Behavior 0 nonsingular Morse Smale flows on S^3

Previous Article
Dynnikov and train track transition matrices of pseudo-Anosov braids
DCDS Home
This Issue
Next Article
Radial sign-changing solution for fractional Schrödinger equation

January 2016, 36(1): 509-540. doi: 10.3934/dcds.2016.36.509

Behavior $0$ nonsingular Morse Smale flows on $S^3$

1.	Department of Mathematics, Tongji University, Shanghai 200092

Received June 2014 Revised December 2014 Published June 2015

Abstract
Related Papers

In this paper, we first develop the concept of Lyapunov graph to weighted Lyapunov graph (abbreviated as WLG) for nonsingular Morse-Smale flows (abbreviated as NMS flows) on $S^3$. WLG is quite sensitive to NMS flows on $S^3$. For instance, WLG detect the indexed links of NMS flows. Then we use WLG and some other tools to describe nonsingular Morse-Smale flows without heteroclinic trajectories connecting saddle orbits (abbreviated as behavior $0$ NMS flows). It mainly contains the following several directions:
1. we use WLG to list behavior $0$ NMS flows on $S^3$;
2. with the help of WLG, comparing with Wada's algorithm, we provide a direct description about the (indexed) link of behavior $0$ NMS flows;
3. to overcome the weakness that WLG can't decide topologically equivalent class, we give a simplified Umanskii Theorem to decide when two behavior $0$ NMS flows on $S^3$ are topological equivalence;
4. under these theories, we classify (up to topological equivalence) all behavior 0 NMS flows on $S^3$ with periodic orbits number no more than 4.

Keywords: topologically equivalent classes., links, Behavior $0$ nonsingular Morse-Smale flows, weighted Lyapunov graphs.

Mathematics Subject Classification: Primary: 37D15, 37C15; Secondary: 37C15, 57M2.

Citation: Bin Yu. Behavior $0$ nonsingular Morse Smale flows on $S^3$. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 509-540. doi: 10.3934/dcds.2016.36.509

References:

[1]	D. Asimov, Round handles and non-singular Morse-Smale flows, Ann. of Math. (2), 102 (1975), 41-54. doi: 10.2307/1970972.
[2]	D. Asimov, Homotopy of non-singular vector fields to structurally stable ones, Ann. of Math. (2), 102 (1975), 55-65. doi: 10.2307/1970973.
[3]	C. Bonatti, V. Grines and R. Langevin, Dynamical systems in dimension 2 and 3: conjugacy invariants and classification, in The geometry of differential equations and dynamical systems, Comput. Appl. Math., 20 (2001), 11-50.
[4]	B. Campos and P. Vindel, NMS flows on $S^3$ with no heteroclinic trajectories connecting saddle orbits, J. Dynam. Differential Equations, 24 (2012), 181-196. doi: 10.1007/s10884-012-9247-4.
[5]	J. Franks, Nonsingular Smale flows on $S^{3}$, Topology, 24 (1985), 265-282. doi: 10.1016/0040-9383(85)90002-3.
[6]	V. Grines and O. Pochinka, Morse-Smale cascades on 3-manifolds, (Russian) Uspekhi Mat. Nauk, 68 (2013), 129-188; translation in Russian Math. Surveys, 68 (2013), 117-173.
[7]	A. Hatcher, Notes on Basic 3-Manifold Topology,, Available from: , ().
[8]	J. Morgan, Nonsingular Morse-Smale flows on 3-dimensional manifolds, Topology, 18 (1978), 41-53. doi: 10.1016/0040-9383(79)90013-2.
[9]	J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Translated from the Portuguese by A. K. Manning, Springer-Verlag, New York-Berlin, 1982.
[10]	M. Peixoto, On the classification of flows on 2 -manifolds, in Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971, Academic Press, (1973), 389-419.
[11]	A. Prishlyak, A complete topological invariant of Morse-Smale flows and handle decompositions of 3-manifolds, (Russian) Fundam. Prikl. Mat., 11 (2005), 185-196; translation in J. Math. Sci. (N. Y.), 144 (2007), 4492-4499. doi: 10.1007/s10958-007-0287-y.
[12]	K. de Rezende, Smale flows on the three-sphere, Trans. Amer. Math. Soc, 303 (1987), 283-310. doi: 10.1090/S0002-9947-1987-0896023-7.
[13]	D. Rolfsen, Knots and Links, Corrected reprint of the 1976 original. Mathematics Lecture Series, 7. Publish or Perish, Inc., Houston, TX, 1990.
[14]	C. Robinson, Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, $2^{nd}$ edition, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1999.
[15]	K. Sasano, Links of closed orbits of nonsingular Morse-Smale flows, Proc. Amer. Math. Soc., 88 (1983), 727-734. doi: 10.1090/S0002-9939-1983-0702309-0.
[16]	Y. Umanskii, Necessary and sufficient conditions for topological equivalence of three-dimensional dynamical Morse-Smale systems with a finite number of singular trajectories, (Russian)Mat. Sb., 181 (1990), 212-239; translation in Math. USSR-Sb., 69 (1991), 227-253.
[17]	M. Wada, Closed orbits of nonsingular Morse-Smale flows on $S^3$, J. Math. Soc. Japan, 41 (1989), 405-413. doi: 10.2969/jmsj/04130405.
[18]	K. Yano, The homotopy class of nonsingular Morse-Smale vector fields on 3 -manifolds, Invent. Math., 80 (1985), 435-451. doi: 10.1007/BF01388724.
[19]	B. Yu, The templates of non-singular Smale flows on three manifolds, Ergodic Theory Dynam. Systems, 32 (2012), 1137-1155. doi: 10.1017/S0143385711000150.
[20]	B. Yu, Lyapunov graphs of nonsingular Smale flows on $S^1 \times S^2$, Trans. Amer. Math. Soc., 365 (2013), 767-783. doi: 10.1090/S0002-9947-2012-05636-4.
[21]	B. Yu, A note on homotopy classes of nonsingular verctor fields on $S^3$, C. R. Math. Acad. Sci. Paris, 352 (2014), 351-355. doi: 10.1016/j.crma.2014.01.016.

