Bifurcation transcritical

When condition (2) in the above theorem is not met, a transcritical bifurcation may take place. 

We shall examine properties of the system (5.69) using only the approach based on investigation of eigenvalues of the stability matrix. 

The stationary state xt = 0 is trivial inasmuch as it occurs for each value of the control parameter. A sensitive state appears when a = 0 then — = A2 = 0. 

Let now the control parameter a vary continuously, passing from negative to positive values. When a 0, then the stationary state xx = 0 is stable and the stationary state x2 = a is unstable for a = 0 the stationary state Xj = x2 = 0 is a sensitive state when a 0 the stationary state xt is unstable while the state x2 = a is stable. A phenomenon of this type is called the stability exchange. 

The next figure (Fig. 80) illustrates a bifurcation diagram, that is the plot of the stationary states x(c) as a function of the parameter c. The unstable state is represented by a dashed line. 

There are certain scientific situations where a fixed point must exist for all values of a parameter and can never be destroyed. For example, in the logistic equation and other simple models for the growth of a single species, there is a fixed point at zero population, regardless of the value of the growth rate. However, such a fixed point may change its stability as the parameter is varied. The transcritical bifurcation is the standard mechanism for such changes in stability. 

This looks like the logistic equation of Section 2,3, but now we allow x and r to be either positive or negative. 

Please note the important difference between the saddle-node and transcritical bifurcations in the transcritical case, the two fixed points don t disappear after the bifurcation—instead they just switch their stability. 

Show that the first-order system x = x(l-x )-a(l-e ) undergoes a transcritical bifurcation at x = 0 when the parameters a, b satisfy a certain equation, to be determined. (This equation defines a bifurcation curve in the (a, i ) parameter space.) Then find an approximate formula for the fixed point that bifurcates from X = 0, assuming that the parameters are close to the bifurcation curve. 

Solution Notethatx = 0 isafixedpointforall (a,/ ). This makes it plausible that the fixed point will bifurcate transcritically, if it bifurcates at all. For small x, we find 

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At the critical value a = oi = 1, however, becomes unstable and the a-dependent fixed point becomes stable. This exchange of stability between two fixed points of a map is known as a transcritical bifurcation. By using the same linear-stability analysis as above, we see that remains stable if — 1 < a(l — Xjjj) < 1, or for all a such that 1 < a < 3. Something more interesting happens at a — 3. 

A theoretical framework for considering how the behavior of dynamical systems change as some parameter of the system is altered. Poincare first apphed the term bifurcation for the splitting of asymptotic states of a dynamical system. A bifurcation is a period-doubling, -quadrupling, etc., that precede the onset of chaos and represent the sudden appearance of a qualitatively different behavior as some parameter is varied. Bifurcations come in four basic varieties flip bifurcations, fold bifurcations, pitchfork bifurcations, and transcritical bifurcations. In principle, bifurcation theory allows one to understand qualitative changes of a system change to, or from, an equilibrium, periodic, or chaotic state. 

For both the CSTR and PFR systems, at DaT = (z0 - z4)/z3 two different manifolds of steady states cross each other, in the combined space of state variables and parameters. According to the bifurcation theory, this is a transcritical bifurcation point Here, an exchange of stability takes place for Da < DaT, the trivial solution... 

When nucleation is highly unfavorable (i.e., a l) the polymer system exhibits a biphasic behavior depending on the total monomer concentration A0. In this case there is a sharp phase transition between the all-monomer state for A0 < 1 /K, where l/K is the critical monomer concentration. When A0 exceeds 1 /K the free monomer concentration stays fixed at [A eq = l/K. This type of nonsmooth behavior at x = lforcr = 1 is called a transcritical bifurcation in non-linear dynamics [191]. It is also widely known as phase transition in physics. Figure 10.5 shows that for a less than unity, the transition is smooth. Hence we see that the... 

Figure 3.2.2 shows the bifurcation diagram for the transcritical bifurcation. As in Figure 3.1.4, the parameter r is regarded as the independent variable, and the fixed points x = 0 and x = r are shown as dependent variables. 

Hence a transcritical bifurcation occurs when ab = 1 this is the equation for the bifurcation curve. The nonzero fixed point is given by the solution of l-a -t-(- -a )x 0, i.e,... 

