Bifurcations and selection of equilibria in a simple cosymmetric model of filtrational convection

A three-dimensional set of ordinary differential equations that constitutes a simple abstract model of Darcy convection is investigated. The model reproduces a number of effects that are typical for dynamic systems with nontrivial cosymmetry. Nontrivial cosymmetry can give rise to a continuous family of equilibria where, in this case, the equilibrium stability spectrum varies along the family. The family of equilibria and its stability are examined analytically, and special bifurcations that occur in the system are investigated. It is shown that discrete and continual symmetries, called "flash symmetries," can be present in the system for certain parameter values. Computer experiments on the selection of equilibria in the symmetric and cosymmetric cases have been carried out. They showed that, for initial points that are far enough from a cycle of equilibria, the neighborhood of a single equilibrium is established in the case of cosymmetry, but all the equilibria are equivalent in the case of symmetry. The authors hope that these results, as well as the formulation of the problems and the approach to their solution, will serve as a sample in the investigation of more complex systems in mathematical physics. (c) 1999 American Institute of Physics.