Normal Form, Symmetry and Infinite Dimensional Lie Algebra for System of Ode's

摘要

The normal form for a system of ode's is constructed from its polynomialsymmetries of the linear part of the system, which is assumed to besemi-simple. The symmetries are shown to have a simple structure such asinvariant function times symmetries of degree one called basic symmetries. Wealso show that the set of symmetries naturally forms an infinite dimensionalLie algebra graded by the degree of invariant polynomials. This implies that ifthis algebra is non-commutative then the method of multiple scales with morethan two scaling variables fails to apply.