Oscillation stable periodic

Of particular interest is the long-term behavior of voting-rule systems, which turns out to very strongly depend on the initial density of sites with value cr = 1 (= p). While all such systems eventually become either stable or oscillate with period-two, they approach this final state via one of two different mechanisms either through a percolation or nucleation process. Figure 3.60 shows a few snapshots of a Moore-neighborhood voting rule > 4 for p = 0.1, 0.15, 0.25 and 0.3. 

Controlled chaos may also factor into the generation of rhythmic behavior in living systems. A recently proposed modeL describes the central circadian oscillator as a chaotic attractor. Limit cycle mechanisms have been previously offered to explain circadian clocks and related phenomena, but they are limited to a single stable periodic behavior. In contrast, a chaotic attractor can generate rich dynamic behavior. Attractive features of such a model include versatility of period selection as well as use of control elements of the type already well known for metabolic circuitry. 

In a series of experiments we have tested the type and range of entrainment of glycolytic oscillations by a periodic source of substrate realizing domains of entrainment by the fundamental frequency, one-half harmonic and one-third harmonic of a sinusoidal source of substrate. Furthermore, random variation of the substrate input was found to yield sustained oscillations of stable period. The demonstration of the subharmonic entrainment adds to the proof of the nonlinear nature of the glycolytic oscillator, since this behavior is not observed in linear systems. A comparison between the experimental results and computer simulations furthermore showed that the oscillatory dynamics of the glycolytic system can be described by the phosphofructokinase model. 

At interesting phenomenon occurs in the case of other resonance horns we have studied it for the case of the 3/1 resonance. The torus pattern breaks when the subharmonic periodic trajectories locked on it for small FA decollate from the torus as FA increases. We are left then with two attractors a stable period 3 and a stable quasi-periodic trajectory. This is a spectacular case of multistability (co-existence of periodic and quasi-periodic oscillations). The initial conditions will determine the attractor to which the system will eventually converge. This decollation of the subharmonics from the torus was predicted by Greenspan and Holmes (1984). They also predicted chaotic trajectories close to the parameter values where the subharmonic decollation occurs. 

Since equilibrium is a stable state, reaction systems at, or close to, equilibrium will not oscillate27. Therefore, one necessary condition for oscillation is that the system be far from equilibrium. We are, of necessity, dealing with open, irreversible systems. Second, and equally important, is the existence of a feedback mechanism. Oscillation or periodicity implies a return to some initial state. That this repetitive behavior would imply a particular kind of feedback, however, is not at all obvious. 

For T = 16 s, the single nephron model undergoes a supercritical Hopf bifurcation at a = 11 (outside the figure), fn this bifurcation, the equilibrium point loses its stability, and stable periodic oscillations emerge as the steady-state solution. For a = 19.5, at the point denoted PDla 2 in Fig. 12.5, this solution undergoes a period-... 

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... 

In the previous section we saw that every solution of (3.2) converges to a periodic solution. The goal of this section is to provide sufficient conditions for the existence of a positive periodic solution of (3.2) possessing strong stability properties. Hereafter, a solution (or vector) x(t) = (Xi(0. - 2(0) will be called positive provided both components are positive. A positive stable periodic solution of (3.2) corresponds to the coexistence of both competitors in the chemostat. In this case, each competitor s concentration oscillates between a positive minimum and maximum value. It is important to point out that coexistence of two competitors can take place only if there exists a positive periodic solution of (3.2). If coexistence means that the concentrations of each of the two populations must remain... 

One singular point, two singular points, three singular points and their stability, as well as stable periodic solutions (sustained oscillations). 

Fig. 3.8. Reversible transition between two simultaneously stable periodic regimes. The transitions are elicited by the addition of adequate amounts of substrate, at the appropriate phase of each oscillation, as indicated in fig. 3.7 established for the same parameter values. The curves are obtained by numerical integration of eqns (3.1). In (a), the transition occurs upon increasing a up to the value 100 in t = 2300 s (initial conditions a - 83.9 y = 3.2), whereas the inverse transition occurs in (b) when a is raised up to the value 70 in t = 2220 s, starting from the initial conditions a = 102.9 and y - 2.6 (Moran Goldbeter, 1984).

The transition between the three stable periodic regimes can also occur, as in the case of birhythmidty, in response to a perturbation such as the addition of an appropriate quantity of substrate, provided that the perturbation occurs at the adequate phase of each oscillation (fig. 4.8). 

One very important mathematical result facilitates the analysis of two-dimensional (i.e., two concentration variables) systems. The Poincare Bendixson theorem (Andronov et al., 1966 Strogatz, 1994) states that if a two-dimensional system is confined to a finite region of concentration space (e.g., because of stoichiometry and mass conservation), then it must ultimately reach a steady state or oscillate periodically. The system cannot wander through the concentration space indefinitely the only possible asymptotic solution, other than a steady state, is oscillations. This result is extremely powerful, but it holds only for two-dimensional systems. Thus, if we can show that a two-dimensional system has no stable steady states and that all concentrations are bounded—that is, the system cannot explode—then we have proved that the system has a stable periodic solution, whether or not we can find that solution explicitly. 

Quartz A crystalline mineral that when electrically excited vibrates with a stable period. Quartz is typically used as the frequency-determining element in oscillators and filters. 

Figure 2. Dependence of the solution (y)Qo on the inlet concentration of oxygen in CO oxidation, obtained by the continuation. sSS-stable steady state, uSS-unstable steady state, sP-stable periodic oscillations (minimum and maximum values), Hopf BP-Hopf bifurca-tionpoint unstable periodic solutions are not presented. T=630 K( isothermal), 5=20 pm, Lhm=S0 mol.m, Lqsc-32 mol.m, no diffusiorml resistance in the washcoat.

We observe that the medium tends to the homogeneous bulk oscillation when 0 is small. The smooth curves in Fig.7 A are the wave trajectories for the numerical solution of (1) in the case I = O.O36 and 0 = 0.2. (The bulk reaction dynamics of (1) has a stable periodic oscillation with period Tp = 159 when I = O.O36.) The local disturbance around x = 0 initiates the bulk oscillation in this region which transiently acts as a pacemaker. The leading wave propagates with nearly constant velocity Ca, = 0.57 as it advances into the medium which is lingering for a long time near the (barely) unstable rest state. The next several succeeding waves initially... 

Such reactions are called oscillating or periodic. Nowadays several dozens of homogeneous and heterogeneous oscillating reactions have been explored. Investigations of the kinetic models for these complex processes have allowed formulating a series of general conditions, which are required for the stable oscillations of the reaction rates and intermediate concentrations ... 

Of the 12 possible initial states with 5 live cells, 5 yield the null state within t = A steps, 2 yield a stable state and four lead to the same period two oscillator marking the end of the evolution of state E in figure 3.67 In marked contrast to this rather benign behavior, however, the remaining 5-cell pattern - called the R-pentamino -evolves in a considerably more dramatic fashion. 

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