Bifurcation unfolding

The situation described in Fig. 6 is strongly reminiscent of the phenomenon of bifurcation, whereby new branches of solutions come into play when some suitable control parameters are varied. The difference is of course that in the present case the appea-rence of new branches can only be a transient. We coined the term "bifurcation unfolding in time" [ 7 ] to this phenomenon in order to capture both the similarities and the differences with its more familiar "static" counterpart. 

The bifurcation unfoldings for Cases B and C are identical and shown in Fig. 13.6.4. Here, p is the splitting parameter of the homoclinic loop, and A is the separatrix value. Since in Sec. 13.4 the separatrix value A was defined only when the loop does not belong to we must specify its meaning for Case C. 

Fig. 13.6.4. The bifurcation unfoldings for an orbit- and an inclination-flip bifurcation are identical in the simplest case.

Golubitsky, M. and Langford, W. F. (1981). Classification and unfoldings of degenerate Hopf bifurcations. J. Differ. Equations, 41, 373-415. 

The prediction of the period 2 saddle-node curve terminating at J remains until someone develops a universal unfolding for this co-dimension two bifurcation. 

The specific activity of C443D, C443L, C443M, and C443V was less than 50%, compared to that of the WT (Table 21). The CD spectra of these mutants indicated that they were partly unfolded. The van der Waals volumes of aspartic acid (91 A3) and valine (105 A3) are close to that of cysteine (86 A3). Aspartic acid and valine have bifurcated side chains at the CY and Cp atoms respectively. The bifurcated side chains are assumed to causes denaturation of the enzyme. 

The qualitative features of the local bifurcation diagrams (0 vs. Da) of Eq. (13) in the neighborhoood of any singular point defined by (14) are same as those of its universal unfolding... 

It is important to note that while the selection of the bifurcation variable does not affect the maximal number of steady-state solutions, it affects the number and type of bifurcation diagrams. For example, if we selected the coolant or feed temperature as the bifurcation variable then Eq. (18) would be the universal unfolding for both the adiabatic and the cooled case and no isolas would exist [1 2 ]. 

Classification of catastrophes will be preceded by the centre manifold theorem which is a counterpart to the splitting lemma in elementary catastrophe theory. It will turn out that in the catastrophe theory of dynamical systems such notions of elementary catastrophe theory as the catastrophe manifold, bifurcation set, sensitive state, splitting lemma, codimension, universal unfolding and structural stability are retained. 

Finally, with Eqs. (5.409) and (5.410) back in the founded bifurcation parameters and further into the unfolded catastrophe functions (5.405) and (5.406), the corresponding general Markovian ELF cases are displayed as (Putz, 2005) ... 

It has been established [59] that any activated exothermic reaction in an open system can give rise to bistable, excitable and oscillatory behavior. The principle features can be obtained using just a single unimolecular step. A detailed bifurcation analysis of such a system has been carried out by Vance and Ross [60] the bifurcation set consists of an unfolding of a (codimension-3) Takens-Bogdanov-cusp point [61]. (The bifurcation fine... 

The Rossler system and the new Lorenz system (C.2,19) are remarkable in that both have a doubly degenerate equilibrium state with characteristic exponents equal to (0, iij). The feature of this bifurcation is that the unfolding may contain a torus bifurcation curve along with curves corresponding to homoclinic loops to saddle-foci, and therefore non-trivial dynamics may emerge instantly in a neighborhood of the bifurcating equilibria. ... 

The point NSF a = 0 corresponds to a neutral saddle-focus. At this codimension-two point the dynamics of the trajectories near the homoclinic loops to the saddle-focus becomes chaotic. This bifurcation indeed proceeds the origin of the chaotic double scroll attractor in Chua s circuit. In the general case, this bifurcation was first considered in [29]. The complete unfolding of... 

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