January  2012, 32(1): 41-56. doi: 10.3934/dcds.2012.32.41

Transversal intersections of invariant manifolds of NMS flows on $S^{3}$

1. 

IMAC - Instituto Universitario de Matemáticas y Aplicaciones de Castellón, Departamento de Matemáticas, Universitat Jaume I. Castellón, Spain, Spain

Received  July 2010 Revised  June 2011 Published  September 2011

In this paper NMS flows on $S^{3}$ with a round handle decomposition made up of connected sum of tori are considered. We describe these flows from the corresponding filtrations and obtain conditions for the existence of transversal intersections of invariant manifolds of saddle orbits.
    Moreover, we build the phase portrait for each case in order to obtain a complete description of the flow and to visualize how invariant manifolds of saddles intersect.
Citation: B. Campos, P. Vindel. Transversal intersections of invariant manifolds of NMS flows on $S^{3}$. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 41-56. doi: 10.3934/dcds.2012.32.41
References:
[1]

D. Asimov, Round handles and non-singular Morse-Smale flows,, Annals of Mathematics, 102 (1975), 41. doi: 10.2307/1970972. Google Scholar

[2]

B. Campos, J. Martínez Alfaro and P. Vindel, Bifurcations of links of periodic orbits in non-singular Morse-Smale systems on $S^{3}$,, Nonlinearity, 10 (1997), 1339. doi: 10.1088/0951-7715/10/5/018. Google Scholar

[3]

B. Campos and P. Vindel, Non equivalence of NMS flows on $S^{3}$,, to be published in Acta Mathematica Bohemica., (). Google Scholar

[4]

B. Campos and P. Vindel, NMS flows on $S^{3}$ with no heteroclinic trajectories connecting saddle orbits,, to appear., (). Google Scholar

[5]

J. W. Morgan, Non-singular Morse-Smale flows on 3-dimensional manifolds,, Topology, 18 (1979), 41. doi: 10.1016/0040-9383(79)90013-2. Google Scholar

[6]

L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. O. Chua, Methods of qualitative theory in nonlinear dynamics, Part II,, World Scientific. Series on Nonlinear Science, 5 (1998), 393. Google Scholar

[7]

M. Wada, Closed orbits of non-singular Morse-Smale flows on $S^{3}$,, J. Math. Soc. Japan, 41 (1989), 405. doi: 10.2969/jmsj/04130405. Google Scholar

[8]

K. Yano, The homotopy class of non-singular Morse-Smale vector fields on 3-manifolds,, Invent. Math., 80 (1985), 435. doi: 10.1007/BF01388724. Google Scholar

show all references

References:
[1]

D. Asimov, Round handles and non-singular Morse-Smale flows,, Annals of Mathematics, 102 (1975), 41. doi: 10.2307/1970972. Google Scholar

[2]

B. Campos, J. Martínez Alfaro and P. Vindel, Bifurcations of links of periodic orbits in non-singular Morse-Smale systems on $S^{3}$,, Nonlinearity, 10 (1997), 1339. doi: 10.1088/0951-7715/10/5/018. Google Scholar

[3]

B. Campos and P. Vindel, Non equivalence of NMS flows on $S^{3}$,, to be published in Acta Mathematica Bohemica., (). Google Scholar

[4]

B. Campos and P. Vindel, NMS flows on $S^{3}$ with no heteroclinic trajectories connecting saddle orbits,, to appear., (). Google Scholar

[5]

J. W. Morgan, Non-singular Morse-Smale flows on 3-dimensional manifolds,, Topology, 18 (1979), 41. doi: 10.1016/0040-9383(79)90013-2. Google Scholar

[6]

L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. O. Chua, Methods of qualitative theory in nonlinear dynamics, Part II,, World Scientific. Series on Nonlinear Science, 5 (1998), 393. Google Scholar

[7]

M. Wada, Closed orbits of non-singular Morse-Smale flows on $S^{3}$,, J. Math. Soc. Japan, 41 (1989), 405. doi: 10.2969/jmsj/04130405. Google Scholar

[8]

K. Yano, The homotopy class of non-singular Morse-Smale vector fields on 3-manifolds,, Invent. Math., 80 (1985), 435. doi: 10.1007/BF01388724. Google Scholar

[1]

E. Canalias, Josep J. Masdemont. Homoclinic and heteroclinic transfer trajectories between planar Lyapunov orbits in the sun-earth and earth-moon systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 261-279. doi: 10.3934/dcds.2006.14.261

[2]

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

[3]

Robert W. Ghrist. Flows on $S^3$ supporting all links as orbits. Electronic Research Announcements, 1995, 1: 91-97.

[4]

Alain Jacquemard, Weber Flávio Pereira. On periodic orbits of polynomial relay systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 331-347. doi: 10.3934/dcds.2007.17.331

[5]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881

[6]

Francesca Alessio, Carlo Carminati, Piero Montecchiari. Heteroclinic motions joining almost periodic solutions for a class of Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 569-584. doi: 10.3934/dcds.1999.5.569

[7]

Wenjun Zhang, Bernd Krauskopf, Vivien Kirk. How to find a codimension-one heteroclinic cycle between two periodic orbits. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2825-2851. doi: 10.3934/dcds.2012.32.2825

[8]

Thorsten Hüls, Yongkui Zou. On computing heteroclinic trajectories of non-autonomous maps. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 79-99. doi: 10.3934/dcdsb.2012.17.79

[9]

Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109

[10]

Flaviano Battelli, Ken Palmer. Transversal periodic-to-periodic homoclinic orbits in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 367-387. doi: 10.3934/dcdsb.2010.14.367

[11]

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

[12]

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

[13]

Victoriano Carmona, Emilio Freire, Soledad Fernández-García. Periodic orbits and invariant cones in three-dimensional piecewise linear systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 59-72. doi: 10.3934/dcds.2015.35.59

[14]

Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807

[15]

B. Buffoni, F. Giannoni. Brake periodic orbits of prescribed Hamiltonian for indefinite Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 217-222. doi: 10.3934/dcds.1995.1.217

[16]

Gabriele Benedetti, Kai Zehmisch. On the existence of periodic orbits for magnetic systems on the two-sphere. Journal of Modern Dynamics, 2015, 9: 141-146. doi: 10.3934/jmd.2015.9.141

[17]

Armengol Gasull, Héctor Giacomini, Maite Grau. On the stability of periodic orbits for differential systems in $\mathbb{R}^n$. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 495-509. doi: 10.3934/dcdsb.2008.10.495

[18]

Ming Huang, Cong Cheng, Yang Li, Zun Quan Xia. The space decomposition method for the sum of nonlinear convex maximum eigenvalues and its applications. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-21. doi: 10.3934/jimo.2019034

[19]

Flaviano Battelli, Ken Palmer. Heteroclinic connections in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 431-461. doi: 10.3934/dcdsb.2008.9.431

[20]

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

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]