Hopf bifurcation supercritical

There are no unstable limit cycles in this model, and the oscillatory solution born at one bifurcation point exists over the whole range of stationary-state instability, disappearing again at the other Hopf bifurcation. Both bifurcations have the same character (stable limit cycle emerging from zero amplitude), although they are mirror images, and are called supercritical Hopf bifurcations. 

FlO. 5.4. The birth and growth of oscillatory solutions for the thermokinetic model with the full Arrhenius temperature dependence, (a) The Hopf bifurcations /x and ft are both supercritical, with [12 < 0, and the stable limit cycle born at one dies at the other, (b) The upper Hopf bifurcation is subcritical, with fl2 > 0. An unstable limit cycle emerges and grows as the dimensionless reactant concentration ft increases—at /rsu this merges with the stable limit cycle born at the lower supercritical Hopf bifurcation point ft. ... 

Fig. 8.7. Supercritical Hopf bifurcation for cubic autocatalysis with decay and /) = 0, appropriate for small dimensionless decay rate constant k2 < 9/256. A stable limit cycle emerges and grows as the residence time is increased above t s. At higher residence times, this disappears at rj , by merging with an unstable limit cycle born from a homoclinic orbit at t. (With non-zero autocatalyst inflow, (i0 > 0, the stable limit cycle itself may form a homoclinic orbit at long tres.)...

Fig. 8.13. (a) The division of the fS0 — K1 parameter region into 11 regions by the various loci of stationary-state and Hopf bifurcation degeneracies. The qualitative forms of the bifurcation diagrams for each region are given in fi)—(xi) in (b), where solid lines represent stable stationary states or limit cycles and broken curves correspond to unstable states or limit cycles, (i) unique solution, no Hopf bifurcation (ii) unique solution, two supercritical Hopf bifurcations (iii) unique solution, one supercritical and one subcritical Hopf (iv) isola, no Hopf points (v) isola with one subcritical Hopf (vi) isola with one supercritical Hopf (vii) mushroom with no Hopf points (viii) mushroom with two supercritical Hopf points (ix) mushroom with one supercritical Hopf (x) mushroom with one subcritical Hopf (xi) mushroom with supercritical and subcritical Hopf bifurcations on separate branches. 

Let us imagine a scenario for which a supercritical Hopf bifurcation occurs as one of the parameters, fi say, is increased. For fi < fi, the stationary state is locally stable. At fi there is a Hopf bifurcation the stationary state loses stability and a stable limit cycle emerges. The limit cycle grows as ft increases above fi. It is quite possible for there to be further bifurcations in the system if we continue to vary fi. With three variables we might expect to have period-doubling sequences or transitions to quasi-periodicity such as those seen with the forced oscillator of the previous section. Such bifurcations, however, will not be signified by any change in the local stability of the stationary state. These are bifurcations from the oscillatory solution, and so we must test the local stability of the limit cycle. We now consider how to do this. 

The upper and lower limits in eqn (13.61) correspond to points of supercritical Hopf bifurcation. The stationary state is unstable in the whole of the region, and near to the limits is surrounded by a stable period-1 limit cycle. We now wish to see how the stability of this oscillatory solution, and any higher-order periodicities which might emerge, varies with rN. 

FIGURE 2 The birth and growth of limit cycle oscillations in the I - a, jS, Tr space for a system with non-zero e and k displaying a mushroom stationary-state pattern. Oscillatory behaviour originates from a supercritical Hopf bifurcation along the upper branch and terminates via homoclinic orbit formation. 

In this section we discuss the model predictions for the ketone ethyl acetoacetate (1). With the ketone absent ([Ket]x = 0 mM), the extended model reproduces all previous results with oscillations of all system variables above [Glc]xo > 18.5 mM [53]. Figure 3.6 shows the system s response to a fixed glucose concentration [Glc]xo at 30 mM and an increase of [Ket]x to 1 mM. The oscillations vanish at [Ket]x = 0.23 mM in a supercritical Hopf bifurcation and the steady state is stable for [Ket]x > 0.23 mM. Figure 3.6a shows the minimum and maximum concentrations of NADH as two thick curves, while in all other panels the time averages of the plotted variables are shown, not the minimum and maximum values. Since the addition of ketone provides an alternative mode of oxidation of NADH, the concentration of NADH is decreasing in Fig. 3.6a whereas the fluxes of carbinol production are increasing in Fig. 3.6b. 

For T = 16 s, the single nephron model undergoes a supercritical Hopf bifurcation at a = 11 (outside the figure), fn this bifurcation, the equilibrium point loses its stability, and stable periodic oscillations emerge as the steady-state solution. For a = 19.5, at the point denoted PDla 2 in Fig. 12.5, this solution undergoes a period-... 

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 empty-site requirement in Eq. (28) can be physically interpreted in one of two different ways either the adsorbed A and B have to rearrange prior to reaction, or they are bound to more than one adsorption site. For the latter case, the intermediate concentration is low, thus allowing a pseudo-steady-state assumption. Through the application of bifurcation analysis and catastrophe theory this model was found to predict a very rich bifurcation and dynamic behavior. For certain parameter values, sub- and supercritical Hopf bifurcations as well as homoclinic bifurcations were discovered with this simple model. The oscillation cycle predicted by such a model is sketched in Fig. 6c. This model was also used to analyze how white noise would affect the behavior of an oscillatory reaction system... 