show all references

References:

[1]	D. Asimov, Round handles and non-singular Morse-Smale flows, Ann. of Math. (2), 102 (1975), 41-54. doi: 10.2307/1970972.
[2]	D. Asimov, Homotopy of non-singular vector fields to structurally stable ones, Ann. of Math. (2), 102 (1975), 55-65. doi: 10.2307/1970973.
[3]	C. Bonatti, V. Grines and R. Langevin, Dynamical systems in dimension 2 and 3: conjugacy invariants and classification, in The geometry of differential equations and dynamical systems, Comput. Appl. Math., 20 (2001), 11-50.
[4]	B. Campos and P. Vindel, NMS flows on $S^3$ with no heteroclinic trajectories connecting saddle orbits, J. Dynam. Differential Equations, 24 (2012), 181-196. doi: 10.1007/s10884-012-9247-4.
[5]	J. Franks, Nonsingular Smale flows on $S^{3}$, Topology, 24 (1985), 265-282. doi: 10.1016/0040-9383(85)90002-3.
[6]	V. Grines and O. Pochinka, Morse-Smale cascades on 3-manifolds, (Russian) Uspekhi Mat. Nauk, 68 (2013), 129-188; translation in Russian Math. Surveys, 68 (2013), 117-173.
[7]	A. Hatcher, Notes on Basic 3-Manifold Topology,, Available from: , ().
[8]	J. Morgan, Nonsingular Morse-Smale flows on 3-dimensional manifolds, Topology, 18 (1978), 41-53. doi: 10.1016/0040-9383(79)90013-2.
[9]	J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Translated from the Portuguese by A. K. Manning, Springer-Verlag, New York-Berlin, 1982.
[10]	M. Peixoto, On the classification of flows on 2 -manifolds, in Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971, Academic Press, (1973), 389-419.
[11]	A. Prishlyak, A complete topological invariant of Morse-Smale flows and handle decompositions of 3-manifolds, (Russian) Fundam. Prikl. Mat., 11 (2005), 185-196; translation in J. Math. Sci. (N. Y.), 144 (2007), 4492-4499. doi: 10.1007/s10958-007-0287-y.
[12]	K. de Rezende, Smale flows on the three-sphere, Trans. Amer. Math. Soc, 303 (1987), 283-310. doi: 10.1090/S0002-9947-1987-0896023-7.
[13]	D. Rolfsen, Knots and Links, Corrected reprint of the 1976 original. Mathematics Lecture Series, 7. Publish or Perish, Inc., Houston, TX, 1990.
[14]	C. Robinson, Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, $2^{nd}$ edition, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1999.
[15]	K. Sasano, Links of closed orbits of nonsingular Morse-Smale flows, Proc. Amer. Math. Soc., 88 (1983), 727-734. doi: 10.1090/S0002-9939-1983-0702309-0.
[16]	Y. Umanskii, Necessary and sufficient conditions for topological equivalence of three-dimensional dynamical Morse-Smale systems with a finite number of singular trajectories, (Russian)Mat. Sb., 181 (1990), 212-239; translation in Math. USSR-Sb., 69 (1991), 227-253.
[17]	M. Wada, Closed orbits of nonsingular Morse-Smale flows on $S^3$, J. Math. Soc. Japan, 41 (1989), 405-413. doi: 10.2969/jmsj/04130405.
[18]	K. Yano, The homotopy class of nonsingular Morse-Smale vector fields on 3 -manifolds, Invent. Math., 80 (1985), 435-451. doi: 10.1007/BF01388724.
[19]	B. Yu, The templates of non-singular Smale flows on three manifolds, Ergodic Theory Dynam. Systems, 32 (2012), 1137-1155. doi: 10.1017/S0143385711000150.
[20]	B. Yu, Lyapunov graphs of nonsingular Smale flows on $S^1 \times S^2$, Trans. Amer. Math. Soc., 365 (2013), 767-783. doi: 10.1090/S0002-9947-2012-05636-4.
[21]	B. Yu, A note on homotopy classes of nonsingular verctor fields on $S^3$, C. R. Math. Acad. Sci. Paris, 352 (2014), 351-355. doi: 10.1016/j.crma.2014.01.016.

