Saddle-node bifurcation on a limit

We choose the control parameters U and a such that the deterministic system exhibits no oscillations but is very close to a bifurcation thus yielding it very sensitive to noise. The transition from stationarity to oscillations in the system may occur either via a Hopf or via a saddle-node bifurcation on a limit cycle as depicted in the bifurcation diagram of Fig. 5.9. The different nature of these two bifurcations is reflected in the effect noise has in each case. The local character of the Hopf bifurcation is responsible for noise-induced high frequency oscillations of strongly varying amplitude around the stable fixed point. We try to characterize basic features of these oscillations such as coherence and time scales. The need to be able to adjust these features as one wishes will lead to the application of the time-delayed... 

Fig. 5.9. Bifurcation diagram in the (cr, t/) plane. Thick and hatched lines mark the transition from stationary to moving fronts via a Hopf or a saddle-node bifurcation on a limit cycle, respectively. The inset shows a blow-up of a small part of the hatched line revealing its saw-tooth-like structure. Dark and white correspond to stationary and moving fronts, respectively, where the numbers denote the positions of the stationary accumulation front in the superlattice. Upper inset shows the frequency / of the limit cycle which is born above the critical point (marked by a cross in the lower inset) as function of U. [57]...

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. 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...

We assume that the equations (7.200) have a simple hysteresis type static bifurcation as depicted by the solid curves in Figures 10 to 12 (A-2). The intermediate static dashed branch is always unstable (saddle points), while the upper and lower branches can be stable or unstable depending on the position of eigenvalues in the complex plane for the right-hand-side matrix of the linearized form of equations (7.198) and (7.199). The static bifurcation diagrams in Figures 10 to 12 (A-2) have two static limit points which are usually called saddle-node bifurcation points. 

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 proofs of Theorems 10.2, 10.3, and 10.4 are found in [348]. Equation (10.17) is of particular interest. Near the Takens-Bogdanov point, the frequency of the limit-cycle oscillations along the line of Hopf bifurcations, a = 0, is given by >h = see above. On the line of saddle-node bifurcations we have Aj = 0. An equation like (10.17) is expected from simple dimensional arguments. The only intrinsic length scales in reaction-diffusion systems come from the diffusion coefficients. The inverse time is determined by the rate coefficients of the reaction kinetics. Thus (10.17) provides an estimate of the intrinsic length of the Turing pattern near a double-zero point ... 

As the residence time is decreased, we pass the point at = 313 at which the limit cycle burst on the upward journey, but do not now move from the lower branch. Only when there is a saddle-node bifurcation of the lower two stationary states at Tres 312.5 does the system jump back into large oscillations. 

The recent simulations of Pearson [44] on a fed autocatalator [104], on the other hand stages interactions between Turing and saddle-node bifurcations or even situations where a Hopf bifurcation interacts with the two previous ones. Then a plethora of scenarios may result. However, the Turing bifurcations are genuine but they occur on the stable branch of an isola of uniform states and may exist beyond the limits of the isola. 

It is very important to stress that changes in the geometrical shape of the integration domain can induce bifurcations in the drift velocity field [31, 47, 50, 52]. Let us consider, for example, the drift velocity field computed for an elliptical domain with major axis o = 3A and minor axis b = a/1.1. As shown in Fig. 9.13(a), instead of the stable limit cycle of the resonance attractor in the circular domain of radius Rg, = 1.5A we have two pairs of fixed points where the drift velocity vanishes. In each pair, one fixed point is a saddle and the other one is a stable node. Depending on the initial conditions, the spiral wave approaches one of the two stable nodes. Trajectories of the spiral center obtained by numerical integration of the Oregonator model (9.1) are in perfect agreement with the predicted drift... 

The circled zones in figure 4 represent, at the same scale, the corresponding Poincare sections of the limit cycle that exists prior to the bifurcation toward the torus. The aireas of these dashed zones stand for the e3q>erimental scatter of the points. It is isotropic almost everywhere except in the part of the limit cycle which corresponds to the stretching of the wrinkle. Furthermore, we must remark that the limit cycle appears to lie on the surface of the torus. This is cin indication that the bifurcation leading to the torus is of a saddle-node type rather than of a Hopf type (in this latter case the limit cycle should be inside the torus). This character is also confirmed by the abruptness of the transition eind the absence of hysteresis. 

In Chap. 12 we will study the global bifurcations of the disappearance of saddle-node equilibrium states and periodic orbits. First, we present a multidimensional analogue of a theorem by Andronov and Leontovich on the birth of a stable limit cycle from the separatrix loop of a saddle-node on the plane. Compared with the original proof in [130], our proof is drastically simplified due to the use of the invariant foliation technique. We also consider the case when a homoclinic loop to the saddle-node equilibrium enters the edge of the node region (non-transverse case). 

To study such bifurcations one should understand the structure of the limit set into which the periodic orbit transforms when the stability boundary is approached. In particular, such a limit set may be a homoclinic loop to a saddle or to a saddle-node equilibrium state. In another bifurcation scenario (called the blue sky catastrophe ) the periodic orbit approaches a set composed of homoclinic orbits to a saddle-node periodic orbit. In this chapter we consider homoclinic bifurcations associated with the disappearance of the saddle-node equilibrium states and periodic orbits. Note that we do not restrict our attention to the problem on the stability boundaries of periodic orbits but consider also the creation of invariant two-dimensional tori and Klein bottles and discuss briefly their routes to chaos. 

This is a one-dimensional system which may have stable and unstable equilibrium states corresponding to stable and saddle equilibrium states of the entire system (12.4.6) or (12.4.7). The evolution along Meq is either limited to one of the stable points, or it reaches a small neighborhood of the critical values of X. Recall, that we consider x as a governing parameter for the fast system and critical values of x are those ones which correspond to bifurcations of the fast system. In particular, at some x two equilibrium states (stable and saddle) of the fast system may coalesce into a saddle-node. This corresponds to a maximum (or a minimum) of x on Meq, so the value of x cannot further... 

Fig. 13.7.20. The bifurcation diagram for the case where both equilibrium states of the heteroclinic cycle are saddles (see Fig. 13.7.12) provided Ai,2 > 0, i/i > 1, 1/2 < 1 and U1U2 > 1- The system has one limit cycle in regions 1-3, two limit cycles in region 4, none in regions 5-7. A pair of limit cycles are born from a saddle-node on the curve SN the unstable one becomes a homoclinic loop on the curve L2, whereas the stable limit cycle terminates on Li.

As p decreases, the limit cycle r = 1 develops a bottleneck at 8 = njl that becomes increasingly severe as > 1. The oscillation period lengthens and finally becomes infinite at p = 1, when a fixed point appears on the circle hence the term infinite-period bifurcation. For p < 1, the fixed point splits into a saddle and a node. 

---