Bifurcation parameter value

Fig. 7.16. Chaotic trajectory arising from anomalous 2 bifurcations. Parameter values

The periodic windows in the U-sequence exist in the "chaotic regime (e.g., the regime of Section 3.2) that follows the accumulation point of the period doubling sequence [50]. There are no chaotic intervals, yet the set of bifurcation parameter values for which the behavior is chaotic has positive measure. [For a typical map probably about 85% of the measure (of the bifurcation parameter range in which the U-sequence occurs) corresponds to chaotic states, and 15% to periodic states.]... 

The tern intermittency is used to describe a transition from periodic to chaotic behavior characterized by occasional bursts of "noise [51]. For bifurcation parameter values slightly beyond that corresponding to the onset of the bursts, there are long intervals of nearly periodic behavior between the bursts, but further beyond the transition the time intervals between bursts is shorter. With further change in bifurcation parameter the intervals between bursts decrease until ultimately it is impossible to recognize the regular oscillations of the periodic states,... 

Consider some finite-parameter family of smooth systems Xg, where e = ( 1,..., 6p) assumes its values from some region V e W. If is non-rough, then q is said to be a bifurcation parameter value. The set of all such values in V is called a bifurcation set. It is obvious that once we know the bifurcation set, we can identify all regions of structural stability in the parameter space. Hence, the first step in the study of a model is identifying its bifurcation set. This emphasizes a special role of the theory of bifurcations among all tools of nonlinear dynamics. 

The procedure for studying a fc-parameter family is similar to that for the one-parameter case firstly, divide the space of the parameters p into the regions of topologically equivalent behavior of trajectories, and study the system in each of these regions. Secondly, describe the boundaries of these regions (the bifurcation set), and finally study what happens at the bifurcation parameter values. We will see below that in the simplest cases (e.g. an equilibrium state with one zero or a pair of imaginary characteristic exponents, or a periodic orbit with one multiplier equal to 1 or to —1) one can almost always, except for extreme degeneracies, choose a correct bifurcation surface of a suitable codimension and analyze completely the transverse families. Moreover, all of these families turn out to be versal. 

Figure C3.6.6 The figure shows tire coordinate, for < 0, of tire family of trajectories intersecting tire Poincare surface at cq = 8.5 as a function of bifurcation parameter k 2- As tire ordinate k 2 decreases, tire first subhannonic cascade is visible between k 2 0.1, tire value of tire first subhannonic bifurcation to k 2 0.083, tire subhannonic limit of tire first cascade. Periodic orbits tliat arise by tire tangent bifurcation mechanism associated witli type-I intennittency (see tire text for references) can also be seen for values of k 2 smaller tlian tliis subhannonic limit. The left side of tire figure ends at k 2 = 0.072, tire value corresponding to tire chaotic attractor shown in figure C3.6.1(a). Otlier regions of chaos can also be seen.

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... 

Because of nonlinear Interactions between buoyancy, viscous and Inertia terms multiple stable flow fields may exist for the same parameter values as also predicted by Kusumoto et al (M.). The bifurcations underlying this phenomenon may be computed by the techniques described In the numerical analysis section. The solution structure Is Illustrated In Figure 7 In terms of the Nusselt number (Nu, a measure of the growth rate) for varying Inlet flow rate and susceptor temperature. Here the Nusselt number Is defined as ... 

Fig. 9. Linear stability diagram illustrating (21). The bifurcation parameter B is plotted against the wave number n. (a) and (b) are regions of complex eigenvalues < > (b) shows region corresponding to an unstable focus (c) region corresponding to a saddle point. The vertical lines indicate the allowed discrete values of n. A = 2, D, = 8 103 D2 = 1.6 10-3.

In the preceding sections we have analyzed the new solutions that appear at a point of instability and have shown that they can be calculated by the methods of bifurcation theory as long as their amplitude is small. In this section we consider a system of the form (2) whose steady-state solutions can be evaluated straightforwardly without implying any restriction on the parameters value. This allows a complete analysis of these branches of solutions.32... 

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. 

Fig. 3.7. Stationary-state loci a( ) and / (/<) showing changes in local stability when the uncatalysed step is included in the model and ku <, showing two Hopf bifurcation points H and n. Particular numerical values correspond to parameter values in Table 3.1.

In this chapter we give an introduction and recipe for the full Hopf bifurcation analysis for chemical systems. Rather than work in completely general and abstract terms, we will illustrate the various stages by using the thermokinetic model of the previous chapter, with the exponential approximation for simplicity. We can draw many quantitative conclusions about the oscillatory solutions in that model. In particular we will be able to show (i)that the parameter values given by eqns (4.49) and (4.50) for tr(J) = 0 satisfy all the requirements of the. Hopf theorem (ii)that oscillatory behaviour is completely confined to the conditions for which the stationary state is... 

Fig. 5.3. Locus of Hopf bifurcation points in K-fi parameter plane for thermokinetic model with the full Arrhenius temperature dependence and y = 0.21. The nature of the Hopf bifurcation point and, hence, the stability of the emerging limit cycle changes along this locus at k = 2.77 x 10 3. Supercritical bifurcations are denoted by the solid curve, subcritical bifurcations occur along the broken segment, i.e. at the upper bifurcation point for the lowest k. The stationary-state solution is unstable and surrounded by a stable limit cycle for all parameter values within the enclosed region. Oscillatory behaviour also occurs in the small shaded region below the Hopf curve, where the stable stationary state is surrounded by both an unstable and...

