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July  2021, 41(7): 3295-3317. doi: 10.3934/dcds.2020406

Complex planar Hamiltonian systems: Linearization and dynamics

Hua Shi 1, , Xiang Zhang 2,, and  Yuyan Zhang 1,

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

Received  March 2020 Revised  October 2020 Published  December 2020

Fund Project: 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

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.

Keywords: Complex Hamiltonian system, Liouvillian integrability, linearization, global dynamics, invariant tori.
Mathematics Subject Classification: Primary: 37J35, 37C10; Secondary: 37C27, 34C14.
Citation: Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3295-3317. doi: 10.3934/dcds.2020406
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 (third edition), Springer–Verlag, Berlin, 1992.  Google Scholar

[3]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics (third edition), Encyclopaedia of Mathematical Sciences, 3, Springer–Verlag, Berlin, 2006.  Google Scholar

[4]

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

[5]

A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems: Geometry, Topology, Classification, Chapman & Hall/CRC, Boca Raton, 2004. doi: 10.1201/9780203643426.  Google Scholar

[6]

L. Cairó, J. Chavarriga, J. Giné and J. Llibre, A class of reversible cubic systems with an isochronous center, Comput. Math. Appl., 38 (1999), 39-53.  doi: 10.1016/S0898-1221(99)00283-7.  Google Scholar

[7]

J. Chavarriga, I. A. García and J. Giné, Isochronicity into a family of time–reversible cubic vector fields, Appl. Math. Comput., 121 (2001), 129-145.  doi: 10.1016/S0096-3003(99)00267-2.  Google Scholar

[8]

J. Chavarriga, J. Giné and I. A. García, Isochronous centers of a linear center perturbed by fifth degree homogeneous polynomials, J. Comput. Appl. Math., 126 (2000), 351-368.  doi: 10.1016/S0377-0427(99)00364-7.  Google Scholar

[9]

J. Chavarriga, J. Giné and I. A. García, Isochronous centers of a linear center perturbed by fourth degree homogeneous polynomial, Bull. Sci. Math., 123 (1999), 77-96.  doi: 10.1016/S0007-4497(99)80015-3.  Google Scholar

[10]

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

[11]

C. J. Christopher and J. Devlin, Isochronous centers in planar polynomial systems, SIAM J. Math. Anal., 28 (1997), 162-177.  doi: 10.1137/S0036141093259245.  Google Scholar

[12]

A. Cima, A. Gasull and F. Ma$\widetilde{n}$osas, Period function for a class of Hamiltonian systems, J. Differential Equations, 168 (2000), 180-199.  doi: 10.1006/jdeq.2000.3912.  Google Scholar

[13]

A. Cima, F. Mañosas and J. Villadelprat, Isochronicity for several classes of Hamiltonian systems, J. Differential Equations, 157 (1999), 373-413.  doi: 10.1006/jdeq.1999.3635.  Google Scholar

[14]

L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals-elliptic case, Comment. Math. Helv., 65 (1990), 4-35.  doi: 10.1007/BF02566590.  Google Scholar

[15]

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

[16]

A. Gasull, J. Llibre and X. Zhang, One–dimensional quaternion homogeneous polynomial differential equations, J. Mathematical Physics, 50 (2009), 082705. doi: 10.1063/1.3139115.  Google Scholar

[17]

J. Giné and J. Llibre, On the planar integrable differential systems, Z. Angew. Math. Phys., 62 (2011), 567-574.  doi: 10.1007/s00033-011-0116-5.  Google Scholar

[18]

H. Ito, Integrability of Hamiltonian systems and Birkhoff normal forms in the simple resonance case, Math. Ann., 292 (1992), 411-444.  doi: 10.1007/BF01444629.  Google Scholar

[19]

X. Jarque and J. Villadelprat, Nonexistence of isochronous centers in planar polynomial Hamiltonian systems of degree four, J. Differential Equations, 180 (2002), 334-373.  doi: 10.1006/jdeq.2001.4065.  Google Scholar

[20]

L. M. Lerman and Ya. L. Umanskiy, Four–Dimensional Integrable Hamiltonian Systems with Simple Singular Points (topological aspects), Translations of Mathmatical Monographs, American Mathematical Society, Providence, Rhode Island, 1998. doi: 10.1090/mmono/176.  Google Scholar

