Bifurcation scenario

It is worthwhile also mentioning that the origin of homochirality has been viewed in terms of a bifurcation scenario (Kondepudi and Prigogine, 1981 Kondepudi et al, 1985). In this case, the homochirality present on Earth would be a product of contingency. 

Bifurcation Scenario Associated with Transition to Chaos... 

Intramolecular dynamics and chemical reactions have been studied for a long time in terms of classical models. However, many of the early studies were restricted by the complexities resulting from classical chaos, Tlie application of the new dynamical systems theory to classical models of reactions has very recently revealed the existence of general bifurcation scenarios at the origin of chaos. Moreover, it can be shown that the infinite number of classical periodic orbits characteristic of chaos are topological combinations of a finite number of fundamental periodic orbits as determined by a symbolic dynamics. These properties appear to be very general and characteristic of typical classical reaction dynamics. 

We have shown elsewhere that the different bifurcation scenarios can be conveniently discussed in terms of area-preserving mappings generated by the action function [10]... 

Figure 9. Supercritical antipitchfork bifurcation scenario for symmetric XYX molecules on the left, bifurcation diagram in the plane of energy versus position in the center, typical phase portraits in some Poincare section in the different regimes on the right, the fundamental periodic orbits in position space.

Figure 10. Same as Fig. 9 for the subcritical antipitchfork bifurcation scenario for symmetric XYX. 

Other more complicated scenarios may occur if x is close to zero and changes its sign, in which case higher order terms should be included to describe the bifurcation scenario. 

The classical dynamics follows the first bifurcation scenario we discussed above, i.e., involving a supercritical antipitchfork bifurcation. 

This dissociative system, which represents the prototype system for chemical reaction dynamics, has been the object of many studies. Child et al. [143] have carried out a detailed analysis of the classical dynamics in a collinear model based on the Karplus-Porter surface. These authors have introduced the concept of PODS and first observed the subcritical antipitchfork bifurcation scenario in this system. 

The bottom of the exit channels is at -3194 cm-1 if the origin corresponds to the saddle of the Karplus-Porter surface. The pair of tangent bifurcations occur at E = 1670 cm 1, which is followed by the subcritical antipitchfork bifurcation at Ea = 2633 cm 1. The bifurcation scenario is thus similar to the CO2 system, and we may expect a three-branch Smale horseshoe in this system as well. 

The universal aspects described in Sect. 4 are contained in any ITT model that contains the central bifurcation scenario and recovers (20,22). Equation (20) states that spatial and temporal dependences decouple in the intermediate time window. Thus it is possible to investigate ITT models without proper spatial resolution. Because of the technical difficulty to evaluate the anisotropic functionals in (lid, 14), it is useful to restrict the description to few or to a single transient correlator. The best studied version of such a one-correlator model is the I -model. 

The best-known examples of electrochemical oscillators are reactions involving the anodic dissolution of a metal in acidic solution. With the exception of the complex bifurcation scenarios observed during Cu dissolution, they have not yet been discussed in this chapter. This is because their kinetics are much more complicated than those of the examples reviewed. Thus, despite the fact that oscillatory metal dissolution reactions have been an intense subject of research over decades, there does not seem to be a single example where the reaction mechanism is identified unambiguously and understood in depth. This is for the most part due to complicated passivation and reactivation kinetics which involve the for-... 

T. Heil, I. Fischer, W. Elsafier, B. Krauskopf, K. Green, and A. Gavrielides. Delay dynamics of semiconductor lasers with short external cavities Bifurcation scenarios and mechanisms. Phys. Rev. E, 67 066214, 2003. 

The bifurcation scenario discussed above was actually observed in the experiment. Although a good qualitative agreement between theory and experiment was found [40], there are quantitative discrepancies. In the experiment, the measured onset of the nutation-precession motion turns out to be about 20% lower than predicted by theory. Moreover, the slope of the precession frequency versus intensity predicted by theory turned out to be different from that observed in the experiment. One of the two possible reasons could be the use of finite beam size in the experiment (that is typically of the order of the thickness of the layer), whereas in theory the plane wave approximation was assumed. Actually, the ratio 5 between diameter of the beam and the width of the layer is another bifurcation parameter (in the plane wave approximation, 6 oo) and was shown to play crucial role on the orientational dynamics [13]. There and in [44] the importance of the so called walk-off effect was pointed out which consists of spatial separation of Pointing vectors of the ordinary and extraordi-... 

In what follows, we present a brief overview of the bifurcation scenario following [50], which is, in our opinion, complete for large and moderate values of ellipticity. 

Many features become more transparent when formulated in real (position) space in terms of ampbtude (envelope) or Ginzburg-Landau equations (GLE). Then one sees that the important information is really condensed in a few parameters and the universal aspects of the systems become apparent. By model calculations, which can often be performed analytically, stability boundaries and secondary bifurcation scenarios are traced out. The real space formulation is essential when it comes to the description of more complex spatio-temporal patterns with disorder and defects, which have been studied extensively in EHC slightly above threshold (Figs. 13.1b, 13.3b). One introduces a modulation ampbtude y4(x) defined as... 

Therefore, on varying the value of the parameter from negative to positive within the interval — 1 < /r < 1, an inversion of the bifurcation scenario with exchange of stability [94] is observed. The bifurcation occurs at the degenerate critical point for fi = Q. The trajectories are displayed on the left of Fig. 7.5. The flow is characterized by the C2v(Cj) = 2mm magnetic symmetry, see Sect. 7.5. 

Equations (29) possess up to six steady-state solutions. Among these, the solution Si = (05,01 — A iO5,0) may switch as the parameters vary from stable node to stable focus to saddle-focus, and seems to be the organizing centre of the system s periodic and chaotic attractors. Keeping all parameters but k-l fixed one observes the bifurcation scenario summarized in Figure 10. 

To study such bifurcations one should understand the structure of the limit set into which the periodic orbit transforms when the stability boundary is approached. In particular, such a limit set may be a homoclinic loop to a saddle or to a saddle-node equilibrium state. In another bifurcation scenario (called the blue sky catastrophe ) the periodic orbit approaches a set composed of homoclinic orbits to a saddle-node periodic orbit. In this chapter we consider homoclinic bifurcations associated with the disappearance of the saddle-node equilibrium states and periodic orbits. Note that we do not restrict our attention to the problem on the stability boundaries of periodic orbits but consider also the creation of invariant two-dimensional tori and Klein bottles and discuss briefly their routes to chaos. 

In terms of the original variable (p — — ut, the stationary value of (the equilibrium state of system (12.1.9)) corresponds to an oscillatory regime with the same frequency as that of the external force. The periodic oscillations of (the limit cycle in (12.1.9)) correspond to a two-frequency regime. Hence, the above bifurcation scenario of a limit cycle from a homoclinic loop to a saddle-node characterizes the corresponding route from synchronization to beat modulations in Eq. (12.1.7). 

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