,

,

,

where , , are the concentrations and is the temperature, is the Damköhler number, is the activation energy, is the ratio of the two rate constants, is the ratio of activation energies, is the heat transfer coefficient, is the coolant temperature, is the adiabatic temperature rise, and is the ratio of enthalpies of reaction. The equations are solved with and initial conditions . As the heat transfer coefficient is increased, the trajectories change from damped oscillations leading to a steady state to periodic, then to chaotic oscillations; further increases in β lead to damped oscillations and finally to an asymptotic approach to equilibrium.