  Our concern is the study of degenerate Hopf bifurcation of smooth planar
dynamical systems near isolated singular points. To do so, we propose to split
up the definition of degeneracy into two types. Degeneracy of first kind shall
means that no limit cycle surrounding the steady state can emerge after or
before the critical point, with the possible emergence of limit cycles
surrounding the point at infinity. Degeneracy of second kind shall means that
either several limit cycles or semistable cycles as a limiting case, emerge
surrounding the steady state super or subcritically. In degenerate bifurcation
of second kind we also show that the radius of the emerging cycle tends to zero
with an "anomalous" order as the bifurcation parameter tends to the critical
value. Finally, we give a sufficient condition for degenerate bifurcations of
second kind up to 6-jet-equivalence, and show some "typical" forms for
degenerate bifurcations.
