Date of Degree
polynomial vector fields, parameter identifiability, input-output equations, mathematical biology, differential algebra, characteristic set
In the first part, we study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field. One motivating factor is that we can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent commuting vector field. This problem is also central to the question of linearizability of vector fields. We first show that a linear vector field admits a full complement of commuting vector fields. Then we study a type of planar vector field for which there exists an upper bound on the degree of a commuting polynomial vector field. Finally, we turn our attention to conservative Newton systems, which form a special class of Hamiltonian systems, and show the following result. Let f be in K[x], where K is a field of characteristic zero, and d be the derivation that corresponds to the differential equation x'' = f(x) in a standard way. We show that if the degree of f is at least 2, then any K-derivation commuting with d is equal to d multiplied by a conserved quantity. For example, the classical elliptic equation x'' = 6x^2 + a, where a is a complex number, falls into this category.
In the second part, we study structural identifiability of parameterized ordinary differential equation models of physical systems, for example, systems arising in biology and medicine. A parameter is said to be structurally identifiable if its numerical value can be determined from perfect observation of the observable variables in the model. Structural identifiability is necessary for practical identifiability. We study structural identifiability via differential algebra. In particular, we use characteristic sets. A system of ODEs can be viewed as a set of differential polynomials in a differential ring, and the consequences of this system form a differential ideal. This differential ideal can be described by a finite set of differential equations called a characteristic set. The technique of studying identifiability via a set of special equations, sometimes called “input-output” equations, has been in use for the past thirty years. However it is still a challenge to provide rigorous justification for some conclusions that have been drawn in published studies. Our main result is on linear systems, which are a topic of current interest. We show that for a linear system of ODEs with one output, the coefficients of a monic characteristic set are identifiable. This result is then generalized, with additional hypotheses, to nonlinear systems.
Thompson, Peter, "A Differential Algebra Approach to Commuting Polynomial Vector Fields and to Parameter Identifiability in ODE Models" (2019). CUNY Academic Works.