Hopf points

Figure C3.6.7 Cubic (jir = 0) and linear (r = 0) nullclines for tire FitzHugh-Nagumo equation, (a) The excitable domain showing trajectories resulting from sub- and super-tlireshold excitations, (b) The oscillatory domain showing limit cycle orbits small inner limit cycle close to Hopf point large outer limit cycle far from Hopf point.

In the next chapter we will derive additional aspects about the birth and growth of oscillations in this system. The natural frequency a>0 of the oscillations as they are born at one or other of the Hopf points can be evaluated explicitly and is related to the stationary-state dimensionless temperature rise ... 

If n2 is positive, the limit cycle grows as fx increases beyond the Hopf point fx. The magnitude of fx2, as well as its sign, is of significance, governing the growth of oscillatory amplitude which increases as... 

The natural frequency or period of the oscillations at the Hopf point can be calculated from eqn (4.56). The growth in period as we move away from n then follows the leading-order form... 

Development of oscillations away from Hopf points... 

We can see from eqn (8.42) that k2 must be small if there are to be real solutions for the Hopf point. In fact the condition k212 < i is exactly the same as that for the existence of isolas (k2 < iV)- Thus all isolas have a point of Hopf bifurcation along their upper branch. 

We should also consider the behaviour along the top of the isola, on the part of the branch lying at longer residence times than the Hopf point. For Tres > t s, and with k2 still in the above range, the uppermost stationary state is unstable and is not surrounded by a stable limit cycle. The system cannot sit on this part of the branch, so it must eventually move to the only stable state, that of no conversion. Thus we fall off the top of the isola not at the long residence time turning point, but earlier as we pass the Hopf bifurcation point. 

We will see that, in fact, the isola region above the curve A does not have any Hopf points the isola region below A shows similar behaviour to that observed when / 0 = 0, with one Hopf point on the upper shore. 

With mushroom patterns, below A there is one Hopf point, on the uppermost branch (Fig. 8.10(e)). Above A, to the left of the tangency between this curve and the hysteresis line, the mushroom has two points of Hopf bifurcation one on the upper branch, the other on the lowest branch at long residence times (Fig. 8. lOf). Above A, to the right of the tangency, both Hopf points are on the uppermost branch of the mushroom. If we move too far away from A, these two Hopf points may merge and disappear. This latter behaviour is typical of the second form of degeneracy which we consider now. 

We may use the above description to write down the condition for two Hopf points to merge, by requiring that... 

Fig. 8.11. The locus H of degenerate Hopf bifurcation points described by the transversality condition (merging of two Hopf points), eqn (8.51). Below this curve, the stationary-state locus exhibits Hopf bifurcation (dynamic instability) at some residence times above it, the system does...

We have seen that the emerging limit cycle can be stable or unstable, depending on the value of k2, for the case / 0 = 0. The condition for the change in stability is that the exponent / 2 describing the stability of a limit cycle passes through zero at the Hopf point. We can follow this third type of degeneracy as a curve across the parameter plane by specifying that... 

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. 

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. 

For the particular example in Fig. 9.10, the Hopf point occurs for /i0 1.105. The two turning points are located at n0 = 0.72 and 0.636. This means that for reactant concentrations in the range 0.72 < fi0 < 1.105, the system has a unique stationary-state profile which is unstable. Under such conditions, the reaction will exhibit time-dependent as well as spatially dependent solutions, i.e. there is a limit cycle. Some representative non-stationary profiles are shown in Fig. 9.12. 

As an example, consider the horizontal cut obtained by varying r with p = 0.017. This cut does not traverse the region of multiplicity, so the reaction rate varies monotonically with r. Nor do we intersect the curve of Hopf points LK. We do, however, encounter the other Hopf curve, NM. The stationary... 

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). 

FIGURE 8 Detailed bifurcation diagram for variations in the reactant partial pressures (a, and a2) when y, = 0.001 and y2 = 0.002. The middle diagram is the superposition of the four surrounding bifurcation curves. The points K, L, M and N correspond to double zero eigenvalues and the points G and H are metacritical Hopf points. 

FIGURE 11 Phase portraits for the point labelled S in Figure 8 and its four neighbours T, U, V and W. The curves of turning points and Hopf points correspond to different steady states. 

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. 

Stable, limit cycle. The latter occurs in the Salnikov case and the modified bifurcation diagram is shown in Fig. 5.11(b). The stable limit cycle born at the lower Hopf point overshoots the upper Hopf point but is extinguished by colliding with the unstable limit cycle born at the upper Hopf point which also grows in amplitude as )jl is increased. Over a, typically narrow range, then there are two limit cycles, one unstable and one stable around the (stable) steady-state point. If we start with the system at some large value of /r, so we settle onto the steady-state locus, and then decrease the parameter, we will first swap to oscillations at the Hopf point /r - At this point there is a stable limit cycle available as the system departs from the now unstable steady-state, but this stable limit cycle is not born at this point and so already has a relatively large amplitude. We would expect to... 

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...

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