A 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... 

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).

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]. 

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.

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... 

In the 1/1 entrainment region each side of the resonance horn terminates at points C and D respectively. These points are codimension-two bifurcations and correspond to double +1 multipliers. As the saddle-node curve at the right horn boundary rises from zero amplitude towards point D, one multiplier remains at unity (the criterion for a saddle-node bifurcation) as the other free-multiplier of the saddle-node increases until it is also equal to unity upon arrival at point D. The same thing occurs for the left boundary of the resonance horn. The arc CD is also a saddle-node bifurcation curve but is different from those on the sides of the resonance horn. As arc CD is crossed from below, the period 1 saddle combines not with its companion stable node, but with the unstable node that was in the centre of the phase locked torus. As the pair collides, the invariant circle is lost and only the stable node remains. Exactly the same scenario is observed for the 1/2 resonance horn as well. 

As S continues to increase we reach a point marked by an open circle. Here, the equilibrium point undergoes a saddle-node bifurcation. Somewhere before this bifurcation, the unstable focus point has turned into an unstable node with two positive real eigenvalues. In the saddle-node bifurcation, one of these eigenvalues... 

From Fig. 18b it is clear that under galvanostatic conditions the limit cycle coexists with a stationary state at high overpotentials. The latter is the only attractor at large current densities. Hence, when the current density is increased above the value of the saddle-loop bifurcation, the potential jumps to a steady state far in the anodic region. Once the system has acquired the anodic steady state, it will stay on this branch as the current density is lowered until the stationary state disappears in a saddle-node bifurcation. 

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. 

The left side term indicates the interactions between the two component subsystems, while the right term shows the interactions within the subsystem of each component. Thus, even with stable subsystems (in isolation), the system can be unstable if the interactions among subsystems are more significant than the interactions within subsystems. So, the enzymatic parameters and the boundary conditions can be controlled in such a way that systemic instability occurs. This particular phenomenon is known as a saddle-node bifurcation. 

Figure 19. For the value % = 0, tt/2, n, energy of the relative equilibria as a function of J. Note that the three families undergo a saddle-node bifurcation at different energy each.

The prototypical example of a saddle-node bifurcation is given by the first-order system... 

There are several other ways to depict a saddle-node bifurcation. We can show a stack of vector fields for discrete values of r (Figure 3.1.2),... 

---