PoincaréDulac

Poincaré-Dulac refers to a foundational result in local normal form theory for analytic vector fields near a fixed point. It provides a systematic method to simplify the equations by removing nonresonant nonlinear terms through near-identity changes of coordinates, leaving only resonant nonlinearities that reflect the intrinsic structure of the linear part.

Statement and key ideas: Consider a system x' = A x + f(x) in a neighborhood of 0, with f(0) = 0 and Df(0) = 0, and let λ1, ..., λn be the eigenvalues of A. A monomial x^m e_j is called resonant if λ_j = m1 λ1 + ... + mn λn, where m = (m1, ..., mn) is a multi-index. The Poincaré-Dulac theorem asserts that there exists a formal, near-identity change of coordinates transforming the system into a normal form in which the nonlinear part consists only of resonant terms. In this form, all nonresonant terms have been eliminated by the normalization procedure. The resulting normal form is formal in general and is unique up to choices among equivalent representations.

Convergence and implications: In the analytic (convergent) category, convergence of the normalization depends on arithmetic properties of the eigenvalues and on small-divisor phenomena; in many cases the formal transformation diverges, so the normal form is formal rather than analytic. Nevertheless, the Poincaré-Dulac normal form provides a clean, canonical description of the local dynamics near the fixed point and is a central tool for classifying equilibria and studying resonant bifurcations.

See also: normal form theory, linearization, resonance, small divisors.