Hopf bifurcations

According to equation (5.63d), the standard form of the system has the following form (in polar variables r, p, r 0)  

On converting to Cartesian coordinates, x = rcos( p), y = rsin( p) we obtain an alternative (although more complex) standard form 

Linearization of equations (5.80) nearby the stationary state yields xs( = 0exp(Af) (5.82a) 

When the parameter c 0 the stationary state (x,y) = (0, 0) is of a stable focus type for c = 0 we deal with the sensitive state the stationary state is a centre for c 0 a catastrophe takes place, because the stationary state becomes an unstable focus. The phase trajectories for the linearized system (5.83) for c 0, c = 0, c 0 nearby the stationary point are shown in Figs. 65, 66, 67, respectively. 

It remains to be settled what happens to the trajectories escaping from the unstable neighbourhood (for c 0) of a stationary point. The question may be readily answered by examining the standard form (5.79). We can see that for c 0 there exists, besides the stationary state r = 0, also a limit cycle of the form 

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Figure C3.6.5 The first two periodic orbits in the main subhannonic sequence are shown projected onto the (c, C2) plane. This sequence arises from a Hopf bifurcation of the stable fixed point for the parameters given in the text. The arrows indicate the direction of motion, (a) The limit cycle or period-1 orbit at k 2 = 0.11. (b) The first subhannonic or period-2 orbit at k 2 = 0.095.

The quasiperiodic route to chaos is historically important. It arises from a succession of Hopf birfurcations. As already noted, a single Hopf bifurcation results in a limit cycle. The next Hopf bifurcation produces a phase flow tliat can be represented on tire surface of a toms (douglmut). This flow is associated witli two frequencies if tire ratio of tliese frequencies is irrational tlien tire toms surface is densely covered by tire phase trajectory, whereas if... 

This complex Ginzburg-Landau equation describes the space and time variations of the amplitude A on long distance and time scales detennined by the parameter distance from the Hopf bifurcation point. The parameters a and (5 can be detennined from a knowledge of the parameter set p and the diffusion coefficients of the reaction-diffusion equation. For example, for the FitzHugh-Nagumo equation we have a = (D - P... 

One may also observe a transition to a type of defect-mediated turbulence in this Turing system (see figure C3.6.12 (b). Here the defects divide the system into domains of spots and stripes. The defects move erratically and lead to a turbulent state characterized by exponential decay of correlations [59]. Turing bifurcations can interact with the Hopf bifurcations discussed above to give rise to very complicated spatio-temporal patterns [63, 64]. 

This route should already be familiar to us from our discussion of the logistic map in chapter 4, Prom that chapter, we recall that the Feigenbaum route calls for a sequence of period-doubling bifurcations pitchfork bifurcations versus the Hopf bifurcations of the Landau-Hopf route) such that if subharmonic bifurcations are observed at Reynolds numbers TZi and 7 2, another can be expected at TZ determined by... 

The stability of the (lAe)-family is lost at a Hopf bifurcation point denoted by the open circle (o) on Fig. 7, where the real parts of a complex conjugate pair of eigenvalues change sign. No stable time-periodic solutions were found near this point, indicating that the time-periodic states evolve sub-critically in P and are unstable. Haug (1986) predicted Hopf bifurcations for codimension two bifurcations of the form shown in Fig. 7. but did not compute the stability of the time-periodic states. 

THE HOPF BIFURCATION OR THE CHANGING NATURE OF EQUILIBRIUM POINTS PROBLEM OP WALAS... 

Figure 31 shows the largest eigenvalue of the Jacobian at the experimentally observed metabolic state as a function of the parameter 0 TP. Similar to Fig. 28 obtained for the minimal model, several dynamic regimes can be distinguished. In particular, for sufficient strength of the inhibition parameter, the system undergoes a Hopf bifurcation and the pathway indeed facilitates sustained oscillations at the observed state. 

It is well known that self-oscillation theory concerns the branching of periodic solutions of a system of differential equations at an equilibrium point. From Poincare, Andronov [4] up to the classical paper by Hopf [12], [18], non-linear oscillators have been considered in many contexts. An example of the classical electrical non-oscillator of van der Pol can be found in the paper of Cartwright [7]. Poore and later Uppal [32] were the first researchers who applied the theory of nonlinear oscillators to an irreversible exothermic reaction A B in a CSTR. Afterwards, several examples of self-oscillation (Andronov-PoincarA Hopf bifurcation) have been studied in CSTR and tubular reactors. Another... 

From the results presented in this chapter, more advanced studies from the bifurcation theory can be planed. For example, inside the lobe, the behavior of the reactor is self-oscillating, i.e. an Andronov-Poincare-Hopf bifurcation can be researched from the calculation of the first Lyapunov value, in order to know if a weak focus may appear, or the conditions which give a Bogdanov-Takens bifurcation etc. Finally, it is interesting to remark that the previously analyzed phenomena should be known by the control engineer in order to either avoid them or use them, depending on the process type. 

From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

Figure 26.1 shows the mole fraction of H2 just above the surface vs. the surface temperature for a mixture of 10% H2 in air at various pressures. At atmospheric pressure (Fig. 26.1a), the mole fraction of H2 is almost insensitive to surface temperature until a turning point, called an ignition (/i), is reached, where the system jumps from an unreactive state to a reactive one. As the surface temperature decreases from high values, the H2 mole fraction increases, and a Hopf bifurcation (HB) point is first found at 980 K, outside the multiplicity regime. The solution branch between the HBi and the extinction is locally unstable (dashed curve). 

Since the only equilibrium point E(0 po) in the phase plane becomes unstable for i > ic and the infinity is unstable for any i, we conclude that limit cycles must exist around E(G,p0) for > ic. At the same time, the proven nonexistence of the limit cycles for i < ic implies the supercritical nature of the Hopf bifurcation at = ic in the symmetric case /"(0) = 0. 

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. 

To this end, address the equation (6.2.7) with estable limit cycle appears around the equilibrium point, i.e., a Hopf bifurcation takes place. We wish to study the solution that arises in the close vicinity of the bifurcation. Introduce a new time... 

We shall not specify here the initial conditions on C(x,t) since in what follows we shall only be preoccupied with the limit state resulting from a Hopf bifurcation from the following stationary solution of the above system... 

Let us begin by analyzing linear stability of solution (6.3.16) and showing that at some value of parameter I a Hopf bifurcation is indeed occurring. According to the common scheme, we look for a perturbed solution of the form... 

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