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10
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1000
HSS
100
IHLC
IHSS
10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Q
Fig. 3.6. Comparison between the bifurcation analysis (top) and the direct calculation with ran-
dom initial conditions (bottom). Note the logarithmic scale of both ordinates in the two plots.
The oscillatory regime is represented by a yellow solid line (top) and a yellow area (bottom); the
IHLC by solid orange lines (top) and a orange-white chess board pattern (bottom); the IHSS by
solid blue lines (top) and a small blue striped area (bottom); and nally the HSS is illustrated by
a solid black line (top) and a grey area (bottom). Parameters are those of Fig. 3.3.
adapt or to store information. On the other hand, only stable regimes with a suf-
cient basin of attractions play a role in biological systems, an information that
is not in the scope of the bifurcation analysis. The basins of attraction can be
quantied in direct numerical simulations from the probability of occurrence of the
dierent dynamical regimes for a set of randomly and appropriately drawn initial
conditions. In what follows, we show results for 1000 time series with random initial
conditions. Figure 3.6 shows a histogram of the resulting regimes as the bifurcation
parameter Q is varied (bottom), compared with the bifurcation plot resulting from
the continuation analysis described in the previous Section. Both methods indicate
that for small coupling, Q < 0:129, anti-phase self-oscillations are the only stable
regime. At Q = 0:129 the homogenous steady state stabilizes through a limit point
bifurcation (LP
1
in Fig. 3.4), coexisting with an oscillatory solution. The direct cal-
culations reveal the dominance of the single-xed-point solution, which has a larger
basin of attraction: at Q = 0:2, for instance, only about 70 of the total 1000 ran-
dom initial conditions result in the oscillatory state, while the other remaining 930
result in HSS. For Q2[0:2236; 0:3588], direct calculations show the existence of an
inhomogenous limit cycle (orange white chessboard pattern in Fig. 3.6,bottom) that
coincides with the region where a stable IHLC solution was found by the bifurcation
analysis (solid orange line in Fig. 3.6,top). One can see a very good coincidence of
