Hopf bifurcation instability

The question of what happens to the system in the range of instability, and how the concentrations of A and B vary as they move away from the unstable stationary state, leads us to the study of sustained oscillatory behaviour. Before a full appreciation of the latter can be obtained, however, we must rehearse the relevant theoretical background. Fortunately the autocatalytic model is again an exemplary system with which to introduce at least the basic aspects of the Hopf bifurcation, and we will do this in the next section. 

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. 

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).

First, can we expect any oscillatory behaviour Instability is possible only if k < e 2. This requirement is satisfied here. From the data in Table 4.4, the Hopf bifurcation points for this system occur for n = 0.207 and n = 0 058. For our example, the initial value /r0 = 0.5 exceeds the upper bifurcation point, so the system at first has a stable pseudo-stationary state to approach, with dss x 10 and ass x 4.54 x 10 4. From Fig. 4.3 we may also estimate that the approach to this state will be monotonic since the initial conditions lie outside the region of damped oscillations. 

The variation in oscillatory amplitude across the whole range of instability must be completed numerically (a suitable method for this is described in the appendix to this chapter). With the full Arrhenius form there are two possible scenarios, corresponding to the two different types of Hopf bifurcation at p. ... 

Fig. 5.6. Typical tc-g parameter plane for the thermokinetic model with the full Arrhenius temperature depedence a region of stationary-state instability lies within the locus of Hopf bifurcation points (solid curve). Also shown, as broken lines, are the loci corresponding to the maximum and minimum in the g(a, 0) = 0 nullcline (see text).

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...

Fig. 10.7. Representation of conditions in the K-fi parameter plane for instability of the uniform stationary state with respect to spatial perturbations for a system with f = 10. Also shown (broken curve) is the Hopf bifurcation locus, within which the uniform state is itself unstable. The two loci cross at some point on their upper shores.

The point at which the pattern instability and Hopf bifurcation loci cross, for a system with given / , is easily located by taking eqns (10.21) and (10.58). These give... 

In the previous section we have taken care to keep well away from parameter values /i and k for which the uniform stationary state is unstable to Hopf bifurcations. Thus, instabilities have been induced solely by the inequality of the diffusivities. We now wish to look at a different problem and ask whether diffusion processes can have a stabilizing effect. We will be interested in conditions where the uniform state shows time-dependent periodic oscillations, i.e. for which /i and k lie inside the Hopf locus. We wish to see whether, as an alternative to uniform oscillations, the system can move on to a time-independent, stable, but spatially non-uniform, pattern. In fact the... 

The size of the matrix as it operates on the perturbation vector is directly related to the eigenvalues of J (or of B). The eigenvalues of J are known as the Floquet multipliers fit the eigenvalues of B are the Floquet exponents / ,. In general the former are easier to evaluate, although we should identify the parameter p2 introduced in chapter 5 with the Hopf bifurcation formula as a Floquet exponent for the emerging limit cycle (then P2 < 0 implies stability, P2 > 0 gives instability, and P2 = 0 corresponds to a bifurcation between these two cases). 

When the HSS solution of the chemical rate equations (la)—(lc) first becomes unstable as the distance from equilibrium is increased (by decreasing P, for example), the simplest oscillatory instability which can occur corresponds mathematically to a Hopf bifurcation. In Fig. 1 the line DCE is defined by such points of bifurcation, which separate regions of stability (I,IV) of the HSS from regions of instability (II,III). Along section a--a, for example, the HSS becomes unstable at point a. Beyond this bifurcation point, nearly sinusoidal bulk oscillations (QHO, Fig. 3a) increase continuously from zero amplitude, eventually becoming nonlin-... 

Equation (12.93) shows that Det > 0 as long as B > k4/k2 + (k3/k2)X2 and the system remains stable, otherwise instability arises, and the system undergoes a Hopf bifurcation at... 

For different systems, we have different signs of the real and imaginary part of Landau coefficient /. Here, we will keep our attention focused to flow past a circular cylinder, that works as a prototypical model for bluff-body flow instability. This instability begins as a linear temporal instability and its first appearance with respect to the Reynolds number is referred to as Hopf bifurcation. Thus, the Reynolds number at which the first bifurcation occurs is given by Rccr- Thus, above Rccr the value of <7 > 0 signifies linear instability. One of the most important aspect of this linear instability is the subsequent non-linear saturation that can be adequately explained by the Landau s equation, if only R is positive. We will focus upon this type of flow only in the next. 

Vortex shedding behind a circular cylinder is explained theoretically as a Hopf bifurcation which is a consequence of linear temporal instability of the flow. In this point of view, the above temporal instability is moderated by nonlinearity of the system, that is quite adequately explained by Landau equation, as given in Landau (1944) and Drazin Reid (1981). Earlier numerical investigations by Zebib (1987), Jackson (1987) and Morzynski Thiele (1993) have identified the onset of vortex shedding to be at a critical Reynolds number (Rccr) between 45 and 46. 

