Dangerous stability boundary

For the remainder of this section we consider only safe and dangerous stability boundaries of codimension one. This allows us to use only one bifurcation parameter. We therefore assume that at 5 = 0, the system... 

The notion of safe and dangerous boimdaries of stability was suggested by Bautin [24] who studied the stability boundaries for equilibrium states. 

Definition 14.3. A point cq on the stability boundary of an equilibrium state Oe is said to be dangerous if Oeo unstable in the sense of Lyapunov. 

For the case of the safe/dangerous points on a stability boundary where periodic trajectories do not exist (the case of global bifurcations), the situation becomes less definite and caimot yet be well specified in general. However, it is well understood in the main cases (see below). 

Summary The set of principal stability boundaries of equilibrium states consists of surfaces of three kinds Si, Sr and Ss. Only the Si-like boundaries are safe. As for periodic orbits, there are nine types of principal stability boundaries among them Se, Sg, Sio, Sn are dangerous, while S2, S3, S4 S5 and Si, S2 2ire safe (the latter two correspond to the subcritical Andronov-Hopf and flip bifurcations, respectively). 

On the stability boundary, the inequality cr > b- -1 is fulfilled. Upon substituting (j = a + 6 +1, the expression for B becomes a polynomial of tr and b with positive coefficients. Hence, if g >0 and 6>0, then Li > 0. Thus, both equilibria 0x 2 are imstable (saddle-foci) on the stability boundary. The boimdary itself is dangerous in the sense of the definition suggested in Chap. 14. Therefore, the corresponding Andronov-Hopf bifurcation of Oi 2 is sub-critical. ... 

We should notice here that in the case of safe boundaries, a slow drift of the parameters back into the stability region brings a system back into the original response, whereas in the dangerous case this is generally impossible. 

Obviously, safe and dangerous boimdaries are distinguished mainly by the stability or instability of the corresponding equilibrium state, or periodic trajectory, on the boundary. 

Bautin, N. N. and Shilnikov, L. P. [1980] Suplement I Safe and dangerous boundaries of stability regions, The Hopf Bifurcation and Its Applications Russian translation of the book by Marsden, J. E. and McCracken, M. (Mir Moscow). 

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