Global bifurcation diagram

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 12 Global bifurcation diagrams for eq. (18) near a wigwam singularity. 

The Lorenz equations have a simple structure and contain two nonlinear terms only. Let us briefly consider the main bifurcations in the system (42) (a more detailed analysis can be found in Ref. 181). We fix the parameters ct = 10, b = and vary the parameter r in this case two global bifurcations take place (see the bifurcation diagram in Fig. 20). For r = 1, a supercritical... 

Fig. 2.7 (a) Temporal variation of the membrane potential V and the intracellular calcium concentration S in the considered simple model of a bursting pancreatic cell, (b) Bifurcation diagram forthe fast subsystem the black square denotes a Hopf bifurcation, the open circles are saddle-node bifurcations, and the filled circle represents a global bifurcation, (c) Trajectory plotted on top of the bifurcation diagram. The null-cline forthe slow subsystem is shown dashed. 

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.)...

The number of bifurcation points does not change as crosses this hypersurface, but the relative position of the bifurcation points changes as illustrated by Figures l.d and l.e. These three hypersurfaces divide the global parameter space into different regions in each of which a different type of bifurcation diagram exists. 

Eq. (18) can have for any N, either zero, two, four,. . . or 2N bifurcation points. All the possible local bifurcation diagrams can be constructed by a method described in [1]. Moreover, it can be proven [JJ that any global bifurcation diagram of Eq. (13) must be similar to one of the local bifurcation diagrams of Eq. (18). 

The Isola and Double Limit varieties do not exist in this case. The Hysteresis variety (a =0) divides the a. space into two regions (a > 0 and a. < 0) corresponding to the two bifurcation diagrams shown in Figures 2.a and 2.b. These two are also the only possible global bifurcation diagrams (0 vs. Da) for Eq. (13) as the Hysteresis variety (B 4) divides the B. space into two regions. 1 1... 

It was shown in [6J that the Hysteresis and Isola varieties divide the (a-.o jCX-) space into seven regions with different bifurcation diagrams. A construction of the Hysteresis and Isola varieties of the steady-state Eq. (11) has shown that the seven bifurcation diagrams shown in Figure 4 are the only ones that exist in the global parameter space (a,B,9, y) [4J. 

The steady-state equations describing lumped parameter systems in which several reactions occur simultaneously contain a very large number of parameters. Thus, it is impractical to conduct an exhaustive parametric study to determine their features. The new technique presented here predicts qualitative features of these systems such as the maximum number of solutions, parameter values for which these solutions exist and all the local bifurcation diagrams. Construction of the three varieties enables the division of the global parameter space into regions with different bifurcation diagrams. 

In the bifurcation diagram shown in Fig. 85, the plane of control parameters was divided into regions of a qualitatively different character of phase trajectories (the shapes of these trajectories are given in the respective regions) and the lines on which occur sensitive states of the Hopf bifurcation and the saddle bifurcation were marked. The diagram also depicts the line of sensitive states of the global bifurcation the appearance of a cycle from the branches of saddle separatrices. 

Fig. 3.6. A global bifurcation diagram of rotating Archimedean spirals with rotation frequency oj > 0 and normal tip velocity (7(0).

The above discussion focused on the suppression of chaos by periodic oscillations. As shown by the bifurcation diagram in fig. 6.3, other dynamic transitions may be brought about by the coupling of two populations endowed with distinct d5mamic properties the global behaviour... 

There is a straightforward relation between lu or ul transitions and concentration shifts. If we denote lu transition for the fth species with respect to change in the inflow of the yth species by + and the opposite transition by —, then we obtain a matrix equal to AXsymb. provided that the dependence of concentration of each species on any constraint is monotonous along the upper and lower branch [9], While the concentration shift matrix reflects local sensitivity of the steady state with respect to constraints, the bifurcation diagram is global in nature. This implies that any two columns of AXgymb determined from the lu/ul transitions are either the same or opposite, and the corresponding tilt is... 

The bifurcation diagrams are shown in Figs. 13.7.20-13.7.23. The sepa-ratrix values A and A2 are defined as derivatives of the global maps near the heteroclinic orbits Fi and F2 on the two-dimensional invariant manifold. Note that the other cases of combinations of the signs of Ai,2 and of saddle values can be obtained similarly by a reversal of time and a permutation of the sub-indices 1 and 2 . 

Fig. 7. (a) The amplitudes and (b) the period T of the oscillations found from integrating Eq. (27), N= 3, for varying values of n. Except for the value n = 4, global limit cycle attractors were found. The arrows on the right-hand side of the diagrams represent the theoretical values for the piecewise linear equation in the limit The arrow on the left-hand ride of (b) is the period predicted by the Hopf bifurcation theorem. 

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