Point stable focus

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. 

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. 

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... 

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. ... 

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. 

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.

Isoelectric focusing-FFF is appropriate for the separation of amphoteric particles, since their electrophoretic mobility depends on pH and is zero at the isoelectric point. If a stable pH gradient is formed in the FFF channel due to an applied electrical field [317], the amphoteric solute will be focused into the position of its isoelectric point. Isoelectric-focusing-FFF was first proposed as a concept and experimentally verified five years later [314,315,318-321]. 

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)-... 

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. III.20. Multiple limit cycles, one stable and two unstable, and multiple singular points, two stable focus and one saddle point. (From Kaimachnikov and Sel kov (1975))...

Since for P0 > 0, these roots have negative real parts, this singular point is a stable focus, and the steady state values given by Equation 40 are approached either by a damped sinusoid or an exponential (63). Note that for P0 — 0, the classical case, the roots are purely imaginary, and the oscillation persists indefinitely. 

Fig. 65. Phase trajectories in the vicinity of the stationary point of a stable focus type.

The notions of a limit point and a limit set will be first exemplified by the linear systems considered in Section 5.1. In the case of a stable node (al), (a2), (a3) and a stable focus (d) the limit set (attractor) consists of one point, the stationary point, which is approached by all trajectories. 

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 system has the spatially homogeneous stationary point (x, y) of a stable focus type. At the point k02 the first catastrophe takes place the imaginary part of the eigenvalues vanishes, both the eigenvalues in this sensitive state being equal, Ax = A2. Such a catastrophe is not related to a loss of stability by the point (x, y) only a global phase portrait changes... 

When 0stationary state (1, 1) is a stable node. When a = otj equation (6.71) has two equal real roots, kr = k2 < 0. The value kt is determined from the requirement of vanishing of the discriminant A of the quadratic equation (6.71). This is a sensitive state when a increasing exceeds the value ax, a change in the phase portrait of the system (6.67) takes place the stationary state is, for < a < a0 = (y — 1) a stable focus (the real part of kx >2 is negative). In accordance with what we established in Chapter 5, the catastrophe does not alter a phase portrait nearby a stationary point but is of a global character. 

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