Period-doubling bifurcation

The jS-cell model displays chaotic dynamics in the transition intervals between periodic spiking and bursting and between the main states of periodic bursting. A careful description of the bifurcation diagram involves a variety of different transitions, including Hopf and saddle-node bifurcations, period-doubling bifurcations, transitions to inter-mittency, and homoclinic bifurcations. 

The next problem to consider is how chaotic attractors evolve from tire steady state or oscillatory behaviour of chemical systems. There are, effectively, an infinite number of routes to chaos [25]. However, only some of tliese have been examined carefully. In tire simplest models tliey depend on a single control or bifurcation parameter. In more complicated models or in experimental systems, variations along a suitable curve in the control parameter space allow at least a partial observation of tliese well known routes. For chemical systems we describe period doubling, mixed-mode oscillations, intennittency, and tire quasi-periodic route to chaos. 

Period-n Limit Cycles As a increases, the system undergoes an infinite sequence of successive period-doublings via pitchfork bifurcations. In general,... 

We know that as a is increased, / undergoes a sequence of period-doubling bifurcations on its way towards chaos. 

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... 

A theoretical framework for considering how the behavior of dynamical systems change as some parameter of the system is altered. Poincare first apphed the term bifurcation for the splitting of asymptotic states of a dynamical system. A bifurcation is a period-doubling, -quadrupling, etc., that precede the onset of chaos and represent the sudden appearance of a qualitatively different behavior as some parameter is varied. Bifurcations come in four basic varieties flip bifurcations, fold bifurcations, pitchfork bifurcations, and transcritical bifurcations. In principle, bifurcation theory allows one to understand qualitative changes of a system change to, or from, an equilibrium, periodic, or chaotic state. 

With this identification, the stable stationary-state behaviour (found for the cubic model with 1 < A < 4) corresponds to oscillations for which each amplitude is exactly the same as the previous one, i.e. to period-1 oscillatory behaviour. The first bifurcation (A = 4 above) would then give an oscillation with one large and one smaller peak, i.e. a period-2 waveform. The period doubling then continues in the same general way as described above. The B-Z reaction (chapter 14) shows a very convincing sequence, reproducing the Feigenbaum number within experimental error. 

Let us imagine a scenario for which a supercritical Hopf bifurcation occurs as one of the parameters, fi say, is increased. For fi < fi, the stationary state is locally stable. At fi there is a Hopf bifurcation the stationary state loses stability and a stable limit cycle emerges. The limit cycle grows as ft increases above fi. It is quite possible for there to be further bifurcations in the system if we continue to vary fi. With three variables we might expect to have period-doubling sequences or transitions to quasi-periodicity such as those seen with the forced oscillator of the previous section. Such bifurcations, however, will not be signified by any change in the local stability of the stationary state. These are bifurcations from the oscillatory solution, and so we must test the local stability of the limit cycle. We now consider how to do this. 

Fig. 13.17. Floquet multipliers lying within the unit circle, indicating a stable periodic motion if the CFM leaves the unit circle through — 1 a period doubling occurs if it goes out through + 1 there is a saddle-node bifurcation with the disappearance of the periodic solution.

This study [14] has shown that a period-doubling bifurcation associated with the Fermi resonance occurs in this subsystem at the energy E = 3061.3 cm-1 (with Egp = 0). Below the Fermi bifurcation, there exist edge periodic orbits of normal type, which are labeled by ( i, 22 -)normai- At the Fermi bifurcation, a new periodic orbit of type (2,1°, ->Fermi appears by period doubling around a period of 2T = 100 fs. This orbit is surrounded by an elliptic island that forms a region of local modes in phase space. Therefore, another family of edge periodic orbits of local type are bom after the Fermi bifurcation that may be labeled by the integers (n n , -)iocai- They are distinct... 

At still higher energies, the elliptic island undergoes a typical cascade of bifurcations in which subsidiary elliptic islands of periods 6, 5, 4, 3 are successively created, which leads to the global destruction of the main elliptic island to the benefit of the surrounding chaotic zone. The cascade ends with a period-doubling bifurcation at Ed, above which the periodic orbit 0 is hyperbolic with reflection, and the main elliptic island has disappeared... 

The most important characteristic in our test cases, however, is that within the 1/1 and the 2/1 resonance horns the torus will break as FA increases. In all models this happens when the unstable source period 1 that existed within the torus hits the saddle-periodic trajectories that lie on the torus. This occurs through a saddle-node bifurcation in the 1/1 resonance horn [Fig. 8(d)], and through an unstable period doubling in the 2/1 resonance [Fig. 8(c)]. After these bifurcations the basic structure of the torus has collapsed, and we are left only with the stable entrained periodic trajectories. 

The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-peri-odic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations. 

The three standard local codimensional-one bifurcations are the saddle-node, Hopf, and period doubling bifurcations and several have been continued numerically for this model and appear in figure 2. We have chosen not to show the curves of focus-node transitions because they do not represent any changes in stability, only changes in the approach to the steady behaviour. The saddle-node bifurcations that occur during phase locking of the torus at low amplitudes continue upward and either close upon themselves as in the case of the period 3 resonance horns or the terminate in some codimension-two bifurcation. 

FIGURE 4 Illustration of the three qualitatively different period doublings that occur on the segments FUE, EJ and JF. Point E has two Floquet multipliers at — I and point J is a metacritical period doubling bifurcation point. 

The excitation diagram was found to contain saddle-node, Hopf, period doubling, and homoclinic bifurcations for the stroboscopic map. In addition, many of these co-dimension one bifurcation curves were found to meet at the following co-dimension two bifurcation points Bogdanov points (double +1 multipliers), points with double -1 multipliers, points with multipliers at li and H, metacritical period-doubling points, and saddle-node cusp points. 

Guevara, M., Glass, L. Shrier, A. 1981 Phase locking, period-doubling bifurcations and irregular dynamics in periodically stimulated cardiac cells. Science, Wash. 24, 1350. 

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