Oscillations bifurcation point

The two jumps at the bifurcation points form a kind of hysteresis loop, which B. van der Pol and E. Appleton, who discovered the phenomenon, call oscillation hysteresis. ... 

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. 

Fig. 26.1a). At first, multistage ignitions and extinctions occur followed by a relaxation (long period) mode [7]. Oscillations die a few degrees below the ignition temperature at a saddle-loop infinite-period homoclinic orbit bifurcation point. This is an example where both ignition and extinction are oscillatory. 

The size and period of the oscillations, or of the corresponding limit cycle, varies with the dimensionless reactant concentration pi. We may determine this dependence in a similar way to that used in 2.5. Close to the Hopf bifurcation points we can in fact determine the growth analytically, but in general we must employ numerical computation. For now we will merely present the basic result for the present model. The qualitative pattern of response is the same for all values of ku < g. 

As the dimensionless concentration of the reactant decreases so that pi just passes through the upper Hopf bifurcation point pi in Fig. 3.8, so a stable limit cycle appears in the phase plane to surround what is now an unstable stationary state. Exactly at the bifurcation point, the limit cycle has zero size. The corresponding oscillations have zero amplitude but are born with a finite period. The limit cycle and the amplitude grow smoothly as pi is decreased. Just below the bifurcation, the oscillations are essentially sinusoidal. The amplitude continues to increase, as does the period, as pi decreases further, but eventually attains a maximum somewhere within the range pi% < pi < pi. As pi approaches the lower bifurcation point /zf from above, the oscillations decrease in size and period. The amplitude falls to zero at this lower bifurcation point, but the period remains non-zero. 

The period of the oscillations at the bifurcation point is given by 2n/co0. Thus the period is shorter at the upper bifurcation than at the lower one. 

Some typical oscillatory records are shown in Fig. 4.6. For conditions close to the Hopf bifurcation points the excursions are almost sinusoidal, but this simple shape becomes distorted as the oscillations grow. For all cases shown in Fig. 4.6, the oscillations will last indefinitely as we have ignored the effects of reactant consumption by holding /i constant. We can use these computations to construct the full envelope of the limit cycle in /r-a-0 phase space, which will have a similar form to that shown in Fig. 2.7 for the previous autocatalytic model. As in that chapter, we can think of the time-dependent... 

First, can we expect any oscillatory behaviour Instability is possible only if k < e 2. This requirement is satisfied here. From the data in Table 4.4, the Hopf bifurcation points for this system occur for n = 0.207 and n = 0 058. For our example, the initial value /r0 = 0.5 exceeds the upper bifurcation point, so the system at first has a stable pseudo-stationary state to approach, with dss x 10 and ass x 4.54 x 10 4. From Fig. 4.3 we may also estimate that the approach to this state will be monotonic since the initial conditions lie outside the region of damped oscillations. 

When we come to look at the stability of the limit cycle which is born at the Hopf bifurcation point, we shall meet a quantity known as the Floquet multiplier , conventionally denoted p2, which plays a role similar to that played for the stationary state by the eigenvalues and k2. If / 2 is negative, the limit cycle will be stable and should correspond to observable oscillations if P2 is positive the limit cycle will be unstable. 

Fig. 10.3. The locus of Hopf bifurcation points indicating the conditions for loss of local stability for the spatially uniform stationary-state solution. Inside this region the system may show spatially uniform time-dependent oscillations.

With p = 0.019, the traverse cuts both Hopf curves, so the stationary-state locus has four Hopf bifurcation points, as shown in Fig. 12.6(c), each one supercritical. There are two separate ranges of the partial pressure of R over which a stable limit cycle and hence sustained oscillations occur. 

FIGURE I Bifurcation diagrams of the autonomous system for y, = 0.001, y2 = 0.002. (a) Multiple-steady states are found inside the finger-shaped region, and limit cycles are born upon crossing the Hopf curves, (b) Steady-state and limit-cycle branches for < 1 = 0.019. The location of the average value of a2 used for forced oscillations is 0.028, which is in the oscillatory region and a distance of Ao from the left Hopf bifurcation point labelled c. 

