Lotka-Volterra system, we give an example of such a system with three limit cycles.