Bifurcation boundary

This bifurcation boundary is plotted in Fig. C.2.1. The corresponding expression for q is q = 2a /9 > 0. Therefore, at / = 0, the equilibria Oi,2 have a pair of pure imaginary characteristic exponents, namely,... 

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... 

The lower a graph is more interesting. While initially the Poincar6 phase portrait looks the same as before (point E, inset 2c) an interval of hysteresis is observed. The saddle-node bifurcation of the pericxiic solutions occurs off the invariant circle, and a region of two distinct attractors ensues a stable, quasiperiodic one and a stable periodic one (Point F, inset 2d). The boundary of the basins of attraction of these two attractors is the one-dimensional (for the map) stable manifold of the saddle-type periodic solutions, SA and SB. One side of the unstable manifold will get attract to one attractor (SC to the invariant circle) while the other side will approach die other attractor (SD to die periodic solution). 

FIGURE7-1 Schematic of flow in daughter brandies of bifurcation model for steady inspiratory flow with flat profile in parent bran<. Velocity profiles in plane of bifurcation (—) and in normal plane (—) are indicated in right branch. Orientation of secondary flows and position of laminar boundary layer are shown in left branch. Redrawn wifo permisnon from Bdl. ... 

To calculate more precisely the average uptake or the local variation in uptake in each airway, the local variations in velocity and concentration profiles must be taken into account. For example, thin momentum and concentration boundary layers occur at bifurcations and gradually increase in thickness with distance downstream. Bell and Friedlander showed that particle and gas transfer to the airway wall is greatest where the boundary layers are thinnest, e.g., at the carina or apex of bifurcations. 

Ka can be defined as a gas-phase transfer coefficient, independent of the liquid layer, when the boundary concentration of the gas is fixed and independent of the average gas-phase concentration. In this case, the average and local gas-phase mass-transfer coefficients for such gases as sulfur dioxide, nitrogen dioxide, and ozone can be estimated from theoretical and experimental data for deposition of diffusion-range particles. This is done by extending the theory of particle diffusion in a boundary layer to the case in which the dimensionless Schmidt number, v/D, approaches 1 v is the kinematic viscosity of the gas, and D is the molecular diffusivity of the pollutant). Bell s results in a tubular bifurcation model predict that the transfer coefficient depends directly on the... 

Figure 26.56 is the corresponding plot for 12% inlet H2 in air. In this case, there is an extinction at about 1000 K for both reactors. The qualitative features are similar to that of the PSR discussed above for 28% H2 in air. For such fuel-lean mixtures, the flame is attached to the surface. As a result, the thermal coupling between the surface and the gas phase is strong, and reduction in surface temperature affects the entire thermal boundary layer resulting in significant reduction of NOj,. These results indicate that the bifurcation behavior, in terms of extinction, determines the role of flame-wall thermal interactions in emissions. 

One of the main themes of this volume is the influence of the environment on chemical reactivity. Such a question is of special interest for chemical systems in far from equilibrium conditions. It is today well known that, far from equilibrium, chemical systems involving catalytic mechanisms may lead to dissipative structures.1-2 It has also been shown—and this is one of the main themes of this discussion—that dissipative structures are very sensitive to global features characterizing the environment of chemical systems, such as their size and form, the boundary conditions imposed on their surface, and so on. All these features influence in a decisive way the type of instabilities, called bifurcations, that lead to dissipative structures. 

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.

There are two more important advantages of these models. One is that it is possible under some conditions to carry out an exact stability analysis of the nonequilibrium steady-state solutions and to determine points of exchange of stability corresponding to secondary bifurcations on these branches. The other is that branches of solutions can be calculated that are not accessible by the usual approximate methods. We have already seen a case here in which the values of parameters correspond to domain 2. This also happens when the fixed boundary conditions imposed on the system are arbitrary and do not correspond to some homogeneous steady-state value of X and Y. In that case Fig. 20 may, for example,... 

