Stability analysis oscillations

An important outcome of these simulations is the location of HB points (largely ignored in previous work), which is important for the development of extinction theory. In particular, the turning point E lies on a locally unstable stationary solution branch and does not coincide with the actual extinction, as previously thought. The actual extinction point is the termination point of oscillations. Thus, local stability analysis is essential to properly analyze flame stability and develop extinction theory. 

We have now seen how local stability analysis can give us useful information about any given state in terms of the experimental conditions (i.e. in terms of the parameters p and ku for the present isothermal autocatalytic model). The methods are powerful and for low-dimensional systems their application is not difficult. In particular we can recognize the range of conditions over which damped oscillatory behaviour or even sustained oscillations might be observed. The Hopf bifurcation condition, in terms of the eigenvalues k2 and k2, enabled us to locate the onset or death of oscillatory behaviour. Some comments have been made about the stability and growth of the oscillations, but the details of this part of the analysis will have to wait until the next chapter. 

The interest in periodically forced systems extends beyond performance considerations for a single reactor. Stability of structures and control characteristics of chemical plants are determined by their responses to oscillating loads. Epidemics and harvests are governed by the cycle of seasons. Bifurcation and stability analysis of periodically forced systems is especially important in the... 

In the present paper we study common features of the responses of chemical reactor models to periodic forcing, and we consider accurate methods that can be used in this task. In particular, we describe an algorithm for the numerical computation and stability analysis of invariant tori. We shall consider phenomena that appear in a broad class of forced systems and illustrate them through several chemical reactor models, with emphasis on the forcing of spontaneously oscillating systems. 

The dynamic behavior of this system was examined using linear stability analysis [8] and is thoroughly discussed in the review articles [9, 10], The results can be best summarized in a so-called two-parameter bifurcation diagram, in which, similar to phase diagrams, regions with qualitatively different behavior (states) are indicated. The dominant regimes of the N-NDR oscillator (Eqs. (38a,b)) are depicted in Fig. 11... 

It is well established that under appropriate circumstances, delayed negative feedback mechanisms can produce oscillations. To illustrate this point we continue with the stability analysis. 

How can the result of unique steady state be consistent with the observed oscillation in Figure 5.9 The answer is that the steady state, which mathematically exists, is physically impossible since it is unstable. By unstable, we mean that no matter how close the system comes to the unstable steady state, the dynamics leads the system away from the steady state rather than to it. This is analogous to the situation of a simple pendulum, which has an unstable steady state when the weight is suspended at exactly at 180° from its resting position. (Stability analysis, which is an important topic in model analysis and in differential equations in general, is discussed in detail in a number of texts, including [146].)... 

Instead of using this equation, in the literature, there are few models proposed by which the frequency or Strouhal number of the shedding is fixed. Koch (1985) proposed a resonance model that fixes it for a particular location in the wake by a local linear stability analysis. Upstream of this location, flow is absolutely unstable and downstream, the flow displays convective instability. Nishioka Sato (1973) proposed that the frequency selection is based on maximum spatial growth rate in the wake. The vortex shedding phenomenon starts via a linear instability and the limit cycle-like oscillations result from nonlinear super critical stability of the flow, describ-able by Eqn. (5.3.1). 

They treated the system much like a CSTR, with the balance for the gas-phase concentration substituted by the coverage equation for the catalyst. Ray and Hastings then applied the analytical treatment that they had developed for the CSTR in this same publication. Stability analysis revealed that the critical Lewis numbers for oscillations were in a range that did not allow for oscillations on normal nonporous catalytic surfaces. However, as Jensen and Ray 243) showed, a certain model for catalytic surfaces, the fuzzy wire model, with the assumption of a very rough surface with protrusions is able to produce Lewis numbers in the proper range for the occurrence of oscillations. This model, however, included both mass and heat balances as well as coverage equations, thus combining the two classes of reactor-reaction models discussed above. 