Theoretically, flow criticality is related to the onset of global linear instability, performed numerically by Jackson (1987), Zebib (1987), Morzynski Thiele (1993), who all have reported 45 < Rccr < 46. For steady flows, we will identify this critical Reynolds number as Rccn, for the ease of future discussion. Similarly, we will identify the critical Reynolds number value indicated in Homann s experiment as Rccr Hopf bifurcation describes the passage of a dynamical system from a steady state to a periodic state as a typical bifurcation parameter is varied, that in this case is the Reynolds number (Golubitsky Schaefer (1984)). The results of the numerical investigations mentioned above, relate to study of the flow system unimpeded by noise or perturbations- barring numerical errors. 

The discussion following Eqn. (5.1.8) imply a single Hopf bifurcation when Reynolds number increases beyond Rccr It is interesting to note that Landau (1944) talked about further instabilities following the nonlinear saturation of the primary instability mode. This is akin to Floquet analysis of the resulting time periodic system (Bender Orszag (1978)). The possibility of multiple bifurcation was also mentioned in Drazin Reid (1981) who stated that in more complete models of hydrodynamic stability we shall see that there may he further bifurcations from the solution A = 0, e.g. where the next least stable mode of the basic flow becomes unstable, and from the solution A = Ae- To the knowledge of the present authors, no theoretical analysis exist that showed multiple bifurcation before for this flow. Here,... 

The examples above illustrate the benefits gained by unsteady operation. They are, however, only partially related to the phenomena dealt with in this review. The instabilities described above are externally introduced by forcing operation parameters, whereas oscillatory states in heterogeneous catalysis are inherently unstable. Because these autonomous oscillations usually arise as a Hopf bifurcation, wherein the stable state is completely lost. 

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. 

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 stationary state (x2, y2, z2) will be stable when all the roots of equation (6.106) have negative real parts. We will investigate the conditions under which this stationary state loses stability, that is under which at least one solution with a positive real part appears. Next, in the region of control parameters corresponding to instability of the state (x2, y2, z2) we shall examine possible catastrophes of codimension 2. It follows from the classification given in Section 5.5 that the bifurcations of codimension one and two of a sensitive state corresponding to the requirement = 0 are theoretically possible the Hopf bifurcation for which a sensitive state is of... 

Quantity C is always positive. In the absence of inhibition by the substrate (0 = 1 or c < 1 with 0 < 1), quantity A is also positive. The roots real part becomes positive. The steady state therefore becomes unstable as a focus (Hopf bifurcation) the passage through the critical point of instability corresponds to the occurrence of sustained oscillations in the course of time. 

Figure 5. Bifurcation diagram on the plane of the two control parameters p and a. The solid lines 1 and 2 mark the primary instability, where the homogeneous homeotropic orientation becomes unstable. At 1, the bifurcation is a stationary (pitchfork) bifurcation, and a Hopf one at 2. The two lines connect in the Takens-Bogdanov (TB) point. The solid lines 3 and 4 mark the first gluing bifurcation and the second gluing bifurcation respectively. The dashed lines 2b and 3b mark the lines of the primary Hopf bifurcation and the first gluing bifurcation when calculated without the inclusion of flow in the equations.

An interesting situation also came to light in the limit of normal incidence. This case was impossible to analyze in the framework of the approximate model, as the modes become large quickly and violate the initial assumptions. It turned out that for a = 0 (which is a peculiar case, since the external symmetry breaking in the x direction vanishes), another stationary instability precedes the secondary Hopf bifurcation that spontaneously breaks the reflection symmetry with respect to x. It is shown by point A in Fig. 18. It is also seen from this figure, that the secondary pitchfork bifurcation is destroyed in tbe case of oblique incidence, which can be interpreted as an imperfect bifurcation with respect to the angle a [43]. 

Figure 12 Constraint diagram showing parameter values of Turing bifurcation locus (left line) and Hopf bifurcation locus (right line). The Turing instability occurs for parameter values in the middle region. (Reprinted with permission from Ref. 34.)...

The stationary bifurcation and the Hopf bifurcation typically occur as one parameter is varied and are therefore known as codimension-one bifurcations. They represent the generic ways in which a steady state of a two-variable system can become unstable. It is sometimes possible to make the stationary and Hopf instability threshold coalesce by varying two parameters. Such an instability, where T = A = 0, is known as a Takens-Bogdanov bifurcation or a double-zero bifurcation, since Ai = A.2 = 0 at such a point [175], This bifurcation is a codimension-two bifurcation, since it requires the fine-tuning of two system parameters. 

Remark 10.3 The analysis of all three approaches to two-variable reaction-transport systems with inertia establishes that the Turing instability of reaction-diffusion systems is structurally stable. The threshold conditions are either the same, HRDEs and reaction-Cattaneo systems, or approach the reaction-diffusion Turing threshold smoothly as the inertia becomes smaller and smaller, t 0. Further, inertia effects induce no new spatial instabilities of the uniform steady state in the diffusive regime, T small. A spatial Hopf bifurcation to standing wave patterns can only occur in the opposite regime, the ballistic regime. 

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