Improved Adjoint Operator Method and Normal Form of Nonlinear Dynamical System

An improved adjoint operator based on the adjoint operator concept of linear operator and S-N decomposition is proposed to calculate the normal forms of k order general nonlinear dynamic systems.Firstly, the whole polynomial solution space of homogeneous nilpotent partial differential equation are obtained.Secondly, the polynomial solution mentioned above is introduced into homogeneous semi-simple partial differential equation to find the whole polynomial solution space of a homogeneous linear partial differential equation Therefore, more polynomial first integrals need not be found and the simplest normal form of nonlinear dynamical system can be obtained easily. The example shows that the method is very effective.