On lopsided systems

Abstract

In this work, we study the Liapunov quantities, the problem of the center, and the local limit cycles of the lopsided systems ẋ = y, ẏ = −x + pk(x, y), when k = 5, 7 and when k is odd. In general, the Liapunov quantities are derived from the focal values η2k+2, but when k = 5, we show that they are derived from the focal values η4k+2. Moreover, when k = 5, the origin is a center if and only if the system is time-reversible and if it is not, no more tan five local limit cycles can bifurcate out of the origin. When k = 7, we show that the origin is a center if and only if the system is time-reversible and if it is not, no more than seven local limit cycles can bifurcate out of the origin under certain conditions. In general, when k is odd, we conjecture that the origin is a center if and only if the system is time-reversible.