American Institute of Mathematical Sciences

Advanced
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 - A, 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

[1]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002
[2]

Yancong Xu, Lijun Wei, Xiaoyu Jiang, Zirui Zhu. Complex dynamics of a SIRS epidemic model with the influence of hospital bed number. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021016
[3]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301
[4]

Hui Zhao, Zhengrong Liu, Yiren Chen. Global dynamics of a chemotaxis model with signal-dependent diffusion and sensitivity. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021011
[5]

Shujing Shi, Jicai Huang, Yang Kuang. Global dynamics in a tumor-immune model with an immune checkpoint inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1149-1170. doi: 10.3934/dcdsb.2020157
[6]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045
[7]

Rong Chen, Shihang Pan, Baoshuai Zhang. Global conservative solutions for a modified periodic coupled Camassa-Holm system. Electronic Research Archive, 2021, 29 (1) : 1691-1708. doi: 10.3934/era.2020087
[8]

Hai-Yang Jin, Zhi-An Wang. Global stabilization of the full attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3509-3527. doi: 10.3934/dcds.2020027
[9]

Sze-Bi Hsu, Yu Jin. The dynamics of a two host-two virus system in a chemostat environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 415-441. doi: 10.3934/dcdsb.2020298
[10]

Shigui Ruan. Nonlinear dynamics in tumor-immune system interaction models with delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 541-602. doi: 10.3934/dcdsb.2020282
[11]

Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322
[12]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003
[13]

Honglei Lang, Yunhe Sheng. Linearization of the higher analogue of Courant algebroids. Journal of Geometric Mechanics, 2020, 12 (4) : 585-606. doi: 10.3934/jgm.2020025
[14]

Sergio Zamora. Tori can't collapse to an interval. Electronic Research Archive, , () : -. doi: 10.3934/era.2021005
[15]

Xueli Bai, Fang Li. Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3075-3092. doi: 10.3934/dcds.2020035
[16]

Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269
[17]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326
[18]

Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021002
[19]

Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024
[20]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

2019 Impact Factor: 1.338

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences