TWO GENERAL CENTRE PRODUCING SYSTEMS FOR THE POINCAR&#201; PROBLEM

G. R. Nicklason

doi:10.11948/2015026

2015 Volume 5 Issue 3

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G. R. Nicklason. TWO GENERAL CENTRE PRODUCING SYSTEMS FOR THE POINCARÉ PROBLEM[J]. Journal of Applied Analysis & Computation, 2015, 5(3): 284-300. doi: 10.11948/2015026

Citation:

G. R. Nicklason. TWO GENERAL CENTRE PRODUCING SYSTEMS FOR THE POINCARÉ PROBLEM[J]. Journal of Applied Analysis & Computation, 2015, 5(3): 284-300. doi: 10.11948/2015026

TWO GENERAL CENTRE PRODUCING SYSTEMS FOR THE POINCARÉ PROBLEM

G. R. Nicklason

Department of Mathematics, Physics and Geology, Cape Breton University, Sydney B1P 6L2, Nova Scotia, Canada

Abstract

Abstract

We consider the polynomial system dx/dt=-y -ax^s+3y^n-s-3 -bx^s+1y^n-s-1, dy/dt=x+ cx^s+2y^n-s-2 + dx^sy^n-s where n ≥ 3 is an odd integer and s=0,…, n -3 is an even integer. We calculate the first three nonzero Lyapunov coefficients for the system and obtain a Gröbner basis for the ideal generated by them. Potential centre conditions for the system are obtained by setting the basis elements equal to zero and solving the resulting system. This gives five basic solutions and within this set we find two well known classes of centres and three new centre producing systems. One of the three is a variant of one of the other new systems, so we obtain two general independent systems which produce multiple centre conditions for each n ≥ 5.
- Centre-focus problem /
- Lyapunov coefficients /
- Gröbner basis /
- hypergeometric function
MSC: 34A05;34C25

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