Information Technology Reference
In-Depth Information
TR
2
10
TR
1
HB
4
1
0.1
0
0.2
0.4
0.6
0.8
1
1.2
Q
Fig. 3.5. Bifurcation diagram versus coupling Q, focusing on the stable anti-phase oscillations
(thick yellow line). Parameters are those of Fig. 3.3.
corresponds to anti-phase oscillations. As shown on Fig. 3.5, this state belongs
to a branch of periodic orbits originating at the Hopf bifurcation HB
4
. Fig. 3.5
illustrates in detail the bifurcation structure of the antiphase dynamics when Q is
being varied. Stable anti-phase oscillations are observed between HB
4
(Q = 1:253)
and TR
1
(torus bifurcation for Q = 1:137), and from Q = 0 until TR
2
(Q = 0:5848).
As demonstrated, this solution loses its stability for 0:5848 < Q < 1:137. Direct
numerical simulations revealed the existence of complex behavior in the latter range
of Q values, which we discuss briey in Sec. 3.2.2.3.
In contrast to the case of positively coupled repressilators [Garc a-Ojalvo et al.
(2004)], where coupling was seen to provide coherence enhancement, investigations
of the dynamical structure of the system with phase-repulsive coupling by means
of direct calculations [Ullner et al. (2007)] did not reveal the presence of a stable
in-phase regime (synchronous oscillations over the entire cell population). The
present bifurcation analysis conrms this result: a branch of synchronous periodic
oscillations is in fact seen to emanate from HB
3
, but it is unstable (data not shown,
see Ullner et al. (2008)). The bifurcation analysis conrmed that the in-phase
regime is unstable for all values of and Q studied, in contrast to the anti-phase
limit cycle oscillations, which arise even for small values. The existence of this
anti-phase (or phase-shifted) solution is a clear manifestation of the phase repulsive
character of the AI-mediated coupling, which enhances the phase dierence between
the oscillators in the model, until the maximal phase dierence of
2
is reached.
3.2.2.2. Comparison between bifurcation analysis and direct calculations
Bifurcation analyses reveal all solutions, their stability, and the connecting bifurca-
tion points. Special interest evokes the ranges of multi-stability, i.e. the coexistence
of dynamical regimes, because it oers opportunities of the biological system to
