AN ORGANIZING CENTER FOR WAVE BIFURCATION IN MULTIPHASE FLOW MODELS

We consider a one-parameter family of nonstrictly hyperbolic systems o f conservation laws modeling three-phase flow in a porous medium. For a particular value of the parameter, the model has a shock wave soluti on that undergoes several bifurcations upon perturbation of its left a nd right states and the parameter. In this paper we use singularity th eory and bifurcation theory of dynamical systems, including Melnikov's method, to find all nearby shock waves that are admissible according to the viscous profile criterion. We use these results to construct a unique solution of the Riemann problem for each left and right state a nd parameter value in a neighborhood of the unperturbed shock wave sol ution; together with previous numerical work, this construction comple tes the solution of the three-phase flow model. In the bifurcation ana lysis, the unperturbed shock wave acts as an organizing center for the waves appearing in Riemann solutions.