[21]

J. Llibre, Integrability of Polynomial Differential Systems, Handbook of differential equations, Elsevier/North-Holland, Amsterdam, 2004,437–532.  Google Scholar

[22]

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

[23]

J. Llibre, C. Valls and X Zhang, The completely integrable differential systems are essentially linear differential systems, J. Nonlinear Sci., 25 (2015), 815-826.  doi: 10.1007/s00332-015-9243-z.  Google Scholar

[24]

W. S. Loud, Behaviour of the period of solutions of certain plane autonomous systems near centers, Contrib. Differential Equations, 3 (1964), 21-36.   Google Scholar

[25]

P. Mardešić, C. Rousseau and B. Toni, Linearization of isochronous centers, J. Differential Equations, 121 (1995), 67-108.  doi: 10.1006/jdeq.1995.1122.  Google Scholar

[26]

J. Moser, On the generalization of a theorem of A. Liapounoff, Comm. Pure Appl. Math., 11 (1958), 257-271.  doi: 10.1002/cpa.3160110208.  Google Scholar

[27]

H. Rũssmann, Ũber das Verhalten analytischer Hamiltonscher Differentialgleichungen in der Nãhe einer Gleichgewichtslõsung, Math. Ann., 154 (1964), 285-300.  doi: 10.1007/BF01362565.  Google Scholar

[28]

J. Vey, Sur certains systèmes dynamiques séparables, Amer. J. Math., 100 (1978), 591-614.  doi: 10.2307/2373841.  Google Scholar

[29]

S. Vũ Ngoc, On semi-global invariants for focus-focus singularities, Topology, 42 (2003), 365-380.  doi: 10.1016/S0040-9383(01)00026-X.  Google Scholar

[30]

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

[31]

X. Zhang, Liouvillian integrability of polynomial differential systems, Trans. Amer. Math. Soc., 368 (2016), 607-620.  doi: 10.1090/S0002-9947-2014-06387-3.  Google Scholar

[32]

X. Zhang, Integrability of Dynamical Systems: Algebra and Analysis, Developments in Mathematics, Vol. 47, Springer, Singapore, 2017. doi: 10.1007/978-981-10-4226-3.  Google Scholar

[33]

N. T. Zung, Convergence versus integrability in Birkhoff normal form, Ann. Math., 161 (2005) 141–156. doi: 10.4007/annals.2005.161.141.  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 (third edition), Springer–Verlag, Berlin, 1992.  Google Scholar

[3]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics (third edition), Encyclopaedia of Mathematical Sciences, 3, Springer–Verlag, Berlin, 2006.  Google Scholar

[4]

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

[5]

A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems: Geometry, Topology, Classification, Chapman & Hall/CRC, Boca Raton, 2004. doi: 10.1201/9780203643426.  Google Scholar

[6]

L. Cairó, J. Chavarriga, J. Giné and J. Llibre, A class of reversible cubic systems with an isochronous center, Comput. Math. Appl., 38 (1999), 39-53.  doi: 10.1016/S0898-1221(99)00283-7.  Google Scholar

[7]

J. Chavarriga, I. A. García and J. Giné, Isochronicity into a family of time–reversible cubic vector fields, Appl. Math. Comput., 121 (2001), 129-145.  doi: 10.1016/S0096-3003(99)00267-2.  Google Scholar

[8]

J. Chavarriga, J. Giné and I. A. García, Isochronous centers of a linear center perturbed by fifth degree homogeneous polynomials, J. Comput. Appl. Math., 126 (2000), 351-368.  doi: 10.1016/S0377-0427(99)00364-7.  Google Scholar

[9]

J. Chavarriga, J. Giné and I. A. García, Isochronous centers of a linear center perturbed by fourth degree homogeneous polynomial, Bull. Sci. Math., 123 (1999), 77-96.  doi: 10.1016/S0007-4497(99)80015-3.  Google Scholar

[10]

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

[11]

C. J. Christopher and J. Devlin, Isochronous centers in planar polynomial systems, SIAM J. Math. Anal., 28 (1997), 162-177.  doi: 10.1137/S0036141093259245.  Google Scholar

[12]