As in the model of Section 2, the problem can be studied on its omega limit set with three rest points Eq,Ei,E2. A local stability analysis and, for some special cases, the asymptotic behavior of solutions were given in [E]. However, the populations cannot invade each other simultaneously El and E2 cannot be simultaneously unstable), so the persistence theory does not hold [E]. Moreover, for Michaelis-Menten dynamics, when one of the boundary rest points is locally stable and the other unstable, the locally stable one is globally stable [HWE]. In particular, the oscillation observed in the case of system (3.2) does not occur with (3.4). Indeed, the delayed system seems to behave much like the simple chemostat. 

THERMOKINETIC FEEDBACK OSCILLATIONS AND LOCAL STABILITY ANALYSIS... 

Using these methods, several resonant quaslperlodlc and periodic orbits were computed and plotted In the Internal coordinate space. These orbits exhibit resonant energy transfer between local (dressed) vlbra-tlon-bend oscillations In the entrance and exit regions of the collision complex. Frequencies and actions from the periodic orbits were then used in the arbitrary-trajectory semlclasslcal quantization scheme (19). The lowest resonance energy predicted for the J=0 reaction was In good agreement with all available quantal and adiabatic results. Further properties of both types of orbit, including those obtained from a stability analysis, will be presented elsewhere (21). 

Steady-state B) is the interesting one, of course, since steady-state (A) corresponds to complete washout of cells. Stability analysis has shown that the two cannot coexist at the same holding time either (A) is stable and (B) is unstable, or (S) is stable and (A) is unstable. Moreover, since (B) is a node if it is stable, we see that Monod s model will not predict oscillations— even damped ones—about a steady state of nonzero cell concentration. Hence, in this sense, there has been no improvement over the Verhulst-Pearl model. 

The linear stability analysis of eqns (2.30) shows that the domain of instability of the unique steady state is larger than in the case of a linear sink of product. The main effect of the Michaelian sink of product is, however, to allow the occurrence of sustained oscillations in the absence of enzyme cooperativity - but not of autocatalytic regulation (Goldbeter Dupont, 1990). 

Let us now see how the nullcline deformation due to recycling gives rise to birhythmicity. Shown in fig. 3.6 are the bifurcation diagrams obtained as a function of parameter v for eight increasing values of the maximum rate of recycling, a , for a fixed value of constant K. In each part, the steady-state value of the substrate ao is indicated, as well as the maximum value qim reached by the substrate during oscillations. Solid lines denote stable steady-state or periodic solutions both types of solution are indicated by dashed lines when unstable. The stability properties of the steady state were determined by linear stability analysis of eqns... 

The stability analysis of the unique steady state admitted by eqns (5.1) yields parameter values for which this state becomes unstable (see also Odell, 1980). The numerical integration of the kinetic equations then shows that the system acquires periodic behaviour in the course of time (fig. 5.17) large-amplitude oscillations are observed for intracellular cAMP (/3) their spike shape is reminiscent of that of the experimentally observed oscillations (fig. 5.12). Oscillations of extracellular cAMP (y), of reduced amplitude, closely follow the periodic evolution of variable j8. Moreover, these oscillations are accompanied by a significant variation of the ATP concentration (a). 

The systems of eqns (5.9a-d) and (5.12) have been subjected to linear stability analysis, in order to determine the conditions in which the regulation of the cAMP-synthesizing system gives rise to an instability of the steady state followed by sustained oscillations (Martiel Goldbeter, 1987a). Two examples of a typical stability diagram established as a function of parameters Li and L2 are shown in fig. 5.29. These parameters denote here the ratio between the kinetic constants of dephosphorylation and phosphorylation of the receptor, free or com-plexed with the ligand, respectively (see table 5.3 for further details on the definition of the parameters). 

Fig. 7.2. Developmental path of the cAMP signaUing system in the parameter space formed by adenylate cyclase and phosphodiesterase activity. The stability diagram is established by linear stability analysis of the steady state admitted by the three-variable system (5.1) governing the dynamics of the allosteric model for cAMP signalling in D. discoideum (see section 5.2). In domain C sustained oscillations occur around an unstable steady state. In domain B, the steady state is stable but excitable as it amplifies in a pulsatile manner a suprathreshold perturbation of given amplitude. Outside these domains the steady state is stable and not excitable. The arrow crossing successively domains A, B and C represents the developmental path that the system should follow in that parameter space to account for the observed sequence of developmental transitions no relay relay oscillations (Goldbeter, 1980).

