Point unstable focus

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. 

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. 

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... 

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]). 

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.

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... 

In region II, y /4 < A, and therefore < 0, but k 0. This region relates to stable focuses where the system evolution toward the initial point is described by a spiral curve. Unstable focuses and nodes are arranged in regions III and IV > 0)> respectively, and also are separated by curve y /4 = A. On axis y = 0, there are center type points for which k = 0, 7 0, and ki 2 = i ir Region V relates to unstable exceptional points of the saddle type. Here, = 0 and k have different signs > 0, 2 0)-... 

The instability arises and evolves owing to thermodynamic fluctua tion (3.29). Such a fluctuation may cause complete system state decay (see, e.g., region V of unstable saddles in Figure 3.4). Flowever, it may also happen that the arising instability creates a new state of the system to be stabilized in time and space. An example is the formation of the limit (restricted) cycle in a system that involves the exceptional point of the unstable focus type. The orbital stability of such a system means exactly the existence of certain time stabilized variations in the thermody namic parameters (for example, the concentrations of reactants) that are... 

Figure 3. The stable and unstable manifolds of the critical points A and B on the TCM for Z = 2. (a) The collinear eZe configuration (a = ti). The critical points A and B on the TCM are hyperbolic fixed points, (b) The Wannier ridge configuration (x = ti/2). The critical points A and B on the TCM are a stable focus and an unstable focus, respectively.

The form of the solutions to the simplified model were analysed by examining the existence and types of the pseudo-stationary points of the equations for d0/dr = d 3/dr = 0 and values of e in the range 0—1 (r = Figure 29 shows the oscillation of a multiple-cool-flame solution about the locus of such a pseudo-stationary point, Sj. The initial oscillation is damped while Si is a stable focus. The changing of Si into a unstable focus surrounded by a stable limit cycle leads to an amplification of the oscillation which approaches the amplitude of the limit cycle. When Si reverts to a stable focus, and then a stable node, the solution approaches the locus of the pseudo-stationary point. In this way an insight may be gained into the oscillatory behaviour of multiple cool flames. 

Fig. 5.8. Phase plane portraits of different possible steady-state singularities (i) stable node, trajectories approach singular point without overshoot (ii) stable focus showing damped oscillatory approach (iii) unstable focus showing divergent oscillatory departure (iv) unstable node showing direct departure (v) saddle point x showing insets and outsets and...

Fig. 66. Phase trajectories in the vicinity of the stationary point of an unstable focus type.

The analysis of linearized sytem thus allows, when conditions (1)—(3) are met, us to find the shape of phase trajectories in the vicinity of stationary (singular) points. A further, more thorough examination must answer the question what happens to trajectories escaping from the neighbourhood of an unstable stationary point (unstable node, saddle, unstable focus). In a case of non-linear systems such trajectories do not have to escape to infinity. The behaviour of trajectories nearby an unstable stationary point will be examined in further subchapters using the catastrophe theory methods. 

When the parameter c < 0 the stationary state (x,y) = (0, 0) is of a stable focus type for c = 0 we deal with the sensitive state the stationary state is a centre for c > 0 a catastrophe takes place, because the stationary state becomes an unstable focus. The phase trajectories for the linearized system (5.83) for c < 0, c = 0, c > 0 nearby the stationary point are shown in Figs. 65, 66, 67, respectively. 

Hence, the stationary point (0,0) for small (as regards the absolute value) e is a stable focus and for small positive e is an unstable focus. When the parameter e changes sign, a catastrophe — a change in the nature of trajectories, takes place in the system. In addition, At 2(0) = +i hence, the state of the system corresponding to e = 0 is a sensitive state typical for the Hopf bifurcation. 

The type of steady state illustrated by the Brusselator example is called a focus because it is the pivot point for the spiraling trajectories that move toward or away from it. As we will see, the existence of a focus is often the prerequisite for the existence of oscillatory solutions to the full equations of motion. In particular, we look for an unstable focus (one for which the real part of the stability eigenvalues is positive) because the trajectories that spiral away from the focus may eventually reach a stable cyclic path surrounding that focus called a limit cycle. 

Figure 26 Generation of a torus attractor via two Hopf bifurcations. The first Hopf bifurcation converts a stable fixed point (a focus) into an unstable focus. A stable limit cycle generally originates at this bifurcation point. A second Hopf bifurcation occurs, rendering the limit cycle unstable, and giving rise to a stable torus. Each Hopf bifurcation results in one additional frequency of oscillation in the system.

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