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

Dynamics of locally linearizable complex two dimensional cubic Hamiltonian systems

Yangyou Pan 1, , Yuzhen Bai 2, and  Xiang Zhang 3,,

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.

Keywords: Complex cubic Hamiltonian system, linearization, heteroclinic orbits, global dynamics, invariants.
Mathematics Subject Classification: Primary: 37J35, 37C10; Secondary: 37C27, 34C14.
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:
[1]

M. J. Alvarez, A. Gasull and R. Prohens, Topological classification of polynomial complex differential equations with all the critical points of centre type, J. Difference Equ. Appl., 16 (2010), 411-423. doi: 10.1080/10236190903232654. Google Scholar

[2]

V. I. Arnold, Ordinary Differential Equations, 3rd edition, Springer-Verlag, Berlin, 1992. Google Scholar

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J. C. Artés and J. Llibre, Quadratic Hamiltonian vector fields, J. Differential Equations, 107 (1994), 80-95. doi: 10.1006/jdeq.1994.1004. Google Scholar

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J. Chavarriga and M. Sabatini, A survey of isochronous centers, Qual. Theory Dyn. Syst., 1 (1999), 1-70. doi: 10.1007/BF02969404. Google Scholar

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A. Cima and J. Llibre, Bounded polynomial vector fields, Trans. Amer. Math. Soc., 318 (1990), 557-579. doi: 10.1090/S0002-9947-1990-0998352-5. Google Scholar

[6] F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer, Berlin, 2006. Google Scholar
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A. Garijo, A. Gasull and X. Jarque, Local and global phase portrait of equation z' = f(z), Discrete Contin. Dyn. Syst., 17 (2007), 309-329. doi: 10.3934/dcds.2007.17.309. Google Scholar

[8]

A. Gasull, J. Llibre and X. Zhang, One-dimensional quaternion homogeneous polynomial differential equations, J. Math. Phys., 50 (2009), 082705, 17 pp. doi: 10.1063/1.3139115. Google Scholar

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L. M. Lerman and Ya. L. Umanskiy, Four-dimensional Integrable Hamiltonian Systems with Simple Singular Points (Topological Aspects), Transl. Math. Monographs, American Math. Soc., Providence, Rhode Island, 1998. Google Scholar

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J. Llibre and V. G. Romanovski, Isochronicity and linearizability of planar polynomial Hamiltonian systems, J. Differential Equations, 259 (2015), 1649-1662. doi: 10.1016/j.jde.2015.03.009. Google Scholar

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J. Llibre and C. Valls, Darboux integrability of 2-dimensional Hamiltonian systems with homogenous potentials of degree 3, J. Math. Phys., 55 (2014), 033507, 12 pp. doi: 10.1063/1.4868701. Google Scholar

[12]

J. Llibre and C. Valls, Liouvillian first integrals for a class of generalized Liénard polynomial differential systems, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 1195-1210. doi: 10.1017/S0308210515000906. Google Scholar

[13]

J. Llibre and X. Zhang, On the Darboux integrability of polynomial differential systems, Qual. Theory Dyn. Syst., 11 (2012), 129-144. doi: 10.1007/s12346-011-0053-x. Google Scholar

[14]

A.J. Maciejewski, M. Przybylska and H. Yoshida, Necessary conditions for the existence of additional first integrals for Hamiltonian systems with homogeneous potential, Nonlinearity, 25 (2012), 255-277. doi: 10.1088/0951-7715/25/2/255. Google Scholar

[15]

Y. P. Martnez and C. Vidal, Classification of global phase portraits and bifurcation diagrams of Hamiltonian systems with rational potential, J. Differential Equations, 261 (2016), 5923-5948. Google Scholar

[16]

V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhäuser, Boston, 2009. doi: 10.1007/978-0-8176-4727-8. Google Scholar

[17]

H. Shi, X. Zhang and Y. Zhang, Linearization and dynamics of complex planar Hamiltonian systems, Preprint.Google Scholar

[18]

C. Valls, Rikitake system: Analytic and Darbouxian integrals, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1309-1326. doi: 10.1017/S030821050000439X. Google Scholar

[19]

X. Zhang, Global structure of quaternion polynomial differential equations, Comm. Math. Phys., 303 (2011), 301-316. doi: 10.1007/s00220-011-1196-y. Google Scholar

[20]

