Semenov developed a model that describes the thermal ignition phenomenon quantitatively. Assume that the temperature, , of the reacting system is constant and uniform across the volume of the system and assume that the walls of the system and the ambient external temperature, , are the same. The steady-state temperature is obtained by setting the heat production term (i.e., the left-hand side of the equation) equal to the heat losses term, which is mainly due to convection between the system and the surroundings (i.e., the right-hand side of the equation):
where is a dimensionless temperature, depends on the overall heat transfer coefficient, and depends on the activation energy.
The system presents multiple steady states, as you can see from the bifurcation diagram. Plotting the heat losses (shown in magenta) and the heat production term (shown in blue) makes it clear that three steady states (i.e., three intersection points identified by the red dots) can be obtained for some values of and .
The bifurcation diagram is obtained using two different techniques: (1) using the arc length continuation method; and (2) using the Mathematica built-in command ContourPlot.
Perfect agreement is obtained using these two methods. Finally, turning points are shown by the green dots.