Unstable focus limit cycle

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. 

Stationary-state solutions correspond to conditions for which both numerator and denominator of (3.54) vanish, giving doc/dp = 0/0, and so are singular points in the phase plane. There will be one singular point for each stationary state each of the different local stabilities and characters found in the previous section corresponds to a different type of singularity. In fact the terms node, focus, and saddle point, as well as limit cycle, come from the patterns on the phase plane made by the trajectories as they approach or diverge. Stable stationary states or limit cycles are often refered to as attractors , unstable ones as repellors or sources . The different phase plane patterns are shown in Fig. 3.4. 

Note here, without proof, one of the synergetic theorems about limit cycles [14, 15] a stable limit cycle contains at least one singular point or the unstable node of focus-type exists. 

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. 

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. 

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. 

Fig. 5. Possible types of attractors inside 3-simplex node (1, 2), focus (3,4), saddle (5) and limit cycle (6, 7). Stable (/, 3,6) and unstable (2,4,5, 7) attractors...

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... 

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. 

Figure 23. Singular trajectories of system of equations (104) with existence of two singulars at different values of fi (a), 10 (fe), 1 (c), 0.35 (d), 0.30 (e), 0.27. y = 1, i = 0.5, tj - 10, Singularities are site-saddle (a,b), stable focus-saddle (c), unstable focus with stable limiting cycle-saddle (d), and unstable focus-saddle (e).

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))...

Two singular solutions, one saddle (S2), one unstable focus (St) and a stable limit cycle. 

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 9 Limit cycle oscillations calculated by numerically solving the dimensionless Brusselator Eq. [32] with the parameter values a - and b = 2.2 and a variety of initial conditions. The unstable focus is located at m = 2.2. Solved using the...

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.

Rigorous analysis of stability was performed by means of Liapunov s theorem, which was written concisely as the inequalities Eq. (1.19). This analysis showed that the initial and end points A and C are stable nodes while the middle B is usually an instable saddle point. This saddle point can sometimes transform into an unstable node or focus, thus allowing for birth of a limit cycle and self-oscillation behaviour. Unexpectedly, the mathematic analysis showed also the possibility of a stable intermediate state in some narrow region of parameters values (such that the maximum in Fig. 5.15 is not very far from the straight line W = ). This result differs from the intuitive physical considerations above. 

The theory of nonlinear oscillations can describe the periodic solution that appears beyond the instability of the steady state. Stable states exist before the instability. The perturbations correspond to complex values of the normal mode frequencies and spiral toward the steady state to a focus. As soon as the steady state becomes unstable, a stable periodic process called the limit cycle occurs. This behavior is independent of the initial conditions, and the system approaches in time the same periodic solution determined by the nonlinear differential equations. The periodic solution is characterized by its period and amplitude. The limit cycle is unique and stable with respect to small fluctuations. 

A more generally applicable approach, known as quenching, has been developed by Hynne and Sorensen (1987). In most chemical oscillators, oscillation arises via a supercritical Hopf bifurcation that produces a saddle focus surrounded by a limit cycle. The unstable steady state that lies inside the oscillatory trajectory in phase space has an unstable manifold, that is, a surface along which trajectories move away from the steady state and toward the limit cycle, and a stable manifold, that is, a surface (possibly just a line) along which trajectories move toward the steady state. The situation is illustrated in Figure 5.9. 

Figure 7.11 shows the phase plane for this case when /x = /X2. In this case, there is an unstable limit cycle surrounding a stable focus and the unstable limit cycle is surrounded by a stable limit cycle. The behavior of the typical trajectories is as shown in Figure 7.11. 

Figure 7.10 Phase plane for fx = fii m Figure 7.9A. Solid curve stable limit cycle trajectories unstable saddle stable steady state (node or focus) O unstable steady state (node or focus) dashed curve separatix. 

Points at which a stable focus becomes unstable correspond to points of Hopf bifurcation [8]. These are the conditions at which limit cycles emerge in the a-3 phase-plane. The line H in figure 2 divides the 3q pa inieter space into... 

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