Turing bifurcation, instability

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... 

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],... 

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. 

This notion of bifurcation point, connected with those of instability and fluctuations, is the basis of this branch of science of self-organization in out-of-equilibrium systems. The story begins with Alan Turing, who, in search of the chemical basis of... 

As discussed in Sect. 2.2 the diffusion equation has the well-known unrealistic feature that localized disturbances spread infinitely fast, though with heavy attenuation, through the system. In that section we described three approaches to address the unphysical behavior of the diffusion equation and reaction-diffusion equation. Since the Turing instability is a diffusion-driven instability, it is of particular interest to explore how this bifurcation depends on the characteristics of the transport process. In this section, we address the effects of inertia in the dispersal of particles or individuals on the Turing instability. Does the finite speed of propagation of perturbations in such systems affect Turing instabilities We determine the stability properties of the uniform steady state for the three approaches presented in Sect. 2.2. 

Remark 10.3 The analysis of all three approaches to two-variable reaction-transport systems with inertia establishes that the Turing instability of reaction-diffusion systems is structurally stable. The threshold conditions are either the same, HRDEs and reaction-Cattaneo systems, or approach the reaction-diffusion Turing threshold smoothly as the inertia becomes smaller and smaller, t 0. Further, inertia effects induce no new spatial instabilities of the uniform steady state in the diffusive regime, T small. A spatial Hopf bifurcation to standing wave patterns can only occur in the opposite regime, the ballistic regime. 

These results lead to the conclusion that the USS of the array is more susceptible to the Hopf instability than to the Turing instability. If one reactor, i.e., half the system, contains no substrate, 02 = 1, then the Hopf bifurcation occurs first, no matter how large the substrate concentration oy in the other reactor. In fact, for a = 50.0 the USS of the inhomogeneous two-reactor array will undergo a transition to oscillations first as b is decreased as long as 02 < 5.41038. Further, the... 

This model was used to study the bifurcation of a stable steady state to a limit cycle oscillation, as well as Turing instability leading to spatial structures. 

In (ll.l), the summation is restricted to wavenumbers close to q<3. In the case of the R.B instability, the quadratic term is due to non-Boussinesq effects (e.g. temperature dependent transport coefficients) a similar term always appears in the case of the Marangoni or the Turing instability. When these equations have a gradient structure and this property is sometimes satisfied near the bifurcation point, one can define a Lyapunov functional that decreases in any dynamics. In that case if one describes local fluctuations by a gaussian white noise term then the (T.D.G.L.) equation can be written as ... 

For/ = lando > / , the system exhibits a primary homogeneous Hopf bifurcation to an oscillatory state. This is a general feature systems which present Turing instabilities also present homogeneous Hopf instabilities. Interactions between these two instabilities can play an important role which will be discussed further. From the condition d > 1, it follows that the two diffusion coefficients must be different for a Turing instability to occur the activator must diffuse faster than the other species. 

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