Type theory in differential equations

The advances in group-theoretical and categorical approaches to differential equations are very helpful in the development of their typology. For example, the information concerning the symmetry group of a dynamical object—that concerning admissible transformations of variables and parameters—provides also essential information about the model under study, namely, a method for classifying the solutions, a method classifying the equations as dependent on parameter(s) (numerical or functional), a possibility to construct the type of equations with a given symmetry group. The group properties of the diffeomorphisms and vector fields over a manifold may permit us to find the simplest diffeomorphisms from the equivalence classes.