SECOND-ORDER NORMAL FORMS FOR N-DIMENSIONAL SYSTEMS WITH A NILPOTENT POINT

Abstract

Normal forms theory is one of the most powerful tools for the study of nonlinear differential equations, in particular, for stability and bifurcation analysis. Many works paid attention to normal forms associated with nilpotent Jacobian where the critical eigenvalues have algebraic multiplicity k ($ k>1 $) and geometric multiplicity one, and in particular, the case $ k>2 $ is more complicated for determining unfolding. Despite a lot of theoretical results on nilpotent normal forms have been obtained, computation developing can not satisfy practical applications. To our knowledge, no results have been reported on the computation of explicit formulas of the nilpotent normal forms for $ k>3 $ with perturbation parameters. The main difficulty is how to determine the complementary spaces of the Lie transformation. In this paper, we achieve the following results. (1) A simple dimension formula for the complementary space of the Lie transform; (2) a simple direct method to determine a basis of the complementary spaces; (3) a simple direct method to determine the projection of any vector to the complementary spaces. Using this method, the second-order normal forms for any n-dimensional nilpotent systems can be given easily. As an illustrative application, the normal forms for the vector field with triple-zero or four-fold zero singularity and functional differential equation with a triple-zero singularity are presented, and explicit formulas for the normal form coefficients with three or four unfolding parameters are obtained.