Taylor-map-based polynomial approximation of non-homogeneous dynamical systems

Abstract

The paper addresses the construction of a polynomial approximation of the flow of a dynamical system governed by autonomous ordinary differential equations with polynomial right-hand sides that contain non-homogeneous (constant) terms, in the case when the initial state is not close to the origin. The existing matrix Taylor map construction is restricted to homogeneous right-hand sides and to initial states in a neighborhood of zero. The contribution of the paper is twofold. First, an algorithm for constructing the matrix Taylor map is developed that combines (i) expansion in deviations from the actual initial state, Y = X − X0 , and (ii) augmentation of the phase space by a constant auxiliary variable, which reduces the non-homogeneous problem to a homogeneous one; the Taylor map is then obtained from the matrizant of the associated linear system for phase moments. Second, the construction is validated numerically on a Van der Pol oscillator with constant bias: a fifth-order map reproduces the displaced limit cycle over t ∈ [0, 100] with an absolute error of 10−4 –10−3 after the transient, and expansion around X0 remains accurate in the regime where the expansion around zero degrades. Unlike numerical integration, the construction yields an explicit parametric representation of the evolution operator whose coefficients are determined by the physical parameters of the system, and, unlike purely data-driven machine learning models, it exploits the governing equations. Since the coefficients enter linearly and have the same monomial structure as a dynamic polynomial regression model, the map can serve as a physically informed initial approximation for data-driven refinement; this application is motivated by prior work and is not evaluated experimentally here.