The behaviour at the upper Hopf point is also that of a supercritical Hopf bifurcation although the loss of stability of the steady-state and the smooth growth of the stable limit cycle now occurs as the parameter is reduced. This is sketched in Fig. 5.10(b). We can join up the two ends of the limit cycle amplitude curve in the case of this simple Salnikov model to show that the amplitude of the limit cycle varies smoothly across the range of steady-state instability, as indicated in Fig. 5.11(a). The limit cycle born at one Hopf point survives across the whole range and dies at the other. Although this is the simplest possibility, it is not the only one. Under some conditions, even for only very minor elaboration on the Salnikov model [16b], we encounter a subcritical Hopf bifurcation. At such an event, the limit cycle that is born is not stable but is unstable. It still has the form of a closed loop in the phase plane but the trajectories wind away from it, perhaps back in towards the steady-state as indicated in Fig. 

Fig. 5.10. The four possible types of Hopf bifurcation (a) a stable steady-state (sss) becomes unstable (uss) as a parameter fi is increased through the bifurcation point (/x ) and a stable limit cycle (sic) emerges - the growth of the limit cycle is indicated by plotting the maximum and minimum of the variable as it undergoes the oscillatory motion around the limit cycle (b) the scenario is reversed, with the steady-state losing stability and a stable limit cycle emerging as the parameter is reduced (a) and (b) are termed supercritical Hopf bifurcations. In (c) and (d) there is an unstable limit cycle emerging to surround the stable part of the steady-state branch this is characteristic of a subcritical Hopf bifurcation.

Fig. 5.11. Variation of the oscillatory (limit cycle) solution with ju for the simple Salnikov model showing that the stable limit cycle born at one supercritical Hopf bifurcation exists over the whole range of the unstable steady-state, shrinking to zero amplitude at the other Hopf point (b) in this case, each Hopf point gives rise to a different limit cycle, with a stable limit cycle born at fi growing as increases and an unstable limit cycle born at /x also increasing in size as fx increases. At some fx> fx the two limit cycles collide and are...

Fig. 5.24. Variation of oscillatory waveform in vicinity of boundary between oscillatory and steady ignition showing characteristic nature of a supercritical Hopf bifurcation. (Reprinted with permission from reference [33], Royal Society of Chemistry.)...

A simple example of a supercritical Hopf bifurcation is given by the following... 

Our idealized case illustrates two rules that hold generically for supercritical Hopf bifurcations ... 

We begin the analysis of (4), (5) by constructing a trapping region and applying the Poincare-Bendixson theorem. Then we ll show that the chemical oscillations arise from a supercritical Hopf bifurcation. 

The universal description of reaction-diffusion systems near a supercritical Hopf bifurcation is provided by the complex Ginzburg-Landau equation [11]. Action of global periodic forcing on the systems described by this... 

Further increase in the rate of product recycling gives rise to a supercritical Hopf bifurcation corresponding to the creation of a stable, small-amplitude limit cycle. This phenomenon occurs at the border of the stability domain induced by recycling, inside the domain of existence of the large-amplitude limit cycle (fig. 3.6d). Birhythmicity arises from the coexistence of these two stable limit cycles. 

In this experiment, a constant inflow of one species at a time is added to the system at steady state near a supercritical Hopf bifurcation. This inflow additional to the inflow terms in eq. (11.2) should not be large enough to shift the system from one dynamics regime to another, for example, from a stationary state to an oscillatory state. The response of the concentrations of as many species as possible should be determined after the addition of each species and compared to the steady-state concentrations of the unperturbed system. These measurements allow the construction of an experimental shift matrix, which is directly related to the Jacobian matrix. If we approximate the... 

A system near a supercritical Hopf bifurcation maybe either at a steady state or oscillatory. Starting from this point, the inflow of a species is varied (increased or decreased) until the transition from oscillations to steady state or vice versa occurs. There are two possibilities ... 

In this experiment, a pulsed perturbation of one species at a time is applied to a system at a stable steady state near a supercritical Hopf bifurcation. The relaxation to the steady state is measured. Two purposes of these experiments can be distinguished, either to discriminate among essential and nonessential species or to estimate sign-symbolic Jacobian matrix elements. 

The variation in time of concentrations of species in an oscillatory reaction can be stopped temporarily by application of a pulse perturbation of a species to the system. If the system is close to a supercritical Hopf bifurcation, an interpretation of the addition leading to quenching of oscillations provides information on nonessential species, and on the Jacobian matrix. In quenching experiments, the phase of oscillation at which the perturbation is added and the amount of perturbant added are varied for each perturbing... 

Vance and Ross [63] show that there is a close relation between the variation of the relative phase shift across the entrainment band and variation of the phase response curve provided that limit cycle oscillations are close to a supercritical Hopf bifurcation. 

This general analysis provides useful insights but needs to be verified for a given set of rate coefficients and constraints by calculating the quantities corresponding to (some of) the experimental tests for classification. For example, we can vary constraints (inflows of the nonenzymatic species in this case) so as to find a supercritical Hopf bifurcation, then calculate the Jacobian matrix and determine (1) its inverse to find concentration shifts,... 

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