Bifurcation diagrams

Figure A3.14.3. Example bifurcation diagrams, showing dependence of steady-state concentration in an open system on some experimental parameter such as residence time (inverse flow rate) (a) monotonic dependence (b) bistability (c) tristability (d) isola and (e) musliroom.

Changing the constants in the SCF equations can be done by using a dilferent basis set. Since a particular basis set is often chosen for a desired accuracy and speed, this is not generally the most practical solution to a convergence problem. Plots of results vs. constant values are the bifurcation diagrams that are found in many explanations of chaos theory. 

Fig. 44. Bifurcational diagram for the potential (6.13) in the (Qo, P) plane. Domains (i), (ii) and (iii) correspond to Arrhenius dependence (stepwise thermally activated transfer), two-dimensional instanton and one-dimensional instanton (concerted transfer), respectively.

The bifurcational diagram (fig. 44) shows how the (Qo,li) plane breaks up into domains of different behavior of the instanton. In the Arrhenius region at T> classical transitions take place throughout both saddle points. When T < 7 2 the extremal trajectory is a one-dimensional instanton, which crosses the maximum barrier point, Q = q = 0. Domains (i) and (iii) are separated by domain (ii), where quantum two-dimensional motion occurs. The crossover temperatures, Tci and J c2> depend on AV. When AV Vq domain (ii) is narrow (Tci — 7 2), so that in the classical regime the transfer is stepwise, while the quantum motion is a two-proton concerted transfer. This is the case when the tunneling path differs from the classical one. The concerted transfer changes into the two-dimensional motion at the critical value of parameter That is, when... 

Figure 6-12 is the bifurcation diagram, in which the quantity q is the bifurcation parameter. The ordinates of the curve represent radii of cycles, and stability and instability is indicated by and o, respectively. If one starts with negative values of q, the origin 0 is unstable, and the... 

We discuss some characteristics of the bifurcation diagram of the K-S for low values of the parameter a (for details see (7)). We will work in one dimension and with periodic boundary conditions. In what follows, we always subtract the mean drift... 

Expanding the sample size to 2Xc admits the other shape families shown on Fig. 6 into the analysis and leads to additional codimension-two interactions between the shapes is the (1A<.)- family and shapes with other numbers of cells in the sample. The bifurcation diagram computed for this sample size with System I and k = 0.865 is shown as Fig. 11. The (lAc)- and (Ac/2)-families are exactly as computed in the smaller sample size, but the stability of the cell shapes is altered by perturbations that are admissible is the larger sample. The secondary bifurcation between the (lAc)- and (2Ae/3)-families is also a result of a codimension two interaction of these families at a slightly different wavelength. Two other secondary bifurcation points are located along the (lAc)-family and may be intersections with the (4Ac and (4A<./7) families, as is expected because of the nearly multiple eigenvalues for these families. 

Leaving the details, the equation describing the motion of one particle in two electrostatic waves allows perturbation methods to be applied in its study. There are three main types of behavior in the phase space - a limit cycle, formation of a non-trivial bounded attracting set and escape to infinity of the solutions. One of the goals is to determine the basins of attraction and to present a relevant bifurcation diagram for the transitions between different types of motion. 

Figure 38, Chapter 3. A bifurcation diagram for the model of the Calvin cycle with product and substrate saturation as global parameters. Left panel Upon variation of substrate and product saturation (as global parameter, set equalfor all irreversible reactions), the stable steady state is confined to a limited region in parameter space. All other parameters fixed to specific values (chosen randomly). Right panel Same as left panel, but with all other parameters sampled from their respective intervals. Shown is the percentage r of unstable models, with darker colors corresponding to a higher percentage of unstable models (see colorbar for numeric values).

Figure 39, Chapter 3. Bifurcation diagrams for the model of the Calvin cycle for selected parameters. All saturation parameters are fixed to specific values, and two parameters are varied. Shown is the number of real parts of eigenvalues larger than zero (color coded), with blank corresponding to the stable region. The stability of the steady state is either lost via a Hopf (HO), or via saddle node (SN) bifurcations, with either two or one eigenvalue crossing the imaginary axis, respectively. Intersections point to complex (quasiperiodic or chaotic) dynamics. See text for details.

Besides the two most well-known cases, the local bifurcations of the saddle-node and Hopf type, biochemical systems may show a variety of transitions between qualitatively different dynamic behavior [13, 17, 293, 294, 297 301]. Transitions between different regimes, induced by variation of kinetic parameters, are usually depicted in a bifurcation diagram. Within the chemical literature, a substantial number of articles seek to identify the possible bifurcation of a chemical system. Two prominent frameworks are Chemical Reaction Network Theory (CRNT), developed mainly by M. Feinberg [79, 80], and Stoichiometric Network Analysis (SNA), developed by B. L. Clarke [81 83]. An analysis of the (local) bifurcations of metabolic networks, as determinants of the dynamic behavior of metabolic states, constitutes the main topic of Section VIII. In addition to the scenarios discussed above, more complicated quasiperiodic or chaotic dynamics is sometimes reported for models of metabolic pathways [302 304]. However, apart from few special cases, the possible relevance of such complicated dynamics is, at best, unclear. Quite on the contrary, at least for central metabolism, we observe a striking absence of complicated dynamic phenomena. To what extent this might be an inherent feature of (bio)chemical systems, or brought about by evolutionary adaption, will be briefly discussed in Section IX. 

