  Limit cycles of planar polynomial vector fields have been an active area of
research for decades; the interest in periodic-orbit related dynamics comes
from Hilbert's 16th problem and the fact that oscillatory states are often
found in applications. We study the existence of limit cycles and their
coexistence with invariant algebraic curves in two families of Kukles systems,
via Lyapunov quantities and Melnikov functions of first and second order. We
show center conditions, as well as a connection between small- and
large-amplitude limit cycles arising in one of the families, in which the first
coefficients of the Melnikov function correspond to the first Lyapunov
quantities. We also provide an example of a planar polynomial system in which
the cyclicity is not fully controlled by the first nonzero Melnikov function.
