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10
LP 5
HB 3
HB 2
HB 4
LP 3
HB 1
LP 2
LP 4
BP 2
LP 5
1
HB 2
BP 1
LP 5
HB 1
LP 1
LP 2
0.1
0
0.2
0.4
0.6
0.8
1
1.2
Q
Fig. 3.4. Bifurcation diagram obtained by variation of Q, illustrating the stable steady state
regimes (HSS and IHSS) and the inhomogenous limit cycle (IHLC). For parameters values see
Fig. 3.3. Here, thin solid lines denote the HSS, thick blue solid lines the IHSS, thick solid orange
line the stable IHLC, and dashed lines denote the unstable steady states especially the dashed
orange line the unstable IHLC. The same bifurcation diagram is valid for the second repressilator.
the set of parameters used here, and is stable until LP 2 at Q = 0:5548. The IHSS
solution coexists in the Q parameter space with the HSS (Fig. 3.4). For example,
for Q = 0:37 there is a coexistence of 9 steady state solutions, 3 of them stable and
6 unstable.
The next step of the bifurcation analysis is to study the limit cycles that arise
from the Hopf bifurcations found on the basic continuation curve. In particular,
the Hopf bifurcation HB 1 gives rise to a branch of stable inhomogeneous periodic
solutions, known in the literature as inhomogeneous limit cycle (IHLC) [Tyson and
Kauman (1975)]. The manifestation of this regime is however dierent in dierent
systems: for two identical diusively coupled Brusselators, e.g., it is dened to be
a periodic solution of the system of oscillators rotating around two spatially non
uniform centers [Tyson and Kauman (1975); Volkov and Romanov (1995)]. For the
model investigated here, the manifestation of the IHLC is somewhat dierent: the
IHLC is characterized by a complex behavior, where one of the oscillators produces
very small oscillations of the protein level, whereas the other one oscillates in the
vicinity of the steady state with an amplitude 4 times smaller than that of an
isolated oscillator [see Fig. 3.3(b)]. The IHLC is stable for values of Q between
HB 1 and LP 5 (Fig. 3.4). In the case of the two-oscillator system considered here,
each oscillator has the same probability to occupy and stay in the upper or lower
state, due to the symmetry of the system. The initial conditions are the only factor
determining the separation of the oscillators.
For coupling values smaller than a given critical value Q crit < 0:129, the system
is characterized by a self-oscillatory solution. For two coupled oscillators, this regime
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