Fig. 9.6. Stability diagram established as a function of the threshold constant for release, and of the total (basal plus signal-triggered) influx of Cif into the cytosol (vq + Vij8). The diagram is obtained by linear stability analysis of eqns (9.1)-(9.2) around the unique steady-state solution admitted by these equations. Parameter values are Vm2 = 100 fi-M/s, = 1 mM/s, m = n=p = 2, Xj = 1 p,M, Xa = 2.5 j,M, k =2s" kf = 0. The steady state is unstable in the dotted domain sustained oscillations of Ca occur under these conditions (Dupont Goldbeter, 1989).

Under these conditions system (9.1) still admits a unique steady state, but linear stability analysis shows that the latter is always stable (Goldbeter Dupont, 1990) this rules out the occurrence of sustained oscillations around a nonequilibrium unstable steady state. This result holds with previous studies of two-variable systems governed by polynomial kinetics these studies indicated that a nonlinearity higher than quadratic is needed for limit cycle oscillations in such systems (Tyson, 1973 Nicolis Prigogine, 1977). Thus, in system (9.1), it is essential for the development of Ca oscillations that the kinetics of pumping or activation be at least of the Michaelian type. Experimental data in fact indicate that these processes are characterized by positive cooperativity associated with values of the respective Hill coefficients well above unity, thus favouring the occurrence of oscillatory behaviour. 

Fig. 9.19. Concentration of cytosolic as a function of the stimulation level (jS) in the one- and two-pool models based, respectively, on the IPj-sensitive and IPj-insensitive Ca -induced Ca release. The solid lines represent the stable level of cytosolic Ca or the maximum and minimum cytosolic Ca concentration reached during oscillations the dashed line indicates the steady-state level of cytosolic Ca in the domain of j8 values where this state is unstable and oscillations occur. Parameter values are A = 10 min", Af - 1 min , n = m - 2 and p - 4. Moreover, for the upper (lower) panel, Vq -1 (1.7) jtM/min, Vj = 7.3 (1.7) jAM/min, V z - 65 (25) pM/min, = 500 (325) p.M/min, = 1 (0-5) jxM, = 2 (1) jiM, Kf = 0.9 (0.45) p,M. The lower values considered for some parameters in the one-pool model have been adjusted so as to limit the amplitude of the first Ca spike to the 1-2 pM range. The concentrations of intravesicular and cytosolic Ca are defined with respect to the total cell volume the actual intravesiculcu Ca concentration is therefore larger than on the given scale. The curves are established by linear stability analysis and numeric integration of eqns (9.1) the expression of Vj in the two-pool and one-pool versions of the model is given by eqns (9.2) and (9.8), respectively (Dupont Goldbeter, 1993).

Fig. 10.8. Stability diagram established as a function of the reduced Michaelis constants X, of the first cycle of the minimal cascade model of fig. 10.4, versus the reduced Michaelis constants (Kj, K4) of the second cycle. The domain of oscillations corresponds to the domain of instability of the unique steady state admitted by eqns (10.1). The stabiUty properties of the steady state are determined by linear stability analysis. The diagrams are established for (a) equal or (b) unequal values of (X K2) on the one hand, and (X3, 4) on the other. Parameter values are as in fig. 10.6 (Guilmot Goldbeter, 1995).

Linear stability analysis of model for birhythmicity, 95 of model for cAMP oscillations, 180 of model for glycolytic oscillations,... 

Stability diagrams for Ca oscillations, 364 for cAMP oscillations, 203,245 for developmental transitions, 287,290 for glycolytic oscillations, 51 for mitotic oscillator, 431,440,441,443 Start, checkpoint in cell cycle, 413 Steady state, stable or unstable, 49,56,62, 120,121,122,141,203,253-6,288,366, 438 see also Bifurcation diagram Bistability Hard excitation Linear stability analysis Stability diagrams Tristability... 

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