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 and Continuous Dynamical Systems, 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), 41-54. doi: 10.2307/1970972.

[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), 1339-1355. doi: 10.1088/0951-7715/10/5/018.

[3]

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

[4]

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

[5]

J. W. Morgan, Non-singular Morse-Smale flows on 3-dimensional manifolds, Topology, 18 (1979), 41-53. doi: 10.1016/0040-9383(79)90013-2.

[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, i-xxiv and 393-957.

[7]

M. Wada, Closed orbits of non-singular Morse-Smale flows on $S^{3}$, J. Math. Soc. Japan, 41 (1989), 405-413. doi: 10.2969/jmsj/04130405.

[8]

K. Yano, The homotopy class of non-singular Morse-Smale vector fields on 3-manifolds, Invent. Math., 80 (1985), 435-451. doi: 10.1007/BF01388724.

show all references

References:
[1]

D. Asimov, Round handles and non-singular Morse-Smale flows, Annals of Mathematics, 102 (1975), 41-54. doi: 10.2307/1970972.

[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), 1339-1355. doi: 10.1088/0951-7715/10/5/018.

[3]

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

[4]

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

[5]

J. W. Morgan, Non-singular Morse-Smale flows on 3-dimensional manifolds, Topology, 18 (1979), 41-53. doi: 10.1016/0040-9383(79)90013-2.

[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, i-xxiv and 393-957.

[7]

M. Wada, Closed orbits of non-singular Morse-Smale flows on $S^{3}$, J. Math. Soc. Japan, 41 (1989), 405-413. doi: 10.2969/jmsj/04130405.

[8]

K. Yano, The homotopy class of non-singular Morse-Smale vector fields on 3-manifolds, Invent. Math., 80 (1985), 435-451. doi: 10.1007/BF01388724.

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