Is there any
period in the light curve? In this section, we try to answer this
question using the powerful Jurkevich 
method (Jurkevich 1971).
The Jurkevich method is based on the expected mean square deviation. It
tests a run of trial periods around which the data are folded. All
data are assigned to m groups according to their phases around each
trial period. The variance 
of each group and the sum
of all groups are computed. For a trial period equal to the
true one, if any, 
reaches its minimum, and a "good'' period
will give a much reduced variance relative to those given by other
false trial periods and with almost constant values. No firm rule
exists for assessing the significance of a minimum in the 
plot.
As in Kidger et al. (1992) and Liu et al. (1995), we consider the
parameter
f,
where 
is the normalized value. In the normalized plot, a value
of 
implies that f=0 and hence there is no periodicity at
all. The best periods can be identified from the plot. A value 
generally indicates that a strong periodicity exists in the data,
whilst f < 0.25 usually indicates that the periodicity, if genuine, is a
weak one. A further test is the relationship between the depth of the
minimum and the noise in the "flat'' section of the 
curve
close to the adopted period. If the absolute value of the relative
change of the minimum to the "flat'' section is larger than ten times
the standard error of this "flat'' section, the periodicity in the
data can be considered as significant and the minimum as highly
reliable. In the Jurkevich test the parameter m can be modified: more
groups give higher sensitivity, but fewer data points per group introduce
a larger noise in the plot. So we analyze the data sample mainly
using m=10, which gives us over 50 points per group. To search for
short time scale periods, we choose a small interval between two
successive trial periods.
The result of the analysis with m=10 is shown in Fig. 2 (click here). A minimum
of 
(f=0.532) is significant at a trial period of
years. A similar analysis with m=20 shows that 
(f=0.627) at the period of 23.5-year. In addition to the
period of 
years, the broad minimum at P = 15.5 years is
also significant with 
and f=0.427 but not as certain as
the one obtained at P = 23.1 years. We have considered the half width at
half minimum as the "formal'' error (cf. Jurkevich 1971) to derive all
effects on the precision,
including random variations in the exact interval between outbursts,
poor coverage of some of the early outbursts and the larger error in some of
the early photographic photometry, the uncertainty of observed data
estimated from
the figures in the literature, random variations in intensity, and the
changing width of the outburst structure.
The errors caused by the conversion from photographic to photo-electric
values and by the measurement with different diaphragms are considered in
this analysis as
random variations in the intensity. They would reduce the depth of
minima and therefore the significance of the periodicity found.
These errors also increase the "formal'' error, and this effect has
been taken into account. The fluctuations
seen around the minimum may also be caused by flickering, which is definitely
non-periodic. The broad width of the minima may also result
from the broad structures of the bursts, the drift of the real period and the
effect of adjacent periods, if present.
Figure 2: Results of the normalized Jurkevich test for the
period search, in Mkn 421. The deepest minimum corresponds to a period
of 23.1 years. The minima corresponding to periods of 3.4 years, 6.7 years, and
15.5 years are also conspicuous
Figure 3: Results of the normalized Jurkevich test for the
period search, in Mkn 421, excluding Miller's data, with m=5. The minima
corresponding to periods of 1.1, 6.0, and 13.7 years are significant
To compensate for the heavy weighting of recent data, we use the 100-day
averaged light curve. This interval is long enough compared to the
possible periods of 15.3 years and 22.8 years and unlikely to prevent a
distribution of
the long term variation findings. The result of the Jurkevich test shows
a larger noise due to fewer points in every group and
flickering effects in the early epochs. However, a minimum of 
near the possible period of 22.8 years is still seen.
Considering the redshift of 0.0308, the period of 23.1 years corresponds to 22.4 years in the rest frame of the source.