A New Approach for Obtaining Normal Forms of Non-Linear Systems

On the Computation of the Coefficients Associated With High Order Normal Forms

“…It presents a major improvement compared to the conventional normal form theory. The examples presented in reference [4] have clearly demonstrated that the above normal form approach provides a number of signi"cant advantages over the existing normal from theory which involves inverting a set of complex matrices (see equation (6)). This may take a long time and occupy a lot of memory in MAPLE calculations, especially in the calculations of high order normal forms, and it becomes increasingly more inconvenient as the order of the normal form increases.…”

“…This procedure is not very convenient and one needs to solve a series of algebraic equations. It becomes increasingly more di$cult as the order of the normal form increases [3,4].…”

“…A modi"ed normal form approach has been presented in references [4}6] which facilitates the calculation of normal forms and the associated coe$cients. The advantages of this approach have been described and demonstrated in reference [4]. However, if high order normal forms are required, the approach introduced in reference [4] is not very convenient to apply, even though it represents a major improvement compared to the conventional normal form theory.…”

“…The advantages of this approach have been described and demonstrated in reference [4]. However, if high order normal forms are required, the approach introduced in reference [4] is not very convenient to apply, even though it represents a major improvement compared to the conventional normal form theory. Some e!ort has been made to calculate the associated coe$cients of &&normal forms''*or certain simpli"ed forms*by other methods, like the method of averaging [7,8] or multiple scale method [8,9].…”

“…In this paper, the modi"ed normal form approach presented in reference [4] is further extended and a new procedure is introduced which lends itself readily to symbolic calculations, like MAPLE. A general MAPLE program is also presented in Appendix A.…”