When the HSS solution of the chemical rate equations (la)—(lc) first becomes unstable as the distance from equilibrium is increased (by decreasing P, for example), the simplest oscillatory instability which can occur corresponds mathematically to a Hopf bifurcation. In Fig. 1 the line DCE is defined by such points of bifurcation, which separate regions of stability (I,IV) of the HSS from regions of instability (II,III). Along section a--a, for example, the HSS becomes unstable at point a. Beyond this bifurcation point, nearly sinusoidal bulk oscillations (QHO, Fig. 3a) increase continuously from zero amplitude, eventually becoming nonlin-... 

Figure 7.28 shows the period of oscillations as the periodic branch approaches the homoclinical bifurcation point the period tends to infinity indicating homoclinical termination of the periodic attractor at Dht = 0.0035 hr 1. 

For normotensive rats, the typical operation point around a = 10—12 and T = 16 s falls near the Hopf bifurcation point. This agrees with the experimental finding that about 70% of the nephrons perform self-sustained oscillations while the remaining show stable equilibrium behavior [22]. We can also imagine how the system is shifted back and forth across the Hopf bifurcation by variations in the arterial pressure. This explains the characteristic temporal behavior of the nephrons with periods of self-sustained oscillations interrupted by periods of stable equilibrium dynamics. 

Figure 12.6a shows the temporal variation of the proximal tubular pressure Pt as obtained from the single-nephron model for a = 12 and T = 16 s. All other parameters attain their standard values as listed in Table 12.1. Under these conditions the system operates slightly beyond the Hopf bifurcation point, and the depicted pressure variations represent the steady-state limit cycle oscillations reached after the initial transient has died out For physiologically realistic parameter values the model reproduces the observed self-sustained oscillations with characteristic periods of 30-40 s. The amplitudes in the pressure variation also correspond to experimentally observed values. Figure 12.6b shows the phase plot Here, we have displayed the normalized arteriolar radius r against the proximal intratubular pressure. Again, the amplitude in the variations of r appears reasonable. The motion... 

There is a voluminous literature on steady-state multiplicity, oscillations (and chaos), and derivation of bifurcation points that define the conditions that lead to onset of these phenomena. For example, see Morbidelli et al. [ Reactor Steady-State Multiplicity and Stability, in Chemical Reaction and Reactor Engineering, Carberry and Varrria (eds), Marcel Dekker, 1987], Luss [ Steady State Multiplicity and Uniqueness... 

A dynamic system may exhibit qualitatively different behavior for different values of its control parameters 0. Thus, a system that has a point attractor for some value of a parameter may oscillate (limit cycle) for some other value. The critical value where the behavior changes is called a bifurcation point, and the event a bifurcation [32]. More specifically, this kind of bifurcation, i.e., the transition from a point attractor to a limit cycle, is referred to as Hopf bifurcation. 

Thermodynamics had been studied both in far-from-equilibrium and in near-equilibrium situations. A near-equilibrium world is a stable world. Fluctuations regress. The system returns to equilibrium. The situation changes dramatically far from equilibrium. Here fluctuations may be amplified. As a result, new space-time structures arise at bifurcation points. We considered the possibility of oscillating reactions as early as in 1954, many years before they were studied systematically. We introduced concepts such as selforganization and dissipative structures, which became very popular. In short, irreversible processes associated to the flow of time have an important constructive role. Therefore, the question that arises is how to incorporate the direction of time into the fundamental laws of physics, be they classical or quantum. 

Hopf bifurcation points from which relaxation oscillations emanate, were obtained as the solution of the system of equations ... 

If attention is restricted to the vicinity of the bifurcation point, then a nonlinear perturbation analysis can be developed for describing analytically the nature of the pulsating mode [111]. In effect, the difference between A and its bifurcation value is treated as a small parameter, say e, and oscillatory solutions for temperature profiles are calculated as perturbations about the steady solution in the form of a power series in /e. The departure of the oscillation frequency from its value at bifurcation is expressed in the same type of series. The methods of analysis possess a qualitative similarity to those of the shock-instability analysis discussed at the end of the previous section. The results exhibit the same general behavior that was found from the numerical integrations [109] for conditions near bifurcation. 

Chaotic oscillations near Zero-Hopf bifurcation point.198... 

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