Turing bifurcation

Now we may state tlie well known conditions for a Turing bifurcation. If A <0 and A22 < 0 we say species A j is tlie activator and speciesis tlie inliibitor. Then, for a Turing bifurcation to occur we must have detB = 0,Tt B> 0 and A 7)2 + > 0. The (unique) wavenumber at tlie bifurcation is... 

Figure C3.6.12 a) Turing spot pattern in the CIMA reaction. (A) Tio-temporal turbulence near the Turing bifurcation. Reproduced by pennission from Ouyang and Swinney [59].

One may also observe a transition to a type of defect-mediated turbulence in this Turing system (see figure C3.6.12 (b). Here the defects divide the system into domains of spots and stripes. The defects move erratically and lead to a turbulent state characterized by exponential decay of correlations [59]. Turing bifurcations can interact with the Hopf bifurcations discussed above to give rise to very complicated spatio-temporal patterns [63, 64]. 

There is another type of bifurcation called Turing bifurcation, which results in a spatial pattern rather than oscillation. A typical example where a new spatial structure emerges from a spatially unique situation is Benard s convection cells. These have been well examined and are formed with increasing heat conduction.53 Prigogine called this type of structure a dissipative structure.54-56... 

A homogeneous steady state undergoes a Turing bifurcation at the following critical value of b (Rudovics et al., 1999)... 

A catastrophe corresponding to the sensitive state (5.128) is sometimes called the Turing bifurcation. Apparently, the question what is the time evolution of the solution of (5.121) type on losing stability by the spatially homogeneous stationary state (x, y) may be settled only by examining the exact equation (5.116). The methods of investigation of this problem will be discussed in the next section. 

Note, however, that in the case of a lack of diffusion the stationary state was unstable, see equation (6.138) for D = 0. In this case the diffusion stabilized, for D < Dcr l, this state. Of considerably more interest is the case of a loss of stability on accounting for diffusion by the stationary state stable in the absence of diffusion. Such an effect of diffusion can be readily shown to be possible only in the case of a system of two equations with diffusion. A catastrophe of this type, called the Turing bifurcation, will be considered for the Brusselator. 

The state (a, b/a) loses stability on crossing by the system, with a continuous variation in the control parameters a, b, Dx, Dy, the sensitive state defined by the equalities in inequalities (6.165). A catastrophe involving the loss of stability of the stationary state (equations without diffusion) due to the effect of diffusion is called the Turing bifurcation. 

Borckmans P, Dewel G, De Wit A and Walgraef D Turing bifurcations and pattern selection Chemical Waves and Patterns eds R Kapral and K Showalter (Dordrecht Kluwer) ch 10, pp 323-63... 

We consider the stability of a general two-variable system, first in the absence of diffusion and then with diffusion terms. Because the Turing bifurcation is a diffusion-induced instability, we will first show that the system is stable... 

For a spatially unbounded system, the onset of instability (i.e., the Turing bifurcation) occurs at 8 = 8 when H(fe ) = 0, because the wavenumbers are dense over the H(k ) curve. For a spatially finite system with no-flux boundary conditions, however, only particular wavenumbers satisfy the boundary conditions because the functional form of the solution must be cos kr). Thus, in a bounded one-dimensional system, perturbations will grow with wavenumbers given by... 

Figure 12 Constraint diagram showing parameter values of Turing bifurcation locus (left line) and Hopf bifurcation locus (right line). The Turing instability occurs for parameter values in the middle region. (Reprinted with permission from Ref. 34.)...

Point B is much farther from the Turing bifurcation, and we anticipate the pattern formation to be more complex. Following the same procedure as described above, we find the following range of wavenumbers over which the diffusion-induced instability occurs... 

Corollary 10.1 No Turing bifurcation can occur in a one-variable reaction-diffusion... 

Theorem 10.2 If a set of dij exists such that a Turing bifurcation occurs for arbitrarily small but nonzero e and the eigenvalues of J remain bounded in modulus,... 

The preceding theorem establishes that the well-mixed system being in the vicinity of a double-zero point is a necessary condition for a Turing bifurcation to occur in (10.1) with nearly equal diffusion coefficients. The next theorem establishes that this is also a sufficient condition. 

