Linear stability analysis oscillations

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... 

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). 

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 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... 

Global StaMlity in the CSTR.— The failure of linear stability analysis to cover the macroscopic behaviour of the CSTR is well illustrated by the oscillatory states computed by Aris and Amundson for such a reactor operating with feedback control. Local stability analysis indicates an unstable equilibrium state but in the large this is surrounded by a stable limit cycle and the resultant behaviour is one of temperatures and concentrations oscillating about an unstable state, rather than approaching a stable one. 

When we studied the emergence of temporal oscillations in Chapter 2, we found that it was useful to examine whether a small perturbation to a steady state would grow or decay. We now attempt a similar linear stability analysis of a system in which diffusion, as well as reaction, can occur. First, consider the general reaction-diffusion equation ... 

Linear stability analysis provides one, rather abstract, approach to seeing where spatial patterns and waves come from. Another way to look at the problem has been suggested by Fife (1984), whose method is a bit less general but applies to a number of real systems. In Chapter 4, we used phase-plane analysis to examine a general two variable model, eqs. (4.1), from the point of view of temporal oscillations and excitability. Here, we consider the same system, augmented with diffusion terms a la Fife, as the basis for chemical wave generation ... 

The results obtained in this study were unequivocal. A combination of linear stability analysis and numerical simulation demonstrates that the first model cannot give oscillatory solutions for any values of the delay time and initial concentrations, while the second model, with an appropriate choice of z, yields oscillations very similar to those found in the three-variable ODE Oregonator model, as shown in Figure 10.6. Thus, the calculation not only shows that delay is essential in this system but also reveals just where in the mechanism that delay plays its role. 

Equation (10.50) has a unique steady state, corresponding to a fixed (and presumably healthy) level of ventilation. Linear stability analysis shows that this state is stable if the delay time is short enough and if the dependence of F on c at the steady state is not too steep, that is, if m is not too large. If these conditions are violated, the steady state becomes unstable, and pathological oscil-... 

Depending on parameter values, there could be multiple states and unique unstable states. Some of the former and all of the latter would lead to sustained oscillations. The usual mathematical methods have been employed in the analysis of oscillatory behavior, including linear stability analysis, Hopf bifurcation analysis and computer simulations. 

Sufficient conditions for instability of the motionless steady initial state of the layer can be obtained by means of a linear stability analysis using normal modes. As the results of this and the preceding energy analysis do not coincide there appear possibilities of subcritical modes of instability (finite amplitude steady states or oscillations) and transient oscillations. Figure 2. gives an illustration of some of the results obtained for the case of vanishing gravitational acceleration (g = 0). 

C. Bizon, M.D. Shattuck, and J.B. Swift. Linear stability analysis of a vertically oscillated granular layer. Physical Review E, 60(6) 7210-7216,1999. 

Multiple Steady States and Local Stability in CSTR.—In the two decades since the seminal work of van Heerden and Amimdson, there has been vast output of papers conoemed with the dynamic behaviour of stirred-tank reactors. Bilous and Amundson put the van He den analysis of local stability of the equilibrium state on a rigorous basis by use of linear stability theory. Their method is similar to the phase-plane treatments of thermokinetic ignitions and oscillations discussed here in Sections 4 and 3 (and preceded them dironologically). The mass and energy balance for the CSTR having a single reactant as feedstock may be expressed as ... 

As was noted in Section 2.1.1, the concentration oscillations observed in the Lotka-Volterra model based on kinetic equations (2.1.28), (2.1.29) (or (2.2.59), (2.2.60)) are formally undamped. Perturbation of the model parameters, in particular constant k, leads to transitions between different orbits. However, the stability of solutions requires special analysis. Assume that in a given model relation between averages and fluctuations is very simple, e.g., (5NASNB) = f((NA), (A b)), where / is an arbitrary function. Therefore k in (2.2.67) is also a function of the mean values NA(t) and NB(t). Models of this kind are well developed in population dynamics in biophysics [70], Since non-linearity of kinetic equations is no longer quadratic, limitations of the Hanusse theorem [23] are lifted. Depending on the actual expression for / both stable and unstable stationary points could be obtained. Unstable stationary points are associated with such solutions as the limiting cycle in particular, solutions which are interpreted in biophysics as catastrophes (population death). Unlike phenomenological models treated in biophysics [70], in the Lotka-Volterra stochastic model the relation between fluctuations and mean values could be indeed calculated rather than postulated. 

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