Fold points and singularities in hall MHD differential algebraic equations

Abstract

We consider the singularity crossing phenomenon in differential- algebraic equations (DAEs) of Hall MHD systems in one spatial dimension. Depending on the dissipative mechanism involved, such systems are described by either DAEs or ODEs. The former have singularities which typically behave as impasse points, singular pseudo-equilibrium points, or singularity induced bifurcation points. Each of these types of points results in different qualitative behavior. The pseudo-equilibrium and singularity induced bifurcation points allow for smooth transitions between the plus (supersonic) and minus (subsonic) Riemann sheets. Within the singular pseudo-equilibrium points there may exist only one analytic trajectory (as in the case of SIB point), two analytic and two Lipschitz trajectories in the case of pseudo- saddle points, or two analytic and infinite number of trajectonot ries of lower smoothness in the case of singular pseudo-nodes. In the paper we analyze qualitative behavior and existence of the above singular points in Hall MHD systems described by DAEs and explain the singularity (forbidden curve3) crossing phenomenon by using the recent developments in the qualitative analysis of DAEs.