Hopf bifurcation plots

Figure 3. Hopf bifurcation plot for k arbie versus Ca showing regions of local stability (SJ, and instability leading to oscillatory dynamics (U).

Figure 4. a) Hopf bifurcation plot of initial MAA doping ((Tq) versus glucose concentration (Cq). b) Time profile of pH in Cell II given different values of ob andCc. kmarbie-Sxlff sec. ... 

Fig. 2.7 (a) Temporal variation of the membrane potential V and the intracellular calcium concentration S in the considered simple model of a bursting pancreatic cell, (b) Bifurcation diagram forthe fast subsystem the black square denotes a Hopf bifurcation, the open circles are saddle-node bifurcations, and the filled circle represents a global bifurcation, (c) Trajectory plotted on top of the bifurcation diagram. The null-cline forthe slow subsystem is shown dashed. 

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. 

Figure 12.6a shows the temporal variation of the proximal tubular pressure Pt as obtained from the single-nephron model for a = 12 and T = 16 s. All other parameters attain their standard values as listed in Table 12.1. Under these conditions the system operates slightly beyond the Hopf bifurcation point, and the depicted pressure variations represent the steady-state limit cycle oscillations reached after the initial transient has died out For physiologically realistic parameter values the model reproduces the observed self-sustained oscillations with characteristic periods of 30-40 s. The amplitudes in the pressure variation also correspond to experimentally observed values. Figure 12.6b shows the phase plot Here, we have displayed the normalized arteriolar radius r against the proximal intratubular pressure. Again, the amplitude in the variations of r appears reasonable. The motion... 

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.

For each of the following systems, a Hopf bifurcation occurs at the origin when jU = 0. Using a computer, plot the phase portrait and determine whether the bifurcation is subcritical or supercritical. 

A catastrophe of this type is called the Hopf bifurcation. It may be represented by a bifurcation diagram in which the position of a stationary state and a limit cycle are plotted as a function of the parameter c (Figs. 83, 84). 

Fig. 6.7. Bifurcation diagrams for stationary solutions. Detuning 5 and coupling strength 77 are plotted. The figures differ by the propagation phase parameter cp. Hopf bifurcations are denoted by dashed lines and saddle-nodes by solid lines. ZH — Zero-Hopf bifurcation CP — cusp GH — generalized Hopf bifurcation (where the Hopf bifurcation changes its criticality)...

Jaisinghani and Ray (40) also predicted the existence of three steady states for the free-radical polymerization of methyl methacrylate under autothermal operation. As their analysis could only locate unstable limit cycles, they concluded that stable oscillations for this system were unlikely. However, they speculated that other monomer-initiator combinations could exhibit more interesting dynamic phenomena. Since at that time there had been no evidence of experimental work for this class of problems, their theoretical analysis provided the foundation for future experimental work aimed at validating the predicted phenomena. Later studies include the investigations of Balaraman et al. (43) for the continuous bulk copolymerization of styrene and acrylonitrile, and Kuchanov et al. (44) who demonstrated the existence of sustained oscillations for bulk copolymerization under non-isothermal conditions. Hamer, Akramov and Ray (45) were first to predict stable limit cycles for non-isothermal solution homopolymerization and copolymerization in a CSTR. Parameter space plots and dynamic simulations were presented for methyl methacrylate and vinyl acetate homopolymerization, as well as for their copolymerization. The copolymerization system exhibited a new bifurcation diagram observed for the first time where three Hopf bifurcations were located, leading to stable and unstable periodic branches over a small parameter range. Schmidt, Clinch and Ray (46) provided the first experimental evidence of multiple steady states for non-isothermal solution polymerization. Their... 

We have shown using a Galerkin type of resolution scheme that when the kinetic constant of the trimolecular step in (1) fluctuates, new pure noise induced transitions become possible. For the sake of clarity let us recall for example that when A = 2, the extrema of the probability density behave as represented in figure 1. u is plotted as a function of a and for the values of e indicated. The curves labelled 1 correspond to the situation typical above the Hopf bifurcation. For small intensities (curve in the lower left corner) one sees that the noise suppresses the extremum corresponding to the usual limit cycle this is the same behavior as in 2.2. Its amplitude which is equal to one (in normalizing with respect to the deterministic limit cycle, i.e. 

These Hopf bifurcations from the RW solutions introduce a second frequency into the spiral dynamics and give rise to the quasiperiodic MRW solutions. Figure 8 shows the bifurcation diagram for MRW states. The (dimensionless) radius ratio ra/ri is plotted as a function of the bifurcation parameter a. Also indicated with a horizontal line are the RW states... 

Fig. 8. Bifurcation diagram for the one-parameter cut shown in Figure 5. The radius ratio, r2/ri, for MRW states is plotted as a function of the parameter a. Also shown as a horizontal line is the branch of RW states (for which r2 = 0) solid indicates stable RW states and dashed indicates unstable RW states. The hollow squares denote Hopf-bifurcation points. Both Hopf bifurcations are supercritical and near the bifurcations the radius ratio scales as the square-root of the distance from the bifurcation points. The radius ratio diverges as a approaches the value where exists a modulated-traveling-wave state.

The first equation gives the value of at the Hopf bifurcation, in terms of ai. The second then gives 02 for the Hopf bifurcation in terms of and a. This Hopf locus is plotted in Figure 10 and is the boundary of the region of modulated waves. 

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.

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].)...

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