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**Unique normal form of the Hamiltonian 1:2 resonance.**
*(English)*
Zbl 0766.58051

Geometry and analysis in nonlinear dynamics, Proc. Workshop Chaotic Dyn. Bifurcations, Groningen/Neth. 1989, Pitman Res. Notes Math. Ser. 222, 56-69 (1992).

[For the entire collection see Zbl 0745.00035.]

The authors extend and apply the theory concerning the computation of unique normal forms for vector fields and differential equations developing the theory by A. Baider [J. Differ. Equations 78, No. 1, 33-52 (1989; Zbl 0689.70005)]. Analyzing the few examples that are completely studied the authors try to extract some general underlying rules in the existing theory (the paper provides a good historical background and an extensive literature on the subject). The general scheme is to determine an unperturbed problem that is in the normal form and uniquely determined, together with those transformations that leave this normal form fixed to \(0\)-th order; to determine the action of transformations that are left, a complement to those actions and the transformations that leave the complement invariant to the first order (calling the elements in this complement to be in the first order normal form) and repeat the procedure to get the second order normal form. And so the procedure continues.

The advanced techniques of graded Lie algebras are used and an example of the Hamiltonian 1:2-resonance is thoroughly considered.

The authors extend and apply the theory concerning the computation of unique normal forms for vector fields and differential equations developing the theory by A. Baider [J. Differ. Equations 78, No. 1, 33-52 (1989; Zbl 0689.70005)]. Analyzing the few examples that are completely studied the authors try to extract some general underlying rules in the existing theory (the paper provides a good historical background and an extensive literature on the subject). The general scheme is to determine an unperturbed problem that is in the normal form and uniquely determined, together with those transformations that leave this normal form fixed to \(0\)-th order; to determine the action of transformations that are left, a complement to those actions and the transformations that leave the complement invariant to the first order (calling the elements in this complement to be in the first order normal form) and repeat the procedure to get the second order normal form. And so the procedure continues.

The advanced techniques of graded Lie algebras are used and an example of the Hamiltonian 1:2-resonance is thoroughly considered.

Reviewer: W.Kryszewski (Torún)

### MSC:

37G05 | Normal forms for dynamical systems |

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |

70H05 | Hamilton’s equations |

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\textit{J. A. Sanders} and \textit{J. C. Van der Meer}, in: Geometry and analysis in nonlinear dynamics. Proceedings of the workshop on chaotic dynamics and bifurcations, held at the University of Groningen, Netherlands, March 1989. Harlow: Longman Scientific \&| Technical; New York: John Wiley \&| Sons, Inc.. 56--69 (1992; Zbl 0766.58051)