Bifurcation diagram, oscillatory

Q dehydrocyclization, 29 311 ring enlargement, 29 311-316 Bifunctional Fisher-Tropsch/hydroformylation catalysts, 39 282 Bifunctional mechanism, 30 4 Bifurcation diagram, oscillatory CO/O, 37 233-234... 

FIGURE I Bifurcation diagrams of the autonomous system for y, = 0.001, y2 = 0.002. (a) Multiple-steady states are found inside the finger-shaped region, and limit cycles are born upon crossing the Hopf curves, (b) Steady-state and limit-cycle branches for < 1 = 0.019. The location of the average value of a2 used for forced oscillations is 0.028, which is in the oscillatory region and a distance of Ao from the left Hopf bifurcation point labelled c. 

Fig. 14. Bifurcation diagram for the oscillatory CO/O, reaction on Pt(IIO) as derived from the experimental data of Fig. 13. (From Ref. 71.)...

Fig. 57. Numerical, one-parameter bifurcation diagram displaying the global current of stable solutions as a function of the applied potential. In the case of the standing waves (SW), the global current is oscillatory and the open circles denote maximum and minimum of the oscillations. (hom.a. = homogeneous active stationary state hom.p. = homogeneous passive stationary state.) (Reproduced from J. Lee, J. Christoph, P. Strasser, M. Eiswirth and G. Ertl, J. Chem. Phys. 115 (2001), 1485 by permission of the American Institute of Physics.)...

Dynamic behavior has been studied under galvanostatic conditions as a function of tbe preset current and tbe bulk concentrations of Cu, Cr, and Br. Since the latter parameters act on the kinetics of the reaction, these bifurcation diagrams are useful when trying to identify the chemical steps of the model. The phase diagrams in the halide concentration/cur-rent-density plane are reproduced in Fig. 16. It can be seen that the critical current density for the onset of oscillatory behavior increases with decreasing halide concentration. In the Cu concentration/current-density plane, the opposite trend was found. Note also that the Br concentration... 

Equation (16) was originally derived to model the reduction of In " from SCN solution on the HMDE. The bifurcation behavior of this system is summarized in the two-parameter bifurcation diagram in Fig. 29. Most remarkably, the two distinct MMO sequences of the model also show up in the experiment. Farey sequences were observed close to the Hopf bifurcation at low values of the series resistance, whereas at the high resistance end of the oscillatory regime, periodic-chaotic mixed-mode sequences were found. Owing to this good agreement of the bifurcation... 

A last type of dynamic phenomenon introduced by the recycling reaction is that of a multiplicity of oscillatory domains as a function of a control parameter. This phenomenon is apparent in the bifurcation diagram of fig. 3.6h. Here again, the interest of the phenomenon stems from its relationship to the behaviour of certain neurons the model provides a straightforward explanation for the neuronal behaviour in terms of phase plane analysis. 

The existence of two distinct oscillatory domains not only depends on the shape of the product nullcline, but also on the time scale structure of the system. Thus, fig. 3.15 shows three bifurcation diagrams obtained for increasing values of parameters q and k, the ratio qtk is held constant so that the product nullcline remains unchanged. As indicated by eqns (3.1), the time evolution of the product becomes more and more... 

Fig. 3.15. Effect of multiple time scales on the oscillatory behaviour of the model. The bifurcation diagram, similar to those of fig. 3.6, is established as a function of parameter (qv/k ) for three increasing values of parameter q. The ratio (q/k ) is fixed in such a manner that the product nullcline, locus of the steady state, remains unchanged. The existence of two distinct domains of oscillations disappears when the product varies more rapidly than the substrate, at elevated values of q (Goldbeter Moran, 1988).

As indicated by the above bifurcation diagrams, the three-variable system (6.3) is capable of displaying different modes of simple or complex oscillatory behaviour. One additional mode is that of birhythmicity for certain values of the parameters, eqns (6.3) admit a coexistence between two simultaneously stable periodic regimes. In the phase plane (pr, a, y), these two regimes correspond to two limit cycles, one of which possesses a smaller amplitude and the second the folded appearance characteristic of bursting (fig. 6.6). 

As a complement, a one-dimensional (ID) bifurcation diagram is a plot of one constraint (or parameter) versus a norm or another characteristic of the displayed dynamic regime. The norm can simply be one of the dynamic variables (time averaged in the case of oscillatory regimes) or a combination of more variables, for example, a Euclidean... 

Schreiber, L Ross, J. Mechanisms of oscillatory reactions deduced from bifurcation diagrams. J. Phys. Chem. 2003,107, 9846-9859. 

Fig. 5.8 Bifurcation diagram of the system (Eqs. 5.28 and 5.29). Region IV corresponds to oscillatory behaviour...

In order to really understand a system, we must study it under a variety of conditions, that is, for many different sets of control parameters. In this way, we will be able to observe whether bifurcations occur and to see how the responses of the system, such as steady-state concentrations or the period and amplitude of oscillations, vary with the parameters. Information of this type, which summarizes the results of a number of time series, is conveniently displayed in a constraint-response plot, in which a particular response, like a concentration, is plotted against a constraint parameter, like the flow rate. If the information is available, for example, from a calculation, unstable states can be plotted as well. Bifurcations appear at points where the solution changes character, and constraint-response plots are sometimes called bifurcation diagrams. An experimental example for a system that shows bistability between a steady and an oscillatory state is shown in Figure 2.13. 

Fig. 12. Nullclines (top) and schematic bifurcation diagram of a thermokinetic model for different parameters exhibiting oscillatory behavior after [57].

Fig. 15. Skeleton bifurcation diagram of the reconstruction model of CO oxidation on Pt(l 10) Equations (4) and of the reduced model Equations (5). Regions A and C are excitable, B oscillatory and D bistable. The latter is subdivided depending on whether one (Di) or two front solutions (Di) exist.

---