The number of small amplitude limit cycles in arbitrary polynomial systems

In this paper, we study the number of small amplitude limit cycles in arbitrary polynomial systems. It is found that almost all the results for the number of small amplitude limit cycles are obtained by calculating Lyapunov constants and determining the order of the corresponding Hopf bifurcation. It is well known that the difficulty in calculating the Lyapunov constants increases with the increasing of the degree of polynomial systems. So, it is necessary and valuable for us to achieve some general results about the number of small amplitude limit cycles in arbitrary polynomial systems with degree mm, which is denoted by M(m)M(m). In this paper, by applying the method developed by C. Christopher and N. Lloyd in 1995, and M. Han and J. Li in 2012, we first obtain the lower bounds for M(6)−M(14)M(6)−M(14), and then prove that M(m)≥m2M(m)≥m2 if m≥23m≥23. Finally, we obtain that M(m)M(m) grows as least as rapidly as 1825⋅12ln2(m+2)2ln(m+2) for all large mm (it is proved by M. Han, J. Li, Lower bounds for the Hilbert number of polynomial systems, J. Differential Equations 252 (2012) 3278–3304 that the number of all limit cycles in arbitrary polynomial systems with degree mm grows as least as rapidly as 12ln2(m+2)2ln(m+2)).