Rough unstable cycle

Such a critical fixed point is called a complex degenerate) saddle. Its stable manifold is y = 0, and the unstable manifold is given by x = 0, as shown in Fig. 10.2.6(b). Here, in the critical case, the trajectories behave qualitatively identical to those nearby the rough unstable cycle shown in Fig. 10.2.7(b). 

The behaviour spectrum of a homogeneous population as a function of parameter (fig. 6.3) reveals a rich variety of dynamic behavioural modes of the cAMP signalling system. Starting from a low initial value of fee (fig- 6.3a), the system evolves toward a stable steady state, represented by the value of the extracellular cAMP concentration, yo-Around k = 2.4 min (fig. 6.3b), a subcritical Hopf bifurcation occurs beyond which the steady state becomes unstable (dashed line) in a range roughly extending from k - 2.2 to 2.4 min in the conditions of fig. 6.3, the system thus admits a coexistence between a stable steady state and a stable limit cycle represented by the upper solid line showing the maximum cAMP level in the course of oscillations, y these two stable solutions are separated by an imstable limit cycle (dashed line). 

To conclude this section, let us elaborate further on the restrictions (D) and (E). In case (D) the surface corresponding to the double cycle is of codimension-one, and therefore, it divides a neighborhood of the non-rough system Xq into two regions and D. Assume that in the double limit cycle is decomposed into two limit cycles, and that it disappears in D. The situation in -D is simple — all systems there are structurally stable and, moreover, of the same type. As for D the situation is less trivial if (D) is violated, then it is obvious that besides structurally stable systems in there are structurally unstable ones whose non-roughness is due to the existence of a heteroclinic trajectory between two saddles, as shown in Fig. 8.1.6(a). Moreover, this picture takes place in any neighborhood of Xq- In other words, in the region, there exists a countable number of the associated bifurcation surfaces of codimension-one which accumulate to In such cases the surface is said to be unattainable from one side. 

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