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Processing

It is know that the periodic normal form at the Limit Point of Cycles (LPC) bifurcation is

ì
ï
ï
í
ï
ï
î
dt

dt

=
1 - x+ a x2 + ¼,
dx

dt

=
b x2 + ¼,
(54)

where t Î [0,T], x is a real coordinate on the center manifold that is transverse to the limitcycle, a,b Î \mathbbR, and dots denote nonautonomous T-periodic O(x3)-terms. For each detected LPC point the normal form coefficient b is computed. A Cusp Point of Cycles (CPC) is detected for b=0.

The periodic normal form at the Period Doubling (PD) bifurcation is

ì
ï
ï
í
ï
ï
î
dt

dt

=
1 + a x2 + ¼,
dx

dt

=
c x3 + ¼,
(55)

where t Î [0,2T], x is a real coordinate on the center manifold that is transverse to the limit cycle, a,c Î \mathbbR, and dots denote nonautonomous 2T-periodic O(x4)-terms. The coefficient c determines the stability of the period doubled cycle in the center manifold and is computed during the processing of each PD point.

The periodic normal form at the Neimark-Sacker (NS) bifurcation is

ì
ï
ï
í
ï
ï
î
dt

dt

=
1 + a |x|2 + ¼,
dx

dt

=
iq

T

x+ d x|x|2 + ¼,
(56)

where t Î [0,T], x is a complex coordinate on the center manifold that is complementary to t, a Î \mathbbR,d Î \mathbbC, and dots denote nonautonomous T-periodic O(|x|4)-terms. The critical coefficient d in the periodic normal form for the NS bifurcation is computed during the processing of a NS point. The critical cycle is stable within the center manifold if Re d < 0 and is unstable if Re d > 0. In the former case, the Neimark-Sacker bifurcation is supercritical, while in the latter case it is subcritical.