Saddle-node

It can be seen from Fig. 15(a) that the atom moves in a stick-slip way. In forward motion, for example, it is a stick phase from A to B during which the atom stays in a metastable state with little change in position as the support travels forward. Meanwhile, the lateral force gradually climbs up in the same period, leading to an accumulation of elastic energy, as illustrated in Fig. 15(fo). When reaching the point B where a saddle-node bifurcation appears, the metastable... 

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... 

The lower a graph is more interesting. While initially the Poincar6 phase portrait looks the same as before (point E, inset 2c) an interval of hysteresis is observed. The saddle-node bifurcation of the pericxiic solutions occurs off the invariant circle, and a region of two distinct attractors ensues a stable, quasiperiodic one and a stable periodic one (Point F, inset 2d). The boundary of the basins of attraction of these two attractors is the one-dimensional (for the map) stable manifold of the saddle-type periodic solutions, SA and SB. One side of the unstable manifold will get attract to one attractor (SC to the invariant circle) while the other side will approach die other attractor (SD to die periodic solution). 

Figure 39, Chapter 3. Bifurcation diagrams for the model of the Calvin cycle for selected parameters. All saturation parameters are fixed to specific values, and two parameters are varied. Shown is the number of real parts of eigenvalues larger than zero (color coded), with blank corresponding to the stable region. The stability of the steady state is either lost via a Hopf (HO), or via saddle node (SN) bifurcations, with either two or one eigenvalue crossing the imaginary axis, respectively. Intersections point to complex (quasiperiodic or chaotic) dynamics. See text for details.

Figure 19. The steady state solutions A0 of the pathway shown in Fig. 18 as a function of the influx vi. For an intermediate influx, two pathways exist in two possible stable steady states (black lines), separated by an unstable state (gray line). The stable and the unstable state annihilate in a saddle node bifurcation. The parameters are k.2 0.2, 3 2.0, K] 1.0, and n 4 (in arbitrary units).

Besides the two most well-known cases, the local bifurcations of the saddle-node and Hopf type, biochemical systems may show a variety of transitions between qualitatively different dynamic behavior [13, 17, 293, 294, 297 301]. Transitions between different regimes, induced by variation of kinetic parameters, are usually depicted in a bifurcation diagram. Within the chemical literature, a substantial number of articles seek to identify the possible bifurcation of a chemical system. Two prominent frameworks are Chemical Reaction Network Theory (CRNT), developed mainly by M. Feinberg [79, 80], and Stoichiometric Network Analysis (SNA), developed by B. L. Clarke [81 83]. An analysis of the (local) bifurcations of metabolic networks, as determinants of the dynamic behavior of metabolic states, constitutes the main topic of Section VIII. In addition to the scenarios discussed above, more complicated quasiperiodic or chaotic dynamics is sometimes reported for models of metabolic pathways [302 304]. However, apart from few special cases, the possible relevance of such complicated dynamics is, at best, unclear. Quite on the contrary, at least for central metabolism, we observe a striking absence of complicated dynamic phenomena. To what extent this might be an inherent feature of (bio)chemical systems, or brought about by evolutionary adaption, will be briefly discussed in Section IX. 

Figure 28. The eigenvalues of the Jacobian of minimal glycolysis as a function of the influence of ATP on the first reaction V (ATP) (feedback strength). Shown is the largest real part of the eigenvalues (solid line), along with the corresponding imaginary part (dashed line). Different dynamic regimes are separated by vertical dashed lines, for > 0 the state is unstable. Transitions occur via a saddle node (SN) and a Hopf (HO) bifurcation. Parameters are v° 1, TP° 1, ATP0 0.5, At 1, and 6 0.8. See color insert.

Figure 29 Bifurcation diagram of the minimal model of glycolysis as a function of feedback strength and saturation 6 of the ATPase reaction. Shown are the transitions to instability via a saddle node (SN) and a Hopf (HO) bifurcation (solid lines). In the regions (i) and (iv), the largest real part with in the spectrum of eigenvalues is positive > 0. Within region (ii), the metabolic state is a stable node, within region (iii) a stable focus, corresponding to damped transient oscillations.

Figure 33. The stability of yeast glycolysis A Monte Carlo approach. A Shown in the distribution of the largest positive real part within the spectrum of eigenvalues, depicted from above (contour plot). Darker colors correspond to an increased density of eigenvalues. Instances with > 0 are unstable. B The probability that a random instance of the Jacobian corresponds to an unstable metabolic state as a function of the feedback strength 0, . The loss of stability occurs either via in a saddle node (SN) or via a Hopf (HO) bifurcation.

While the conditions 1,2 can be verified approximately by simulation, proving the condition 3 is very difficult. Note that in many studies of chaotic behavior of a CSTR, only the conditions 1,2 are verified, which does not imply chaotic d3mamics, from a rigorous point of view. Nevertheless, the fulfillment of conditions 1,2, can be enough to assure the long time chaotic behavior i.e. that the chaotic motion is not transitory. From the global bifurcations and catastrophe theory other chaotic behavior can be considered throughout the disappearance of a saddle-node fixed point [10], [19], [26]. 

