Non-Regular, Incomplete And Singular Dynamical Systems: Impasse Points In Dynamical Systems

Abstract

The non-regular, incomplete and singular differential dynamical systems are discussed from a geometric point of view. The constitutive space of a dynamical system is considered as a subset of the tangent bundle to the manifold (the generalised hypersurface) being the ambient space of the system. The system is the essentially non-regular differential dynamical system if there is observed reduction of the configuration space to the 'proper' configuration space of the system. The dynamical system, which is not observed by the solutions in the entire configuration space, is called the incomplete system. The system such that the dynamic space (the solution space) is not the topologically closed subspace of the ambient space of the system is called the singular dynamical system. The points in the topological closure of the configuration space that are not contained in the solution space of the dynamical system are called the impasse points of the system. The points in the topological closure of the solution space that are not contained in the solution space of the system are called the singularity points of the dynamical system. The systematic classification of the impasse points in dynamical systems is included which extends the classification proposed for the systems defined by the differential algebraic equations. The well posed and the relatively well-posed dynamical systems are defined. These are the systems, which are considered most often in the modelling of real systems. The classification which includes the non-regular, the incomplete, the well posed, and the relatively well-posed dynamical systems is proposed. The trajectory available impasse points have been extended for the general differential dynamical systems. The trajectory available impasse points are important notion in the modelling and in the simulation of real systems which exhibit the discontinuous trajectory behaviour. At the time when the trajectory of the system approached the singularity point, one usually observes the discontinuity in the dynamic behaviour of the system or the 'switching' of the trajectories which is possibly continuous takes place in the system. The examples are enclosed which illustrate the considerations.