Figure 29 shows the bifurcation diagram for different values of the saturation parameter 6 of the ATPase reaction. Qualitatively, the plot shows the same... 

Figure 29 Bifurcation diagram of the minimal model of glycolysis as a function of feedback strength and saturation 6 of the ATPase reaction. Shown are the transitions to instability via a saddle node (SN) and a Hopf (HO) bifurcation (solid lines). In the regions (i) and (iv), the largest real part with in the spectrum of eigenvalues is positive > 0. Within region (ii), the metabolic state is a stable node, within region (iii) a stable focus, corresponding to damped transient oscillations.

Figure 32. Bifurcation diagram of the medium complexity model of glycolysis, analogous to Fig. 29. A The largest real part of the eigenvalues as a function of the feedback strength 0 TP, depicted for increasing saturation of the overall ATPase reaction. B The metabolic state is stable only for an intermediate value of the feedback parameter. For increasing saturation of the ATPase reaction, the stable region decreases.

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... 

In this situation, a periodic variation of coolant flow rate into the reactor jacket, depending on the values of the amplitude and frequency, may drive to reactor to chaotic dynamics. With PI control, and taking into account that the reaction is carried out without excess of inert (see [1]), it will be shown that it the existence of a homoclinic Shilnikov orbit is possible. This orbit appears as a result of saturation of the control valve, and is responsible for the chaotic dynamics. The chaotic d3mamics is investigated by means of the eigenvalues of the linearized system, bifurcation diagram, divergence of nearby trajectories, Fourier power spectra, and Lyapunov s exponents. 

If the cell cycle in amphibian embryonic cells appears to be driven by a limit cycle oscillator, the question arises as to the precise dynamical nature of more complex cell cycles in yeast and somatic cells. Novak et al. [144] constructed a detailed bifurcation diagram for the yeast cell cycle, piecing together the diagrams obtained as a function of increasing cell mass for the transitions between the successive phases of the cell cycle. In these studies, cell mass plays the role of control parameter a critical mass has to be reached for cell division to occur, provided that it coincides with a surge in cdkl activity which triggers the G2/M transition. 

To investigate the effect of composition at high pressures, two-parameter bifurcation diagrams are constructed. An example is shown in Fig. 26.2. Cuts at fixed compositions are shown in Fig. 26.1. A nonextinction regime is found on each side of the stoichiometric point, within which the flame cannot be... 

Fig. 11. Bifurcation diagram for an even wave number and

When <0, the bifurcation diagram is as in Fig. 13. There exists a subcritical region in which three stable steady-state solutions may coexist simultaneously the thermodynamic branch and two inhomogeneous solutions. It must be pointed out that the latter are necessarily located at a finite distance from the thermodynamic branch. As a result, their evaluation cannot be performed by the methods described here. The existence of these solutions is, however, ensured by the fact that in the limit B->0, only the thermodynamic solution exists whereas for B Bc it can be shown that the amplitude of all steady-state solutions remains bounded. 

The bifurcation diagrams are similar to those of Figs. 11 and 13.14 The expressions for x(z) are qualitatively the same whether p is even or odd. However, a remarkable feature is the spontaneous appearance of a macroscopic gradient along the system each time that p is odd, the value of the concentration at the two boundaries 2 = 0 and 2 = 1 being different. 

Fig. 16. Bifurcation diagram of temporal dissipative structures, c (maximal amplitude of the oscillation minus the homogeneous steady-state value) is sketched versus B for a two-dimensional system with zero flux boundary conditions. The first bifurcation occurs at B = Bn and corresponds to a stable homogeneous oscillation. At B, two space-dependent unstable solutions bifurcate simultaneously. They become stable at B a and Bfb. Notice that as it is generally the case Bfa Bfb.

Fig. 19. Bifurcation diagram in which a rotating wavelike solution and the homogeneous steady state would be stable simultaneously.

In Fig. 21 we have drawn the bifurcation diagram of the fundamental steady-state solutions for three values of p [ Kxn is plotted versus UK) as the bifurcation parameter]. There is a subcritical region in the upper or lower branch, depending on the relative height of the peaks in Fig. 20c. The asymptotes K and K" of these branches correspond to half-period solutions of infinite length. When p 2 the asymptote K merges with the w-axis therefore situation 2 above can be viewed as a particular case of situation 3 above, in which the bifurcation point moves to infinity. 

These results are thus in agreement with those of bifurcation theory. In the case of odd wave numbers they demonstrate that in general the bifurcation diagrams have to exhibit a subcritical branch. However, there always exists even for odd wave numbers a value of the parameters such that the bifurcation is soft and this value marks the transition from an upper to a lower subcritical branch (see Fig. 21). This feature was less... 

Fig. 21. Bifurcation diagram of Km for three values of the ratio of diffusion coefficients. As can be seen in Fig. 20, the amplitude of the inhomogeneous solution is proportional to K n.

Fig. 23. Bifurcation diagram of KU2 in the case of Fig. 22. The lower branch of solutions corresponds to the thermodynamic branch. It tends to the asymptotic value K 12, which separates it from nonequilibrium types of solutions.

Figure 9. The two largest Lyapunov exponents (a) and the bifurcation diagram (the maxima of yi) (b) versus the modulation parameter Q. Parameters are/o — 1. — y2 — 0.01 and the initial...

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