American Institute of Mathematical Sciences

Advanced
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

[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

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)
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

[1]

Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097
[2]

Peter Albers, Urs Frauenfelder. Spectral invariants in Rabinowitz-Floer homology and global Hamiltonian perturbations. Journal of Modern Dynamics, 2010, 4 (2) : 329-357. doi: 10.3934/jmd.2010.4.329
[3]

Daniel Wilczak. Abundance of heteroclinic and homoclinic orbits for the hyperchaotic Rössler system. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1039-1055. doi: 10.3934/dcdsb.2009.11.1039
[4]

Viktor L. Ginzburg, Başak Z. Gürel. On the generic existence of periodic orbits in Hamiltonian dynamics. Journal of Modern Dynamics, 2009, 3 (4) : 595-610. doi: 10.3934/jmd.2009.3.595
[5]

Piotr Kokocki. Homotopy invariants methods in the global dynamics of strongly damped wave equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3227-3250. doi: 10.3934/dcds.2016.36.3227
[6]

Junmin Yang, Maoan Han. On the number of limit cycles of a cubic Near-Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 827-840. doi: 10.3934/dcds.2009.24.827
[7]

Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934/dcdsb.2011.15.789
[8]

Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114
[9]

Anna Cima, Armengol Gasull, Francesc Mañosas. Global linearization of periodic difference equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1575-1595. doi: 10.3934/dcds.2012.32.1575
[10]

Héctor E. Lomelí. Heteroclinic orbits and rotation sets for twist maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 343-354. doi: 10.3934/dcds.2006.14.343
[11]

Christopher K. R. T. Jones, Siu-Kei Tin. Generalized exchange lemmas and orbits heteroclinic to invariant manifolds. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 967-1023. doi: 10.3934/dcdss.2009.2.967
[12]

Zhipeng Qiu, Huaiping Zhu. Complex dynamics of a nutrient-plankton system with nonlinear phytoplankton mortality and allelopathy. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2703-2728. doi: 10.3934/dcdsb.2016069
[13]

Fatima Ezzahra Lembarki, Jaume Llibre. Periodic orbits for a generalized Friedmann-Robertson-Walker Hamiltonian system in dimension $6$. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1165-1211. doi: 10.3934/dcdss.2015.8.1165
[14]

Kei Irie. Dense existence of periodic Reeb orbits and ECH spectral invariants. Journal of Modern Dynamics, 2015, 9: 357-363. doi: 10.3934/jmd.2015.9.357
[15]

Kingshook Biswas. Complete conjugacy invariants of nonlinearizable holomorphic dynamics. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 847-856. doi: 10.3934/dcds.2010.26.847
[16]

George Papadopoulos, Holger R. Dullin. Semi-global symplectic invariants of the Euler top. Journal of Geometric Mechanics, 2013, 5 (2) : 215-232. doi: 10.3934/jgm.2013.5.215
[17]

Tifei Qian, Zhihong Xia. Heteroclinic orbits and chaotic invariant sets for monotone twist maps. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 69-95. doi: 10.3934/dcds.2003.9.69
[18]

Guowei Yu. Ray and heteroclinic solutions of Hamiltonian systems with 2 degrees of freedom. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4769-4793. doi: 10.3934/dcds.2013.33.4769
[19]

Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623
[20]

Hans Koch, Héctor E. Lomelí. On Hamiltonian flows whose orbits are straight lines. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2091-2104. doi: 10.3934/dcds.2014.34.2091

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (65)
  • HTML views (397)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences