Smooth equivalence and linearization of reversible systems.

*(English. Russian original)*Zbl 1021.37032
Math. Notes 70, No. 1, 86-96 (2001); translation from Mat. Zametki 70, No. 1, 96-108 (2001).

From the introduction: This article is a continuation of the paper [the author, Smooth equivalence of differential equations and linear automorphisms, Math. Notes 66, 464-473 (2000)], which dealts with systems of ordinary differential equations with linear automorphisms, i.e., systems whose right-hand side is invariant under some linear change of variables.

Simplifying by an appropriate coordinate transformation is one of the most efficient methods for studying differential equations. In this connection, it is important to use transformations that preserve the above-mentioned automorphism of the system. This problem was studied in [A. D. Bryuno, A local method of nonlinear analysis for differential equations, Moskva, “Nauka” (1979; Zbl 0496.34002)] for formal and analytic systems.

The present paper deals with the same problem for smooth reversible systems, i.e., systems equipped with an automorphism that reverses the motion along trajectories.

Simplifying by an appropriate coordinate transformation is one of the most efficient methods for studying differential equations. In this connection, it is important to use transformations that preserve the above-mentioned automorphism of the system. This problem was studied in [A. D. Bryuno, A local method of nonlinear analysis for differential equations, Moskva, “Nauka” (1979; Zbl 0496.34002)] for formal and analytic systems.

The present paper deals with the same problem for smooth reversible systems, i.e., systems equipped with an automorphism that reverses the motion along trajectories.

##### MSC:

37G05 | Normal forms for dynamical systems |

34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |

37L10 | Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems |