Modified Carleman Linearization and its use in oscillators

The standard Carleman linearization states that every analytic n-dimensional nonlinear systems is equivalent to an infinite dimensional linear system. In this paper we truncate this linearization and introduce some modification which reduces the error in the truncation process. We applied this modified Carleman linearization to two van der Pol oscillators with slight different frequencies. Each one is approximate by a 14th order linear system; then we coupled this two linear oscillators and look for a synchronization. We conclude that even the approximation is very good, is not possible synchronize high order linear oscillators as the nonlinear oscillators do.