X. Zhang, Integrability of Dynamical Systems: Algebra and Analysis, Springer, Singapore, 2017. doi: 10.1007/978-981-10-4226-3. Google Scholar

show all references

References:
[1]

M. J. Alvarez, A. Gasull and R. Prohens, Topological classification of polynomial complex differential equations with all the critical points of centre type, J. Difference Equ. Appl., 16 (2010), 411-423. doi: 10.1080/10236190903232654. Google Scholar

[2]

V. I. Arnold, Ordinary Differential Equations, 3rd edition, Springer-Verlag, Berlin, 1992. Google Scholar

[3]

J. C. Artés and J. Llibre, Quadratic Hamiltonian vector fields, J. Differential Equations, 107 (1994), 80-95. doi: 10.1006/jdeq.1994.1004. Google Scholar

[4]

J. Chavarriga and M. Sabatini, A survey of isochronous centers, Qual. Theory Dyn. Syst., 1 (1999), 1-70. doi: 10.1007/BF02969404. Google Scholar

[5]

A. Cima and J. Llibre, Bounded polynomial vector fields, Trans. Amer. Math. Soc., 318 (1990), 557-579. doi: 10.1090/S0002-9947-1990-0998352-5. Google Scholar

[6] F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer, Berlin, 2006. Google Scholar
[7]

A. Garijo, A. Gasull and X. Jarque, Local and global phase portrait of equation z' = f(z), Discrete Contin. Dyn. Syst., 17 (2007), 309-329. doi: 10.3934/dcds.2007.17.309. Google Scholar

[8]

A. Gasull, J. Llibre and X. Zhang, One-dimensional quaternion homogeneous polynomial differential equations, J. Math. Phys., 50 (2009), 082705, 17 pp. doi: 10.1063/1.3139115. Google Scholar

[9]

L. M. Lerman and Ya. L. Umanskiy, Four-dimensional Integrable Hamiltonian Systems with Simple Singular Points (Topological Aspects), Transl. Math. Monographs, American Math. Soc., Providence, Rhode Island, 1998. Google Scholar

[10]

J. Llibre and V. G. Romanovski, Isochronicity and linearizability of planar polynomial Hamiltonian systems, J. Differential Equations, 259 (2015), 1649-1662. doi: 10.1016/j.jde.2015.03.009. Google Scholar

[11]

J. Llibre and C. Valls, Darboux integrability of 2-dimensional Hamiltonian systems with homogenous potentials of degree 3, J. Math. Phys., 55 (2014), 033507, 12 pp. doi: 10.1063/1.4868701. Google Scholar

[12]

J. Llibre and C. Valls, Liouvillian first integrals for a class of generalized Liénard polynomial differential systems, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 1195-1210. doi: 10.1017/S0308210515000906. Google Scholar

[13]

J. Llibre and X. Zhang, On the Darboux integrability of polynomial differential systems, Qual. Theory Dyn. Syst., 11 (2012), 129-144. doi: 10.1007/s12346-011-0053-x. Google Scholar

[14]

A.J. Maciejewski, M. Przybylska and H. Yoshida, Necessary conditions for the existence of additional first integrals for Hamiltonian systems with homogeneous potential, Nonlinearity, 25 (2012), 255-277. doi: 10.1088/0951-7715/25/2/255. Google Scholar

[15]

Y. P. Martnez and C. Vidal, Classification of global phase portraits and bifurcation diagrams of Hamiltonian systems with rational potential, J. Differential Equations, 261 (2016), 5923-5948. Google Scholar

[16]

V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhäuser, Boston, 2009. doi: 10.1007/978-0-8176-4727-8. Google Scholar

[17]

H. Shi, X. Zhang and Y. Zhang, Linearization and dynamics of complex planar Hamiltonian systems, Preprint.Google Scholar

[18]

C. Valls, Rikitake system: Analytic and Darbouxian integrals, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1309-1326. doi: 10.1017/S030821050000439X. Google Scholar

[19]

X. Zhang, Global structure of quaternion polynomial differential equations, Comm. Math. Phys., 303 (2011), 301-316. doi: 10.1007/s00220-011-1196-y. Google Scholar

[20]

X. Zhang, Integrability of Dynamical Systems: Algebra and Analysis, Springer, Singapore, 2017. doi: 10.1007/978-981-10-4226-3. Google Scholar

Figure 1.  Phase portrait of the last two equations of system (5)
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Figure 2.  Phase portrait of the last two equations of system (9)
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