A plot of equilibrium r/e as a function of the external parameter a is schematically presented in Fig. 2.3.5a. The plot in Fig. 2.3.5a is markedly different from that in Fig. 2.3.4a by its lack of bifurcation. (Uniqueness of the appropriate solutions of the Poisson-Boltzmann equation for any values of a is proved in [18].) In the (F, ri) or (F, aeS) plane this corresponds to the existence of solutions of the Poisson-Boltzmann equations with finite F (bounded norm of the appropriate solution with a subtracted singular part due to the effective line charge) only for aeS < with adetermined by Conjecture 2.1. This is schematically illustrated in Fig. 2.3.5b. Note that F as a function of creS is constructed in a single counterion case by solving (2.3.3a) with a = of J e rdr and with the boundary conditions tp(a) = -aeS lna, = 0, and by going to the limit a- 0. 

In the next section we will investigate the behaviour shown by systems which contain both reactant A and autocatalyst B in the inflow. Our aim is to map out the regions of the parameter plane (i.e. the combinations of k2 and / ) for which qualitatively different responses are found. The boundaries between these regions are the loci of degenerate bifurcations, so we wish to follow these as P0 varies. 

Fig. 8.9. The locus A of double-zero eigenvalue degeneracies of the Hopf bifurcation for cubic autocatalysis with decay. Also shown, as broken curves, are the loci of stationary-state degeneracies, corresponding to the boundaries for isola and mushroom patterns. The curve A lies completely within the parameter regions for multiple stationary states.

Fig. 12.6. (a) Hopf bifurcation loci for the Takoudis-Schmidt Aris model with k, = 10-3 and k2 = 2x 10-3. Also shown (broken curves) are the saddle-node boundaries from Fig. 12.6. (b)-(i) The eight qualitative arrangements of Hopf and saddle-node bifurcation points. (Adapted and reprinted with permission from McKarnin, M. A. et al. (1988). Proc. R. Soc., A415,... 

Let us examine more closely what occurs on the right-hand boundary of the 1/1 resonance horn [Fig. 9(a)]. In a sequence of one-parameter bifurcation diagrams with respect to oj/co0, each taken at a successively higher forcing amplitude FA, we observe that as FA increases, the bifurcation point to a torus changes. The point of exit of the Floquet multipliers of the periodic... 

The point S of figure 8 at which the Hopf bifurcation curve crosses the boundary of the multiplicity region is not a double zero degeneracy, for the upper steady state (i.e. that with the larger 0b) is undergoing the Hopf bifurcation at the same time as the lower steady-state undergoes a saddle-node bufurcation, i.e. the conditions trJ = 0 and detJ = 0 apply at different points. It does, however, serve to show the four combinations of the two most common... 

In the 1/1 entrainment region each side of the resonance horn terminates at points C and D respectively. These points are codimension-two bifurcations and correspond to double +1 multipliers. As the saddle-node curve at the right horn boundary rises from zero amplitude towards point D, one multiplier remains at unity (the criterion for a saddle-node bifurcation) as the other free-multiplier of the saddle-node increases until it is also equal to unity upon arrival at point D. The same thing occurs for the left boundary of the resonance horn. The arc CD is also a saddle-node bifurcation curve but is different from those on the sides of the resonance horn. As arc CD is crossed from below, the period 1 saddle combines not with its companion stable node, but with the unstable node that was in the centre of the phase locked torus. As the pair collides, the invariant circle is lost and only the stable node remains. Exactly the same scenario is observed for the 1/2 resonance horn as well. 

Several codimension-two bifurcations have already been mentioned. Although they occur in restricted subspaces of parameter space and would therefore be difficult to locate experimentally, their usefulness lies in their role as centres for critical behaviour. Emanating from each local codimen-sion-two point will be two or more of the above codimension-one bifurcation curves. Their usefulness in studying dynamics is akin to that of the triple point in thermodynamic phase equilibria in which boundaries between three different phases come together at a point in a two-parameter diagram. Because some of these codimension-two points have been studied and classified analytically, finding one can provide clues about what other codimension-one bifurcation curves to expect near by and thus aids in the continuation of all of the bifurcation curves in the excitation diagram. 

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