Salnikov model

Fig. 5.7. Variation of quasi-steady-state temperature excess O s with the dimensionless reactant concentration for the Salnikov model. The steady-state is stable at high and low fi but is unstable over a region fi < fi < fi indicated by the broken section of the locus.

For the Salnikov model, the partial derivatives can be evaluated and the steady-state solution substituted to obtain the Jacobian matrix for this two-variable system in the form... 

The terms node and focus are most easily understood if instead of plotting the perturbations as a function of time, we plot one variable against the other. This gives rise to a trajectory in the y-6 phase plane. The steady-state corresponds to a point on this plane and the trajectory indicates the direction in which the system evolves in the vicinity of this singular point (it is termed singular as the slope of the trajectory which is given by dd/dy = (dd/dT)/ dy/dT) = 0/0 at this point). In the case of a stable steady-state, the local trajectory is directed towards the steady-state point either approaching directly (node) or as an inward spiral (focus) for unstable points the flows are in the opposite direction. The phase portraits associated with the four cases above, and also for a fifth case to be discussed below but not present in the Salnikov model, as shown in Fig. 5.8. 

Returning to the specific case of the Salnikov model, the major qualitative change in behaviour occurs when damped oscillatory decay of the perturbation gives way to oscillatory growth. The condition for this change from case (ii) to case (iii) which is known as a Hopf bifurcation is, in general terms. 

The behaviour at the upper Hopf point is also that of a supercritical Hopf bifurcation although the loss of stability of the steady-state and the smooth growth of the stable limit cycle now occurs as the parameter is reduced. This is sketched in Fig. 5.10(b). We can join up the two ends of the limit cycle amplitude curve in the case of this simple Salnikov model to show that the amplitude of the limit cycle varies smoothly across the range of steady-state instability, as indicated in Fig. 5.11(a). The limit cycle born at one Hopf point survives across the whole range and dies at the other. Although this is the simplest possibility, it is not the only one. Under some conditions, even for only very minor elaboration on the Salnikov model [16b], we encounter a subcritical Hopf bifurcation. At such an event, the limit cycle that is born is not stable but is unstable. It still has the form of a closed loop in the phase plane but the trajectories wind away from it, perhaps back in towards the steady-state as indicated in Fig. 

Fig. 5.11. Variation of the oscillatory (limit cycle) solution with ju for the simple Salnikov model showing that the stable limit cycle born at one supercritical Hopf bifurcation exists over the whole range of the unstable steady-state, shrinking to zero amplitude at the other Hopf point (b) in this case, each Hopf point gives rise to a different limit cycle, with a stable limit cycle born at fi growing as increases and an unstable limit cycle born at /x also increasing in size as fx increases. At some fx> fx the two limit cycles collide and are...

Salnikov, I. Ye. (1948). Thermokinetic model of a homogeneous periodic reaction. Dokl. Akad. Nauk SSSR, 60, 405-8. 

Salnikov, I. E., 1948, Thermokinetic model of homogeneous periodic reactions. DokL AkatL Nauk. 60, 405. 

Another model studied under the name of thermokinetic feedback is due to Salnikov [14,15]. This has two first-order reaction steps which, in general, could both be exothermic and both have an Arrhenius temperature dependence ... 

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