Analyze the dynamics ofx=rlnx-t-x-l near x = 1, and show that the system undergoes a transcritical bifurcation at a certain value of r. Then find new variables X and R such that the system reduces to the approximate normal form X >= RX - X near the bifurcation. 

When Nq < k/G, the fixed point at n = 0 is stable. This means that there is no stimulated emission and the laser acts like a lamp. As the pump strength is increased, the system undergoes a transcritical bifurcation when - k/G. For Ag > k/G, the origin loses stability and a stable fixed point appears at n = (GNq -k)laG > 0, corresponding to spontaneous laser action. Thus Ng = k/G can be interpreted as the laser threshold in this model. Figure 3.3.3 summarizes our results. 

For each of the following exercises, sketch all the qualitatively different vector fields that occur as r is varied. Show that a transcritical bifurcation occurs at a critical value of r, to be determined. Finally, sketch the bifurcation diagram of fixed points X vs. r. 

The next two exercises concern the normal form for the transcritical bifurcation. In Example 3.2.2, we showed how to reduce the dynamics near a transcritical bifurcation to the approximate form X = RX -X + O(X ). Our goal now is to show that the (9(X ) terms can always be eliminated by a suitable nonlinear change of variables in other words, the reduction to normal form can be made exact, not just approximate. 

Imperfect transcritical bifurcation) Consider the system x — h + rx — x. When h-0, this system undergoes a transcritical bifurcation at r-0. Our goal is to see how the bifurcation diagram of x vs. r is affected by the imperfection parameter h. 

The saddle node catastrophe and the Hopf bifurcation may be shown to be structurally stable. Certain additional conditions (see Sections 5.5.2.2, 5.5.2.3) are imposed on the transcritical bifurcation and the pitchfork bifurcation. The system is structurally stable under perturbations not disturbing these additional conditions on the other hand, when arbitrary... 

Note that equation (6.28) has a form of the transcritical bifurcation, (5.69). Equation (5.69) may describe the catastrophe occurring on a change of sign of the parameter a. However, an analogous coefficient in equation (6.28), k, is always non-negative. Therefore, in the chemical system described by (6.28) a catastrophe cannot take place. The stationary state Xj = 0 is unstable and the state x2 = kla/k 1 is stable in the entire range of variability of the control parameters klt k t, a. 

Equation (6.30) has a standard form of the transcritical bifurcation, see (5.69). A transcritical catastrophe may occur since the coefficient k2a — k2 may change sign. However, properties of the system (6.30) differ somewhat from those of a general system (5.69). Note that the stationary states of kinetic equation (6.30) are given by... 

The stationary state x2 has a chemical meaning only if > k2, since concentration is a non-negative quantity. On the basis of this conclusion and the analysis of the transcritical bifurcation, see Section 5.5.2.2, the occurrence of the following phenomena in the system (6.30) can be predicted. 

When kta — k2 < 0, the system has just one stationary state, xx = 0, which is stable. In contrast, when k1a — k2> 0, the system has two stationary states Xj = 0, an unstable state, and x2 = (k1a — k2)/k l, a stable state. Recall that in a standard system of the transcritical bifurcation (5.69) the catastrophe of stability exchange between the two existing states Xj and x2 takes place. On the other hand, in the chemical system (6.30) the catastrophe involves destabilization of the state x1 = 0 with a simultaneous appearance of the stable state x2 > 0. 

Now we consider a more interesting, 2+1 case of a 3D film with a 2D surface whose evolution is described by eq.(5). Note that eq.(5) does not have the symmetry h —h. In this case, the instabihty whose threshold corresponds to a finite wavenumber (see Fig.2) usually results in a hexagonal pattern that occurs via a transcritical bifurcation [13, 14], First we concentrate on the formation of surface structures with hexagonal symmetry. 

Eqs. (12), (13) are similar to those derived previously for several other pattern forming systems with conserved quantifies [16], Note that since a hexagonal pattern occurs via a transcritical bifurcation, eqs.(12) are valid, strictly... 

Moreover, for Da = Daj, the determinant of the Jacobian of Eqs (1) to (5) vanishes. Thus, Da-i defines what is called a transcritical bifurcation [15]. Because this type of bifurcation occurs only in special cases, it is expected that it will disappear under model perturbation, for example for complex kinetics. This aspect is discussed in more detail in Kiss et al. [18]. 

Fig. 11.2.16. Coordinates of equilibrium states at transcritical bifurcations. None of the equilibrium states disappears they exchange their stability.

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