Focus stable

From the standpoint of the classical (analytical) theory with which we were concerned in this review, the situation is obviously absurd since each of these two equations is linear and of a dissipative type (since h > 0) trajectories of both of these equations are convergent spirals tending to approach a stable focus. However, if one carries out a simple analysis (see Reference 6, p. 608), one finds that change of equations for = 0, results in the change of the focus in a quasi-discontinuous manner, so that the trajectory can still be closed owing to the existence of two nonanalytic points on the -axis. If, however, the trajectory is closed, this means that there exists a stationary oscillation and in such a case the system (6-197) is nonlinear, although, from the standpoint of the differential equations, it is linear everywhere except at the two points at which the analyticity is lost. 

Figure 29 Bifurcation diagram of the minimal model of glycolysis as a function of feedback strength and saturation 6 of the ATPase reaction. Shown are the transitions to instability via a saddle node (SN) and a Hopf (HO) bifurcation (solid lines). In the regions (i) and (iv), the largest real part with in the spectrum of eigenvalues is positive > 0. Within region (ii), the metabolic state is a stable node, within region (iii) a stable focus, corresponding to damped transient oscillations.

From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

Complex, real parts — ve Stable focus (damped oscillatory approach)... 

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. 

If the uncatalysed reaction rate increases with respect to the rate of catalyst decay, so that ku becomes larger than gk2, there are no real solutions to eqn (3.60). The stationary state can no longer become unstable as /i is varied. Damped oscillatory responses can still be observed when we have a stable focus, but undamped oscillations will not be found. 

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... 

This is thus a stable node or a stable focus, depending on the size of the wave velocity c. For c > 2, the eigenvalues are real for lower wave velocities with... 

Fig. 2.2. A stable focus. The solution of the same equation as presented in Fig. 2.1, but with...

When pK > 4(32 holds, the singular point remains stable, Reei, 2 < 0, but the roots (2.1.16) have imaginary parts Imei = Im ei. In this case the phase portrait reveals a stable focus - Fig. 2.2. This regime results in damped oscillations around the equilibrium point (2.1.24). The damping parameter pK/(3 is small, for large 3, in which case the concentration oscillation frequency is just ui = y/pK. ... 

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 Hopf bifurcation where a stable focus becomes unstable and sheds or absorbs a periodic solution is an important transition which has received a great deal of attention (for a review see Marsden McCracken 1976). Clearly the lines over which it can take place are the loci of steady-states whose eigenvalues are purely imaginary. These are shown on the sides of the fin in figure 5. Because this is a two-dimensional system we can write down the condition quite explicitly. Writing the equations ... 

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. 

In Figure 15 there is an unstable limit cycle surrounding a stable focus and the unstable limit cycle is surrounded by a stable limit cycle. 

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.

In order to examine the stability of the equilibrium points it is customary to separate the three-dimensional system Eqs. (6) to (11) into a fast subsystem involving V and n and a slow subsystem consisting of S. The z-shaped curve in Fig. 2.7b shows the equilibrium curve for the fast subsystem, i.e. the value of the membrane potential in the equilibrium points (dV/dt = 0, dn/dt = 0) as a function of the slow variable S, which is now to be treated as a parameter. In accordance with common practice, those parts of the curve in which the equilibrium point is stable are drawn with full lines, and parts with unstable equilibrium points are drawn as dashed curves. Starting from the top left end of the curve, the equilibrium point is a stable focus. The two eigenvalues of the fast subsystem in the equilibrium point are complex conjugated and have negative real parts, and trajectories approach the point from all sides in a spiraling manner. 

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. 

These results are supported by the standard stability analysis of Figure 11.2, where A is set to 0.1 and y = 2 (y = k ). The eigenvalues computed by (11.6) are plotted as functions of y. In this figure, unstable and stable equilibrium points are clearly separated by an interval, [0.1974 0.2790], where eigenvalues are complex, leading to a stable focus. With increasing A, this interval becomes narrower and for A > 0.65, the eigenvalues have only real parts. 

The steep negative slope

complex eigenvalues. The frequency of the oscillation increases with the steepness. The operating point in such cases is a stable focus. In contrast, shallow negative slopes... 

Figure 11.3 State space for different initial conditions. The equilibrium point Pi ( o ) is a stable focus, P2 ( ) is an unstable saddle point, and P3 ( ) is stable.

For motion in the x direction the analysis is more involved because X can exist as either a real or complex number, depending on whether < > > l/(4a) or < > < 1/(4a). For the case < > > l/(4a) the plot is of the stable focus type (Fig. 7.11). The right-hand side of Figure 7.11 has been lightly shaded since this is an imaginary zone—this is the space behind the infinite plane. 

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)-... 

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