Hopf bifurcation subcritical

Fig. 4.9. The development of oscillatory amplitude Ae and period T across the range of instability, 4.2 x 10 3 = n < n < jx = 0.0195, for the pool chemical model with k = 2x 10-3 and y = 0.21, typical of a system with a subcritical Hopf bifurcation at which an unstable limit cycle emerges at The broken curves give the limiting forms predicted by eqns (4.59)—(4.61).

Fig. 8.6. Typical arrangement of local stabilities and development of unstable limit cycle, from a subcritical Hopf bifurcation, appropriate to cubic autocatalysis with decay and no autocatalyst inflow and with 9/256 < k2 < 1/16. The unstable limit cycle grows as t, decreases below t m, and terminates by means of the formation of a homoclinic orbit at rf . Stable stationary states, including the zero conversion branch 1 — a, = 0, are indicated by solid curves, unstable states and limit cycles by broken curves.

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. 

Next, consider the case with p = 0.02014. The traverse across Fig. 12.6(a) as r is varied now also cuts the region of multi stability. It passes above the cusp point C (see Fig. 12.5), giving rise to two turning points in the stationary-state locus, but below the double-zero eigenvalue point M. There are still four intersections with the Hopf curve, so there are four points of Hopf bifurcation. The Hopf point at highest r is now a subcritical bifurcation. The dependence of the reaction rate on r for this system is shown in Fig. 12.6(d). 

FIGU RE 10 Illustration of the disappearance of a limit cycle via a turning point on a periodic branch near a subcritical Hopf bifurcation, (a) A stable limit cycle surrounding an unstable focus (b) the unstable focus undergoes a subcritical Hopf bifurcation and leaves an inner unstable limit cycle surrounding a stable focus (c) the two limit cycles combine into a metastable configuration and disappear altogether as the parameter is further increased. 

Subcritical Hopf transitions are found on the segments HM and GL of the Hopf curve and all other transitions are supercritical. The points H and G in figure 8 are located at (< ] = 0.019308, a2 = 0.030686) and ( i = 0.020668, a2 = 0.018330) respectively, and might be called metacritical. They are bifurcations of codimension two so that we expect only isolated metacritical points on the Hopf curve. These have to satisfy not only the conditions of (42), but also ... 

A dynamic bifurcation occurs when the dynamic behavior of the solution to a system undergoes a qualitative change. For example, a subcritical Hopf bifurcation occurs when a dynamic system changes from a stable node to a limit cycle. Again, AUTO can be used to determine parameter changes that cause this bifurcation to occur. 

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.

Subcritical Hopf bifurcations occur in the dynamics of nerve cells (Rinzel and Ermentrout 1989), in aeroelastic flutter and other vibrations of airplane wings (Dowell and Ilgamova 1988, Thompson and Stewart 1986), and in instabilities of fluid flows (Drazin and Reid 1981). 

The behaviour spectrum of a homogeneous population as a function of parameter (fig. 6.3) reveals a rich variety of dynamic behavioural modes of the cAMP signalling system. Starting from a low initial value of fee (fig- 6.3a), the system evolves toward a stable steady state, represented by the value of the extracellular cAMP concentration, yo-Around k = 2.4 min (fig. 6.3b), a subcritical Hopf bifurcation occurs beyond which the steady state becomes unstable (dashed line) in a range roughly extending from k - 2.2 to 2.4 min in the conditions of fig. 6.3, the system thus admits a coexistence between a stable steady state and a stable limit cycle represented by the upper solid line showing the maximum cAMP level in the course of oscillations, y these two stable solutions are separated by an imstable limit cycle (dashed line). 

The OPT occurs at p = 1 via a subcritical Hopf-type bifurcation where the system settles to a uniform precession state with a small reorientation amplitude (A 7T so that 0 system switches back to the unperturbed state at p = p] 0.88 where a saddle-node bifurcation occurs. The trajectory in the ux, Uy) plane is a circle whereas, in a coordinate system that rotates with frequency /o around the z axis, it is a fixed point. The time Fourier spectra of the director n have one fundamental frequency /o, whereas 0 , 4> and A do not depend on time. 

Figure 8.15.3-Figure 8.17.3. The oscillations having small and the ones having large amplitudes are analyzed separately. In the vicinity of the bifurcation point denoting the transition between the stable and unstable steady states, that is, when the chaotic behavior emerges, the approximately constant periods between oscillations and linear response of the squares of amplimdes with respect to control parameter (temperature) can be noted for both type of oscillations. Thus, two intersections between mentioned straight lines and abscissa can be determined for every experimental series (Tc-large and Tc-small). These intersections could be considered as the bifurcation points. The linear response of the squares of amplitudes with respect to control parameter (temperature) was found, but these intersections cannot be simple Hopf bifurcation points since both type of oscillations have the intersections with abscissa at a temperature where the stable steady state is found. They cannot correspond to subcritical Hopf bifurcation point since hysteresis is not obtained. Moreover, the bifurcation point here is a complex one with two kinds of oscillations that emerge from it.

I- "Minimal", Subcritical Hopf Bifurcation Nucleation Induced Transitions [8]... 

Recall that a Hopf bifurcation is termed supercritical if its bifurcation diagram is as shown schematically in Fig. 6.2.2a. Correspondingly, in this case a stable limit cycle is born around the equilibrium, unstable hereon, only at a critical (bifurcation) value of the control parameter A = Ac. In contrast, in the subcritical case (Fig. 6.2.2b), the equilibrium is surrounded by limit cycles already for A < Ac, with an unstable limit cycle separating the stable one from the still stable equilibrium. At the bifurcation A = Ac the unstable limit cycle dies out with the equilibrium, unstable hereon, surrounded by a stable limit cycle. Thus the main feature of the subcritical case (as opposed to the supercritical one) is that a stable equilibrium and a stable limit cycle coexist in a certain parameter range, with a possibility to reach the limit cycle through a sufficiently strong perturbation of the equilibrium. 

Fig. 5.3. Locus of Hopf bifurcation points in K-fi parameter plane for thermokinetic model with the full Arrhenius temperature dependence and y = 0.21. The nature of the Hopf bifurcation point and, hence, the stability of the emerging limit cycle changes along this locus at k = 2.77 x 10 3. Supercritical bifurcations are denoted by the solid curve, subcritical bifurcations occur along the broken segment, i.e. at the upper bifurcation point for the lowest k. The stationary-state solution is unstable and surrounded by a stable limit cycle for all parameter values within the enclosed region. Oscillatory behaviour also occurs in the small shaded region below the Hopf curve, where the stable stationary state is surrounded by both an unstable and...

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. ... 

These requirements specify two loci one of them, labelled DH l in Fig. 8.12, emanates from the points / = 0, k2 = 9/256, as located in 8.3.6. This curve cuts through the parameter space for isola and mushroom patterns, but always lies below the curve A. (In fact it intersects A at the common point P0 = i(33/2 - 5), k2 = rg(3 - /3)4(1 -, /3)2 where the locus H also crosses.) In the vicinity of DH x, the stationary-state curve has only one Hopf point. This changes from a subcritical bifurcation (unstable limit cycle emerging) for conditions to the right of the curve to supercritical (stable limit cycle emerging) to the left. 

Two other points are marked, one along each Hopf curve. These are the degenerate bifurcation points at which the emerging limit cycle changes from stable (supercritical) to unstable (subcritical). These have the locations... 

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