A. Cima, A. Gasull and F. Ma$\widetilde{n}$osas, Period function for a class of Hamiltonian systems, J. Differential Equations, 168 (2000), 180-199.  doi: 10.1006/jdeq.2000.3912.  Google Scholar

[13]

A. Cima, F. Mañosas and J. Villadelprat, Isochronicity for several classes of Hamiltonian systems, J. Differential Equations, 157 (1999), 373-413.  doi: 10.1006/jdeq.1999.3635.  Google Scholar

[14]

L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals-elliptic case, Comment. Math. Helv., 65 (1990), 4-35.  doi: 10.1007/BF02566590.  Google Scholar

[15]

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

[16]

A. Gasull, J. Llibre and X. Zhang, One–dimensional quaternion homogeneous polynomial differential equations, J. Mathematical Physics, 50 (2009), 082705. doi: 10.1063/1.3139115.  Google Scholar

[17]

J. Giné and J. Llibre, On the planar integrable differential systems, Z. Angew. Math. Phys., 62 (2011), 567-574.  doi: 10.1007/s00033-011-0116-5.  Google Scholar

[18]

H. Ito, Integrability of Hamiltonian systems and Birkhoff normal forms in the simple resonance case, Math. Ann., 292 (1992), 411-444.  doi: 10.1007/BF01444629.  Google Scholar

[19]

X. Jarque and J. Villadelprat, Nonexistence of isochronous centers in planar polynomial Hamiltonian systems of degree four, J. Differential Equations, 180 (2002), 334-373.  doi: 10.1006/jdeq.2001.4065.  Google Scholar

[20]

L. M. Lerman and Ya. L. Umanskiy, Four–Dimensional Integrable Hamiltonian Systems with Simple Singular Points (topological aspects), Translations of Mathmatical Monographs, American Mathematical Society, Providence, Rhode Island, 1998. doi: 10.1090/mmono/176.  Google Scholar

[21]

J. Llibre, Integrability of Polynomial Differential Systems, Handbook of differential equations, Elsevier/North-Holland, Amsterdam, 2004,437–532.  Google Scholar

[22]

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

[23]

J. Llibre, C. Valls and X Zhang, The completely integrable differential systems are essentially linear differential systems, J. Nonlinear Sci., 25 (2015), 815-826.  doi: 10.1007/s00332-015-9243-z.  Google Scholar

[24]

W. S. Loud, Behaviour of the period of solutions of certain plane autonomous systems near centers, Contrib. Differential Equations, 3 (1964), 21-36.   Google Scholar

[25]

P. Mardešić, C. Rousseau and B. Toni, Linearization of isochronous centers, J. Differential Equations, 121 (1995), 67-108.  doi: 10.1006/jdeq.1995.1122.  Google Scholar

[26]

J. Moser, On the generalization of a theorem of A. Liapounoff, Comm. Pure Appl. Math., 11 (1958), 257-271.  doi: 10.1002/cpa.3160110208.  Google Scholar

[27]

H. Rũssmann, Ũber das Verhalten analytischer Hamiltonscher Differentialgleichungen in der Nãhe einer Gleichgewichtslõsung, Math. Ann., 154 (1964), 285-300.  doi: 10.1007/BF01362565.  Google Scholar

[28]

J. Vey, Sur certains systèmes dynamiques séparables, Amer. J. Math., 100 (1978), 591-614.  doi: 10.2307/2373841.  Google Scholar

[29]

S. Vũ Ngoc, On semi-global invariants for focus-focus singularities, Topology, 42 (2003), 365-380.  doi: 10.1016/S0040-9383(01)00026-X.  Google Scholar

[30]

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

[31]

X. Zhang, Liouvillian integrability of polynomial differential systems, Trans. Amer. Math. Soc., 368 (2016), 607-620.  doi: 10.1090/S0002-9947-2014-06387-3.  Google Scholar

[32]

X. Zhang, Integrability of Dynamical Systems: Algebra and Analysis, Developments in Mathematics, Vol. 47, Springer, Singapore, 2017. doi: 10.1007/978-981-10-4226-3.  Google Scholar

[33]

N. T. Zung, Convergence versus integrability in Birkhoff normal form, Ann. Math., 161 (2005) 141–156. doi: 10.4007/annals.2005.161.141.  Google Scholar

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