Simultaneous local normal forms of dynamical systems with singular underlying geometric structures

Abstract The aim of this paper is to develop, for the first time, a general theory of simultaneous local normalisation of couples <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">G</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , where X is a dynamical system (vector field) and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">G</mml:mi> </mml:mrow> </mml:mrow> </mml:math> is an underlying geometric structure preserved by X , even if both have singularities. Such couples appear naturally in many problems, e.g. Hamiltonian dynamics, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">G</mml:mi> </mml:mrow> </mml:mrow> </mml:math> is a symplectic structure and one has the theory of Birkhoff normal forms, or constrained dynamics, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">G</mml:mi> </mml:mrow> </mml:mrow> </mml:math> is a smooth, in general singular, distribution of tangent subspaces, etc. In this paper, the geometric structure <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">G</mml:mi> </mml:mrow> </mml:mrow> </mml:math> is of the following types: volume form, symplectic form, contact form, Poisson tensor, as well as their singular versions. The paper addresses mainly the more difficult situations when both X and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">G</mml:mi> </mml:mrow> </mml:mrow> </mml:math> are singular at a point and its results prove the existence of natural simultaneous normal forms in these cases. In general, the normalisation is only formal, but when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">G</mml:mi> </mml:mrow> </mml:mrow> </mml:math> and X are (real or complex) analytic and X is analytically or Darboux integrable, then the simultaneous normalisation is also analytic. Our theory is based on a new approach, called the Toric Conservation Principle , as well as the classical step-by-step normalisation technique, and the equivariant path method.