Carleman approximants for non-linear differential systems

Abstract We examine a method introduced by Torsten Carleman a century ago to approximate the solution of non-linear differential systems. His idea consists of replacing the non-linear system with an infinite-dimensional linear one. Carleman approximants are obtained when solving a truncated version. We describe the linearization process and present two different approaches to apply the Carleman technique. In the first one, which is the common case, the Carleman approximants are centred at an equilibrium point in phase space. The resulting linear system is always homogeneous, and higher orders of approximation are usually analytically solvable. The closer the point of interest is to the equilibrium point, the more accurate the approximate solution. In a way, this approach may be viewed as an extension of the so-called local, or qualitative, analysis of non-linear differential systems. In the second one, the approximants are built up around the initial condition. In this novel approach, there is a gain in accuracy at the price of solving a non-homogeneous linear system that allows for an analytical solution too. We found a noteworthy pattern in the analyticity properties of the approximants that explains this improved behaviour. The connection with the Koopman operator and related methods is commented on. A number of systems are worked out intended to provide evidence of the practical utility of the Carleman approach. We also derive approximate solutions for the Susceptible-Infected-Removed epidemiological model, which leads to an extended classification of the evolutionary stage of an epidemic process.