September  2008, 22(3): 587-604. doi: 10.3934/dcds.2008.22.587

Bott integrable Hamiltonian systems on $S^{2}\times S^{1}$

1. 

Dpt. Matemàtica Aplicada. Universidad Politécnica de Valencia, Cno. de Vera s/n. 46022 Valencia, Spain

2. 

Dpt. Matemàtica Aplicada. Facultat Matemàtiques, Universitat de València. Avda. Dr. Moliner, 50, 46100 Burjassot (Valencia), Spain

3. 

Dpt. Matemàtiques. Universitat Jaume I. Campus Riu Sec., 12071 Castelló, Spain

Received  June 2007 Revised  February 2008 Published  August 2008

In this paper, we study the topology of Bott integrable Hamiltonian flows on $S^{2}\times S^{1}$ in terms of some types of periodic orbits, called NMS periodic orbits. The set of these periodic orbits can be identified by means of some operations applied on global and local links. These operations come from the round handle decomposition of these systems on $S^{2}\times S^{1}.$ We apply the results to obtain a non-integrability criterium.
Citation: Alicia Cordero, José Martínez Alfaro, Pura Vindel. Bott integrable Hamiltonian systems on $S^{2}\times S^{1}$. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 587-604. doi: 10.3934/dcds.2008.22.587
[1]

Sonja Hohloch, Silvia Sabatini, Daniele Sepe. From compact semi-toric systems to Hamiltonian $S^1$-spaces. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 247-281. doi: 10.3934/dcds.2015.35.247

[2]

Raina Raj, Vidyottama Jain. Optimization of traffic control in $ MMAP\mathit{[2]}/PH\mathit{[2]}/S$ priority queueing model with $ PH $ retrial times and the preemptive repeat policy. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022044

[3]

Saikat Mazumdar. Struwe's decomposition for a polyharmonic operator on a compact Riemannian manifold with or without boundary. Communications on Pure and Applied Analysis, 2017, 16 (1) : 311-330. doi: 10.3934/cpaa.2017015

[4]

Xijun Hu, Li Wu. Decomposition of spectral flow and Bott-type iteration formula. Electronic Research Archive, 2020, 28 (1) : 127-148. doi: 10.3934/era.2020008

[5]

Liying Pan, R. Julian R. Abel, Jinhua Wang. Constructions of optimal multiply constant-weight codes MCWC$ (3,n_1;1,n_2;1,n_3;8)s $. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022025

[6]

Valeria Banica, Luis Vega. Singularity formation for the 1-D cubic NLS and the Schrödinger map on $\mathbb S^2$. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1317-1329. doi: 10.3934/cpaa.2018064

[7]

Dongfeng Zhang, Junxiang Xu. On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 635-655. doi: 10.3934/dcds.2006.16.635

[8]

Hiroshi Matano, Ken-Ichi Nakamura. The global attractor of semilinear parabolic equations on $S^1$. Discrete and Continuous Dynamical Systems, 1997, 3 (1) : 1-24. doi: 10.3934/dcds.1997.3.1

[9]

Jaume Llibre, Y. Paulina Martínez, Claudio Vidal. Phase portraits of linear type centers of polynomial Hamiltonian systems with Hamiltonian function of degree 5 of the form $ H = H_1(x)+H_2(y)$. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 75-113. doi: 10.3934/dcds.2019004

[10]

Hahng-Yun Chu, Se-Hyun Ku, Jong-Suh Park. Conley's theorem for dispersive systems. Discrete and Continuous Dynamical Systems - S, 2015, 8 (2) : 313-321. doi: 10.3934/dcdss.2015.8.313

[11]

Zheng-Chao Han, YanYan Li. On the local solvability of the Nirenberg problem on $\mathbb S^2$. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 607-615. doi: 10.3934/dcds.2010.28.607

[12]

Abbas Bahri. Attaching maps in the standard geodesics problem on $S^2$. Discrete and Continuous Dynamical Systems, 2011, 30 (2) : 379-426. doi: 10.3934/dcds.2011.30.379

[13]

Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems and Imaging, 2007, 1 (1) : 159-179. doi: 10.3934/ipi.2007.1.159

[14]

Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032

[15]

Sung-Seok Ko, Jangha Kang, E-Yeon Kwon. An $(s,S)$ inventory model with level-dependent $G/M/1$-Type structure. Journal of Industrial and Management Optimization, 2016, 12 (2) : 609-624. doi: 10.3934/jimo.2016.12.609

[16]

Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks and Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007

[17]

Giovanni Forni, Howard Masur, John Smillie. Bill Veech's contributions to dynamical systems. Journal of Modern Dynamics, 2019, 14: v-xxv. doi: 10.3934/jmd.2019v

[18]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks and Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617

[19]

Alexander Moreto. Complex group algebras of finite groups: Brauer's Problem 1. Electronic Research Announcements, 2005, 11: 34-39.

[20]

Fadia Bekkal-Brikci, Giovanna Chiorino, Khalid Boushaba. G1/S transition and cell population dynamics. Networks and Heterogeneous Media, 2009, 4 (1) : 67-90. doi: 10.3934/nhm.2009.4.67

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (54)
  • HTML views (0)
  • Cited by (0)

[Back to Top]