An asymptotic analysis of the Sal’nikov thermokinetic oscillator

The Sal’nikov thermokinetic oscillator is studied in the limiting case where the dimensionless heat capacity tends to zero. This is equivalent to the ‘no fuel consumption’ approximation in classical thermal explosion theory and is equally revealing in that many exact results can be obtained by simple algebraic methods. Regions in parameter space are found where, although the system is asymptotically stable, a large single excursion occurs before the steady state is approached. These regions border the region of oscillations which in the limiting case are of the relaxation type. All the interesting behaviour requires RT / E < 1/4, an obvious parallel with thermal explosion theory. The unstable limit cycles that occur in the Sal’nikov oscillator disappear in this limiting case. However, the requirements for an unstable limit cycle to exist in the ‘relaxation’ limit are discussed. The homoclinic bifurcation in the limiting case is also examined and it is shown that this bifurcation can (in theory) be calculated exactly. In addition, an extension to the Sal’nikov oscillator scheme in a closed system to include fuel consumption is studied both numerically and in a limiting case. It is shown that the full scheme exhibits finite trains of almost periodic behaviour before monotonically approaching equilibrium.