# American Institute of Mathematical Sciences

October  2019, 12(6): 1761-1774. doi: 10.3934/dcdss.2019116

## Dynamics of locally linearizable complex two dimensional cubic Hamiltonian systems

 1 Department of Mathematics and Computer Science, Chizhou University, Chizhou 274000, China 2 School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China 3 School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: Xiang Zhang

Received  September 2017 Revised  December 2017 Published  November 2018

Fund Project: The third author is partially supported by NNSF of China grant numbers 11671254 and 11871334, and by innovation program of Shanghai Municipal Education Commission grant number 15ZZ012

The aim of this paper is to characterize global dynamics of locally linearizable complex two dimensional cubic Hamiltonian systems. By finding invariants, we prove that their associated real phase space $\mathbb R^4$ is foliated by two dimensional invariant surfaces, which could be either simple connected, or double connected, or triple connected, or quadruple connected. On each of the invariant surfaces all regular orbits are heteroclinic ones, which connect two singularities, either both finite, or one finite and another at infinity, or both at infinity, and all these situations are realizable.

Citation: Yangyou Pan, Yuzhen Bai, Xiang Zhang. Dynamics of locally linearizable complex two dimensional cubic Hamiltonian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1761-1774. doi: 10.3934/dcdss.2019116
##### References:

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##### References:
Phase portrait of the last two equations of system (5)
Phase portrait of the last two equations of system (9)
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