Carleman linearization of nonlinear systems with linear centers: return times and Poincaré-Lyapunov constants

Carleman linearization is applied to a two-dimensional nonlinear system with a linear center. The solution of the Carleman-linearized system is obtained without relying on the Jordan canonical form for the matrix exponential, using Putzer’s method. The solution of the Carleman-linearized system evaluated at the first return time contains the Poincaré-Lyapunov constants. We introduce a constructive procedure to obtain these constants. Examples support the validity of the proposed method.