Dangerous bifurcation

Like pitchfork bifurcations, Hopf bifurcations come in both super- and subcritical varieties. The subcritical case is always much more dramatic, and potentially dangerous in engineering applications. After the bifurcation, the trajectories must jump to a distant attractor, which may be a fixed point, another limit cycle, infinity, or—in... 

Axioms 12. Movement along the bifurcation cascade in case of the system s stability loss is characterized by the movement trajectory displaying increasing the danger of a critical situation (a catastrophe) beginning. 

The latter comment is only heuristic. It reflects the influence of small noise, which always persists in real systems (moreover, it is well-known that fluctuations near bifurcational thresholds are amplified). Therefore, a representative point may break loose from an old regime even before the system reaches the dangerous boundary. 

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). 

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... 

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). 

Both cases have much in common in the sense that the imstable set of both bifurcating equilibrium states is one-dimensional. If the unstable set of the critical equilibrimn state is of a higher-dimension, then the subsequent picture may be completely different. Figure 14.3.1 depicts such a situation. When the imstable cycle shrinks into the equilibrium state we have a dilemma the representative point may jump either to the stable node 0 or to the stable node 02- Therefore this dangerous boundary must be classified as dynamically indefinite. 

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. ... 

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). 

---