Bifurcation mechanisms saddle-node bifurcations

The changes of lateral force F in forward and backward motions follow the curve 1 and 2, respectively. It can be observed that there is one saddle-node bifurcation for the repulsive pinning center, but two bifurcations for the attractive piiming center. This suggests that the interfacial instability results from different mechanisms. On one hand, the asperity suddenly looses contact as it slides over a repulsive pinning center, but in the attractive case, on the other hand, the... 

Recently, Bernard et al. [499] studied oscillations in cyclical neutropenia, a rare disorder characterized by oscillatory production of blood cells. As above, they developed a physiologically realistic model including a second homeostatic control on the production of the committed stem cells that undergo apoptosis at their proliferative phase. By using the same approach, they found a local supercritical Hopf bifurcation and a saddle-node bifurcation of limit cycles as critical parameters (i.e., the amplification parameter) are varied. Numerical simulations are consistent with experimental data and they indicate that regulated apoptosis may be a powerful control mechanism for the production of blood cells. The loss of control over apoptosis can have significant negative effects on the dynamic properties of hemopoiesis. 

It has been shown recently that the vibrational spectra of HCP [33-36], HOCl [36-39], and HOBr [40,41] obtained from quantum mechanical calculations on global ab initio surfaces can be reproduced accurately in the low to intermediate energy regime (75% of the isomerization threshold for HCP, 95% of the dissociation threshold for HOCl and HOBr) with an integrable Fermi resonance Hamiltonian. Based on the analysis of this Hamiltonian, this section proposes an interpretation of the most salient feature of the dynamics of these molecules, namely the first saddle-node bifurcation, which takes place in the intermediate energy regime. 

Saddle-node bifurcations taking place for the reasons just described have been observed for HOBr [41], HOCl [36,38,39], and HCP [34-36]. For HOBr and HOCl, the stable PO bom at the saddle-node bifurcations is called [D] for dissociation, because this PO stretches along the dissociation pathway and scars OBr- or OCl-stretch quantum mechanical wavefunctions (see Fig. lie of Ref. 38, Figs. 3b and 3g of Ref. 41, or Section III.B). In the case of HCP, the stable PO born at the bifurcation is better called [I], for isomerization, because this PO stretches along the isomerization pathway and scars bending quantum mechanical wavefunctions (see Figs. 6b and 6d of Ref. 35 or Figs. 7b and 7d of Ref. 36). 

The saddle-node bifurcation is the basic mechanism by which fixed points are created and destroyed. As a parameter is varied, two fixed points move toward each other, collide, and mutually annihilate. 

The saddle-node bifurcation is the basic mechanism for the creation and destruction of fixed points. Here s the prototypical example in two dimensions ... 

To conclude, noise-induced front motion and oscillations have been observed in a spatially extended system. The former are induced in the vicinity of a global saddle-node bifurcation on a limit cycle where noise uncovers a mechanism of excitability responsible also for coherence resonance. In another dynamical regime, namely below a Hopf bifurcation, noise induces oscillations of decreasing regularity but with almost constant basic time scales. Applying time-delayed feedback enhances the regularity of those oscillations and allows to manipulate the time scales of the system by varying the time delay t. 

Fig. 11.1 Two-dimensional bifurcation diagram calculated by continuation from the Citri-Epstein mechanism for the chlorite-iodide system. Plot of 2D space of constraints is chosen to be the ratio of input concentrations, [CIO ]o/[I ]o> versus the logarithm of the reciprocal resideuce time, logfco-Notation SSI, SS2, region of steady states with high [1 ] and low [I ], respectively Osc, region of periodic oscillations Exc, region of excitahihty snl, sn2, curves of saddle-node bifurcations of steady states hp, curve of Hopf bifiucatious sup, ciuve of saddle-node bifurcations of periodic orbits swt, swallow tail (a small area of tristability) C, cusp point TB, Takens-Bogdanov point (terminus of hp on snl). (From [5].)...

In Section 5.1, we have provided some evidence for the existence of saddle-node bifurcations, as a (local) mechanism accounting for the creation of pairs of solutions of our reaction-diffusion system (3) with Dirichlet boundary conditions. As the normal form theory [65, 66] shows, at least one of these two solutions is unstable. From a physical point of view, one can guess that, in the limit D oo, any stationary solution should be stable. However, as D is decreased, or equivalently, as the size of the system is increased, the system becomes approximately translationally invariant, and the stationary solutions are very likely to be unstable. As shown in this section, this transition involves not only (stationary) saddle-node bifurcations, but (oscillatory) Hopf bifurcations as well [62,104]. 

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