The uniform steady state is stable if all roots have a negative real part for all k. The necessary and sufficient conditions for this to hold are the Routh-Hurwitz conditions, see Theorem 1.2. The Turing bifurcation corresponds to a real root k crossing the imaginary axis for some nonzero kj, i.e., k/ = 0. This occurs if condition (1.35) is violated, i.e., 4 = 0 for some 7 0. A bifurcation to oscillatory patterns, i.e., a spatial Hopf bifurcation, corresponds to a pair of complex conjugate roots crossing the imaginary axis for some nonzero %, i.e., = 0. 

The Turing condition for two-variable DIRWs, (10.85), has the same form as the Turing condition for two-variable reaction-diffusion systems, (10.31). Consequently, the uniform steady state (10.70) of a DIRW undergoes a Turing bifurcation with critical wavenumber... 

To explore if the activation energy always affects the Turing bifurcation at a higher order than the inertia, we consider the kinetic scheme where the termolecular step in the Brusselator proceeds via an activated UV complex instead of an activated U dimer ... 

The coefficients of this equation differ from those of the corresponding equation for the Brusselator DIRW (10.97) by terms that are first order in x, namely the terms with 1 hx and 1/ /Iv, and a term that is second order in r, namely the term with l/(/u,v). In the Brusselator DDRW with activated UV complex, inertia and activation energy both affect the threshold of the Turing bifurcation at first order, in contrast to the DDRW with activated U dimer. We find for the Turing threshold T,uv = o,uv + i,uv + ... 

The parameter C varies over (0, oo) if 1/2 < y < 1 and over (0, l/(y - 1)) if 1 < y < 2. Analysis shows that for y = 1, i.e., the activator and inhibitor have the same anomaly coefficient and which includes the case of standard diffusion, the conditions (10.200) reduce to those of the standard Brusselator RD system, see Sect. 10.1.3. If y < 1, the anomaly coefficient of the inhibitor is larger than that of the activator, and the tail of the jump length PDF of the inhibitor decreases faster than that of the activator, see (3.191). In other words, the activator has a longer range than the inhibitor, which prevents a Turing instability in a standard RD system, hi contrast, a Turing bifurcation can occur in the superdiffusive Brusselator for such a situation, provided a is sufficiently large [164],... 

To understand the precise role of the substrate and the complexation reaction, consider the case d = 1, the case most unfavorable for a Turing bifurcation in... 

Note that the third stability condition corresponds to Aj > 0. According to (1.38), the wavenumber zero mode of the activator-inhibitor-substrate system undergoes a Hopf bifurcation if A2 goes through 0. For mass-action kinetics, (12.20) implies that G or Po- Consequently, the third stability condition can fail, if the total substrate is too low. Then the Turing bifurcation ceases to be the primary instability, and a uniform Hopf bifurcation occurs first in the fiiU system. 

The coefficients C K) of the characteristic polynomial det[J( )—X tl4] = 0 and the Hurwitz determinants A K) are easily obtained using computational algebra software such as Mathematic A (Wolfram Research, Inc., Champaign, IL) or Maple (Waterloo Maple Inc., Waterloo, Ontario). The th mode undergoes a stationary bifurcation when condition (12.41d) is violated, namely c K) = 0, as discussed in Sect. 1.2.3, see (1.36). In other words, a Turing bifurcation of the uniform steady state corresponds to c iki) = 0 with k 0, while the stability conditions (12.41) are satisfied for all other modes with k fej. The feth mode undergoes an oscillatory bifurcation when condition (12.41c) is violated, namely A K) = 0, as discussed in Sect. 1.2.3, see (1.38). A wave bifurcation of the uniform steady state corresponds to A k i) = 0 with k f/ 0, while the stability conditions (12.41) are satisfied for all otha modes with k few As discussed in Sect. 10.1.2, see (10.29), a wave bifurcation cannot occur in a two-variable reaction-diffusion system. 

Given the lengthy expressions for c, (AT) and A,(AT), an exact general derivation of the Turing bifurcation and wave bifurcation thresholds is neither desirable nor analytically feasible for the latter, even with the help of symbolic computation software. Vanag and Epstein have carried out numerical evaluations of the eigenvalues... 

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