Two routes to chaos in condensed phase combustion

The equations governing two models of gasless combustion which exhibit pulsating solutions are numerically solved. The models differ in that one allows for melting of the solid fuel, while the other does not. While both models undergo a Hopf bifurcation from a solution propagating with a constant velocity to one propagating with a pulsating (T-periodic) velocity when parameters related to the activation energy exceed a critical value, the subsequent behavior differs markedly. Numerically both models exhibit a period doubling transition to a 2T solution when the bifurcation parameter for each model is further increased. For the model without melting, a sequence of additional period doublings occurs, after which apparently chaotic solutions are found. For the model with melting, it is found that the 2T solution returns to the T-periodic solution branch. Then two additional windows of 2T behavior are found. After the last such window, the solution no longer returns to the T-periodic solution branch, but rather exhibits intermitency, with long laminar regions interrupted by randomly occurring bursts. Further increasing the bifurcation parameter leads to shorter laminar regions, with the bursts occurring more frequently. Increasing the bifurcation parameter yet further leads to apparently fully chaotic solutions. The numerical results demonstrate two mechanisms for chaotic dynamics in gasless combustion.