The range of parameter values in the Pex-K2 plane for which Hopf bifurcation is possible in the present system is that lying below the line H in Fig. 9.6(f). 

In the previous section we have taken care to keep well away from parameter values /i and k for which the uniform stationary state is unstable to Hopf bifurcations. Thus, instabilities have been induced solely by the inequality of the diffusivities. We now wish to look at a different problem and ask whether diffusion processes can have a stabilizing effect. We will be interested in conditions where the uniform state shows time-dependent periodic oscillations, i.e. for which /i and k lie inside the Hopf locus. We wish to see whether, as an alternative to uniform oscillations, the system can move on to a time-independent, stable, but spatially non-uniform, pattern. In fact the... 

If we consider the well-stirred system, the stationary state has two Hopf bifurcation points at /r 2, where tr(U) = 0. In between these there are two values of the dimensionless reactant concentration /r 1>2 where the state changes from unstable focus to unstable node. In between these parameter values we can have (tr(U))2 — 4det(U) > 0, so there are real roots to eqn (10.76). 

But when the field is nonzero the trivial solution is not allowed. Instead, there is always one real nontrivial solution for all values of the bifurcation parameter X and a pair of other real solutions which exist only for values of X larger than a certain value Xc. However, there exists no bifurcation of new solutions from a given branch. This situation is described in Fig. 10 of the paper by I. Prigogine. It provides the basis for understanding the high sensitivity of the system in the vicinity of X, and the pattern selection introduced by the gravitational field. We come back to this problem in Section IV. 

If we keep ko constant, then a is directly proportional to the volume V, and inversely proportional to the flow rate q. The values of a can reach very large magnitudes due to the nature of ko and its range of values. In general, a is the most widely varying parameter of the reaction. Therefore we usually investigate bifurcation with a chosen as the bifurcation parameter. 

To gain further and broader insights into the bifurcation behavior of nonadiabatic, nonisothermal CSTR systems, we again use the level-set method for nonalgebraic surfaces such as z = /(K,., y). This particular surface is defined via equation (3.14) as follows for a given constant value of yc with the bifurcation parameter Kc ... 

Compute and plot the bifurcation diagram with Kc as the bifurcation parameter for yc = 0.6 and the same value of a that you have chosen above to give multiplicity. Discuss the physical meaning of the diagram. 

As before we use the dilution rate D as the bifurcation parameter for varying values of Cso The role of these two variables can easily be exchanged to achieve similar results with Cso as the bifurcation parameter for varying dilution rates D. 

The most important dynamic bifurcation is Hopf bifurcation. This occurs when Ai and A2 cross the imaginary axis into the right half-plane of C as the bifurcation parameter g changes. At the crossing point both roots are purely imaginary with det(A) > 0 and tr(A) = 0, making Ay2 = i y/det(A). At this value of g, periodic solutions (stable limit cycles) start to exist as depicted in Figures 10 and 11 (A-2). 

Fig. 4 Numerical simulation revealing mirror-symmetry breaking in which k.2 acts as a bifurcation parameter (same conditions as in Fig. 3 except [R]o + [S]o = 0). Each dot represents the final ee of an individual computer simulation. For k2 > 6 x 103 M-1 s-1 the system becomes optically active, where positive and negative values of the resulting ee are equally distributed...

Here, w = m, n, and S. V represents the membrane potential, n is the opening probability of the potassium channels, and S accounts for the presence of a slow dynamics in the system. Ic and Ik are the calcium and potassium currents, gca = 3.6 and gx = 10.0 are the associated conductances, and Vca = 25 mV and Vk = -75 mV are the respective Nernst (or reversal) potentials. The ratio r/r s defines the relation between the fast (V and n) and the slow (S) time scales. The time constant for the membrane potential is determined by the capacitance and typical conductance of the cell membrane. With r = 0.02 s and ts = 35 s, the ratio ks = r/r s is quite small, and the cell model is numerically stiff. The calcium current Ica is assumed to adjust immediately to variations in V. For fixed values of the membrane potential, the gating variables n and S relax exponentially towards the voltage-dependent steady-state values noo (V) and S00 (V). Together with the ratio ks of the fast to the slow time constant, Vs is used as the main bifurcation parameter. This parameter determines the membrane potential at which the steady-state value for the gating variable S attains one-half of its maximum value. The other parameters are assumed to take the following values gs = 4.0, Vm = -20 mV, Vn = -16 mV, 9m = 12 mV, 9n = 5.6 mV, 9s = 10 mV, and a = 0.85. These values are all adjusted to fit experimentally observed relationships. In accordance with the formulation used by Sherman et al. [53], there is no capacitance in Eq. (6), and all the conductances are dimensionless. To eliminate any dependence on the cell size, all conductances are scaled with the typical conductance. Hence, we may consider the model to represent a cluster of closely coupled / -cells that share the combined capacity and conductance of the entire membrane area. 

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