When the determinant of the Jacobian matrix becomes zero, one of the roots of (3.43) also becomes zero. This represents the point at which a node (stable or unstable depending on the sign of tr(J)) is just changing to a saddle point or vice versa. Such saddle-node bifurcations, characterized by... 

For all physically acceptable conditions, the determinant of J is positive, so we will not find saddle points or saddle-node bifurcations. We can, however, expect to find conditions under which nodal states become focal (damped oscillatory responses), i.e. where A = 0, and where focal states lose stability at Hopf bifurcations, i.e. where tr(J) = 0 and where we shall look for the onset of sustained oscillations. 

Fig. 8.3. The approach to, or departure from, stationary-state solutions following small perturbations for simple cubic autocatalysis again showing the instability of the middle branch. The turning points (ignition and extinction) have one-sided stability as perturbations in one direction decay back to the saddle-node point, but those of the opposite sign depart for the other...

The condition tr(J) = det(J) = 0 corresponds to a Hopf bifurcation point moving exactly onto the saddle-node turning point (ignition or extinction point) on the stationary-state locus. Above the curve A the system may have two Hopf bifurcations, or it may have none as we will see in the next subsection. Below A there are two points at which tr (J) = 0, but only one of... 

Fig. 12.4. Stationary-state solutions and limit cycles for surface reaction model in presence of catalyst poison K3 = 9, k2 = 1, k3 = 0.018. There is a Hopf bifurcation on the lowest branch p = 0.0237. The resulting stable limit cycle grows as the dimensionless partial pressure increases and forms a homoclinic orbit when p = 0.0247 (see inset). The saddle-node bifurcation point is at...

Fig. 12.6. (a) Hopf bifurcation loci for the Takoudis-Schmidt Aris model with k, = 10-3 and k2 = 2x 10-3. Also shown (broken curves) are the saddle-node boundaries from Fig. 12.6. (b)-(i) The eight qualitative arrangements of Hopf and saddle-node bifurcation points. (Adapted and reprinted with permission from McKarnin, M. A. et al. (1988). Proc. R. Soc., A415,... 

Fig. 13.17. Floquet multipliers lying within the unit circle, indicating a stable periodic motion if the CFM leaves the unit circle through — 1 a period doubling occurs if it goes out through + 1 there is a saddle-node bifurcation with the disappearance of the periodic solution.

We now consider the phenomenon of entrainment (the development of resonances) on the torus (Meyer, 1983). When (and if) the off-diagonal band in Fig. 6 crosses the diagonal [Figs. 6(c) and 6(f)], there exist points whose images fall on themselves they are fixed points of the map we study. These points lie on periodic trajectories that are locked on the torus. Such trajectories appear in pairs in saddle-node bifurcations and are usually termed subharmonics . When this occurs there is no quasi-periodic attractor winding around the torus surface, but the basic structure of the torus persists the invariant circle is patched up from the unstable manifolds of the periodic saddle-points with the addition of the node-periodic point (Arnol d, 1973, 1982). As we continue changing some system parameter the periodic points may come to die in another saddle-node bifurcation (see Fig. 5). Periodic trajectories thus... 

FIGURE 5 Subharmonic saddle-node bifurcations, (a). The subharmonic period 3 isola for the surface model (o/o0 = 1.4, o0 = 0.001). One coordinate of the fixed points of the third iterate of the stroboscopic map is plotted vs. the varying frequency ratio oi/eio. Six such points (S, N) exist simultaneously, three of them (N) lying on the stable node period 3 and three (S) on the saddle period 3. Notice the two triple turning point bifurcations at o>/o>o = 2.9965 and 3.0286. In (b) the PFM of these trajectories on the isola Is also plotted, (c) shows another saddle-node bifurcation occurring (for a>o = 3) as this time the forcing amplitude is increased to o/o0 = I.6SS. 

The most important characteristic in our test cases, however, is that within the 1/1 and the 2/1 resonance horns the torus will break as FA increases. In all models this happens when the unstable source period 1 that existed within the torus hits the saddle-periodic trajectories that lie on the torus. This occurs through a saddle-node bifurcation in the 1/1 resonance horn [Fig. 8(d)], and through an unstable period doubling in the 2/1 resonance [Fig. 8(c)]. After these bifurcations the basic structure of the torus has collapsed, and we are left only with the stable entrained periodic trajectories. 

As we now change stable periodic trajectories cannot lose stability through a saddle-node bifurcation, since the saddles no longer exist rather they lose stability through a Hopf bifurcation of the stroboscopic map to a torus (Marsden and McCracken, 1976). This phenomenon, as well as the torus resulting from it, is considerably different from the frequency unlocking case. One of the main differences is that the entire quasi-periodic attractor that bifurcates from a periodic trajectory lies close to it [see Figs. 9(c) and 9(d)],... 

The point S of figure 8 at which the Hopf bifurcation curve crosses the boundary of the multiplicity region is not a double zero degeneracy, for the upper steady state (i.e. that with the larger 0b) is undergoing the Hopf bifurcation at the same time as the lower steady-state undergoes a saddle-node bufurcation, i.e. the conditions trJ = 0 and detJ = 0 apply at different points. It does, however, serve to show the four combinations of the two most common... 

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