Focus unstable

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.

We have discussed the case in which the unstable homogeneous steady state is a saddle point, that is, when a real eigenvalue con changes sign. Let us now consider what happens when the uniform steady state is an unstable focus and a time periodic regime sets in. 

Complex, real parts + ve Unstable focus (oscillatory divergence)... 

FiC. 3.4. Representations of the different singular points in the concentration phase plane (a) stable node, sn (b) stable focus, sf (c) unstable focus uf (d) unstable node, un (e) saddle. point, sp. 

When the dimensionless reactant concentration is slightly greater than the lower root of eqn (3.65), i.e. /i > /if, the stationary state must be an unstable focus, which becomes stable as /i passes through /if. Below this the state is first a stable focus then, as /i approaches zero, a stable node. 

Fig. 4.3. The ft-K parameter plane showing loci of changes in local stability or character for the model with the exponential approximation (a) full plane (b) enlargement of region near origin showing, in particular, the locus of Hopf bifurcation (change from stable to unstable focus) and the locus corresponding to the maximum in the ass(/r) curves (broken line).

The locus of these Hopf bifurcation points is also shown in Fig. 4.3 and can be seen to be another closed loop emanating from the origin. It lies in the region between the loci for changes between nodal and focal character, so the condition tr(J) separates stable focus from unstable focus. The curve has a maximum at... 

Fig. 4.8. The h k parameter plane showing changes in local stability and character for y = 0.1 (a) full parameter plane (b) enlargement for small /r and k showing locus of Hopf bifurcation (transition from stable to unstable focus and (as broken lines) the loci for the maximum and...

The condition for Hopf bifurcation, i.e. for a change from stable to unstable focus, is also shown in Fig. 4.8. This is given parametrically by... 

We may also note, for the special case / = 1, that the locus described by eqns (10.58) and (10.59) is exactly that corresponding to the boundary between unstable focus and unstable node for the well-stirred system. This seems to be a general equivalence between the existence of unstable nodal solutions in the well-stirred system and the possibility of diffusion-driven pattern formation in the absence of stirring. We have seen in chapter 5 that unstable nodes are not found in the present model if the full Arrhenius rate law is used and the activation energy is low, i.e. iff <4 RTa. In that case we would also not expect spatial instability. 

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). 

If the dimensionless rate constant satisfies inequality (10.78), the well-stirred system again has two Hopf bifurcation points fi and n. However, within the range of reactant concentrations between these, the uniform state also changes character from unstable focus to unstable node at n and n 2, as shown in Fig. 10.10. 

Statement 2. Substitution into the concentration dynamics (equations (8.2.12) and (8.2.13)) of the reaction rate K — K(Na, Nb), dependent on the current concentrations, changes the nature of the singular point. In particular, a centre (neutral stability) could be replaced by stable or unstable focus. This conclusion comes easily from the topological analysis its illustrations are well-developed in biophysics (see, e.g., a book by Bazikin [30]). 

Fig. 8.1. Phase portraits of the Lotka-Volterra model for d = 3 (a) Unstable focus (re = 0.9) (b) Stable focus (re = 0.2) (c) Concentration oscillations during the steady-state formation (re = 0.1) (d) Chaotic regime (re = 0.05). The values of the distinctive parameter are shown.

The behaviour of the correlation functions shown in Fig. 8.5 corresponds to the regime of unstable focus whose phase portrait was earlier plotted in Fig. 8.1. For a given choice of the parameter k = 0.9 the correlation dynamics has a stationary solution. Since a complete set of equations for this model has no stationary solution, the concentration oscillations with increasing amplitude arise in its turn, they create the passive standing waves in the correlation dynamics. These latter are characterized by the monotonous behaviour of the correlations functions of similar and dissimilar particles. Since both the amplitude and oscillation period of concentrations increase in time, the standing waves do not reveal a periodical motion. There are two kinds of particle distributions distinctive for these standing waves. Figure 8.5 at t = 295 demonstrates the structure at the maximal concentration... 

FIGURE 9 Phase portraits for the system when -yi = 0.001 and -y2 = 0.002. (a) Phase diagram at an apparent triple point (labelled P in Figure 8 a, = 0.025, a2 = 0.026) where only the value of 6t is identical for the three steady-states, (b) Limit cycle surrounding an unstable focus corresponding to the point labelled Q in Figures I and 8 (a, = 0.017, a2 = 0.028). 

FIGU RE 10 Illustration of the disappearance of a limit cycle via a turning point on a periodic branch near a subcritical Hopf bifurcation, (a) A stable limit cycle surrounding an unstable focus (b) the unstable focus undergoes a subcritical Hopf bifurcation and leaves an inner unstable limit cycle surrounding a stable focus (c) the two limit cycles combine into a metastable configuration and disappear altogether as the parameter is further increased. 

This is called Hopf bifurcation. Figure 10 (A-2) shows two Hopf bifurcation points with a branch of stable limit cycles connecting them. Figure 13 (A-2) shows a schematic diagram of the phase plane for this case when g = g. In this case a stable limit cycle surrounds an unstable focus and the behavior of the typical trajectories are as shown. Figure 11 (A-2) shows two Hopf bifurcation points in addition to a periodic limit point (PLP) and a branch of unstable limit cycles in addition to the stable limit cycles branch. 

Fig. 2. Types of stationary points on the plane, (a), (c), (e) Stable nodes (b), (d), (f) unstable nodes (g) saddle point (h) stable focus (i) unstable focus, (k) whirl.

Figure 1 shows Yq, the steady state value of Y, as a function of P with other parameters held constant. Over a range of values of P, three steady states co-exist. The stability of these states has been examined by normal mode analysis ( 5) and unstable states are depicted by the broken line. Over the range of values of P associated with the asterisks, the "upper" steady state has the characteristics of an unstable focus and computer integration of equations (2) and (3) demonstrates the existence of a stable limit cycle around this state. This region can therefore provide the required co-existence of a stable limit cycle and a stationary state. 

With increasing values of S, as we pass the point marked by the black square, the fast subsystem undergoes a Hopf bifurcation. The complex conjugated eigenvalues cross the imaginary axis and attain positive real parts, and the stable focus is transformed into an unstable focus surrounded by a limit cycle. The stationary state, which the system approaches as initial transients die out, is now a self-sustained oscillation. This state represents the spiking behavior. 

As S continues to increase we reach a point marked by an open circle. Here, the equilibrium point undergoes a saddle-node bifurcation. Somewhere before this bifurcation, the unstable focus point has turned into an unstable node with two positive real eigenvalues. In the saddle-node bifurcation, one of these eigenvalues... 

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