Global Existence and Stability of Solutions of Matrix Riccati Equations

We consider a matrix Riccati equation containing two parameters c and . The quantity c denotes the average total number of particles emerging from a collision, which is assumed to be conservative Ži.e., 0 c 1., and Ž0 1. is an angular shift. Let S Žc, . : 0 c 1 and 0 14. Stability analysis for two steady-state solutions Xmin and Xmax are provided. In particular, we prove that Xmin is locally asymptotically stable for S Ž1, 0.4, while Xmax is unstable for S Ž1, 0.4. For c 1 and 0, Xmin Xmax is neutral stable. We also show that such equations have a global positive solution for Žc, . S, provided that the initial value is small and positive.