Complex planar Hamiltonian systems: Linearization and dynamics

Hua Shi; Xiang Zhang; Yuyan Zhang

doi:10.3934/dcds.2020406

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2021, Volume 41, Issue 7: 3295-3317. Doi: 10.3934/dcds.2020406

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solution for a non-linear Hamilton-Jacobi system Next Article Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems

Complex planar Hamiltonian systems: Linearization and dynamics

1.
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
2.
School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China

^* Corresponding author: Xiang Zhang
^* Corresponding author: Xiang Zhang

Received: February 29, 2020

Revised: September 30, 2020

Early access: December 2020

Published: July 2021

The third author is partially supported by NNSF of China grant numbers 11671254, 11871334 and 12071284, and also by Innovation Program of Shanghai Municipal Education Commission

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Abstract

Global dynamics of complex planar Hamiltonian polynomial systems is difficult to be characterized. In this paper, for general complex quadratic Hamiltonian systems of one degree of freedom, we obtain some sufficient conditions on the existence of family of invariant tori. We also complete characterization on locally analytic linearizability of complex planar Hamiltonian systems with homogeneous nonlinearity of degrees either 2 or 3 at a nondegenerate singularity, and present their global dynamics. For these classes of systems we also prove existence of families of invariant tori, together with isochronous periodic orbits.

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Mathematics Subject Classification: Primary: 37J35, 37C10; Secondary: 37C27, 34C14.

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