The chemical rate equations for the Chapman cycle modeling the production of ozone are O2 + hv → O + O, Rate = k1, O2 + O +M → O3 +M, Rate = k2, O3 + hv → O2 + O, Rate = k3, O + O3 → O2 + O2, Rate = k4, where O is a singlet, O2 is oxygen, and O3 is ozone. The reaction rate equations for species x = [O], y = [O2], and z = [O3] are

˙ x = 2k1y + k3z − k2xy[M] − k4xz,

˙ y = k3z + 2k4xz − k1y − k2xy[M],

˙z = k2xy[M] − k3z − k4xz.

This is a stiff system of differential equations. Many differential equations applied in chemical kinetics are stiff. Given that [M] = 9 × 1017, k1 = 3 × 10−12, k2 = 1.22 × 10−33, k3 = 5.5 × 10−4, k4 = 6.86 × 10−16, x(0) = 4 × 1016, y(0) = 2 × 1016, and z(0) = 2 × 1016, show that the steady state reached is [O] = 4.6806 × 107, [O2] = 6.999 × 1016, and

[O3] = 6.5396 × 1012.

168 7. Three-Dimensional Autonomous Systems and Chaos