Implicit discrete-time systems and accessibility

An intrinsic description for dynamic systems, whose evolution along discrete time is governed by (nonlinear) implicit difference equations in one independent variable and zero-order (algebraic) equations, is presented by means of differential geometrical methods, where systems are associated with appropriate geometric objects reflecting their dynamics. Dynamic systems given in implicit form have the peculiarity that they may contain so-called hidden restrictions. A normal form is presented which is characterized by the circumstances that there are no further restrictions. In addition, it is illustrated that such a normal form allows for an equivalent system representation in explicit form. Based on the geometric picture of (implicit) discrete-time systems the qualitative property of accessibility along a fixed trajectory is discussed. By applying symmetry groups of discrete-time systems and studying invariants of these groups a formal approach is provided that allows us to gather local accessibility criteria successively, which can be tested by computer algebra. Several examples illustrate the results.