[1]	Bin Yu. Regular level sets of Lyapunov graphs of nonsingular Smale flows on 3-manifolds. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1277-1290. doi: 10.3934/dcds.2011.29.1277
[2]	Ming-Chia Li. Stability of parameterized Morse-Smale gradient-like flows. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 1073-1077. doi: 10.3934/dcds.2003.9.1073
[3]	Vladislav Kibkalo, Tomoo Yokoyama. Topological characterizations of Morse-Smale flows on surfaces and generic non-Morse-Smale flows. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022072
[4]	Radosław Czaja, Waldyr M. Oliva, Carlos Rocha. On a definition of Morse-Smale evolution processes. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3601-3623. doi: 10.3934/dcds.2017155
[5]	Robert W. Ghrist. Flows on $S^3$ supporting all links as orbits. Electronic Research Announcements, 1995, 1: 91-97.
[6]	Arnaud Goullet, Shaun Harker, Konstantin Mischaikow, William D. Kalies, Dinesh Kasti. Efficient computation of Lyapunov functions for Morse decompositions. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2419-2451. doi: 10.3934/dcdsb.2015.20.2419
[7]	A. Procacci, Benedetto Scoppola. Convergent expansions for random cluster model with $q>0$ on infinite graphs. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1145-1178. doi: 10.3934/cpaa.2008.7.1145
[8]	Wael Bahsoun, Benoît Saussol. Linear response in the intermittent family: Differentiation in a weighted $C^0$-norm. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6657-6668. doi: 10.3934/dcds.2016089
[9]	Konstantin Mischaikow, Marian Mrozek, Frank Weilandt. Discretization strategies for computing Conley indices and Morse decompositions of flows. Journal of Computational Dynamics, 2016, 3 (1) : 1-16. doi: 10.3934/jcd.2016001
[10]	A. Kochergin. Well-approximable angles and mixing for flows on T^2 with nonsingular fixed points. Electronic Research Announcements, 2004, 10: 113-121.
[11]	Qianying Xiao, Zuohuan Zheng. $C^1$ weak Palis conjecture for nonsingular flows. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1809-1832. doi: 10.3934/dcds.2018074
[12]	Keith Burns, Katrin Gelfert. Lyapunov spectrum for geodesic flows of rank 1 surfaces. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1841-1872. doi: 10.3934/dcds.2014.34.1841
[13]	Shin Kiriki, Ming-Chia Li, Teruhiko Soma. Geometric Lorenz flows with historic behavior. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7021-7028. doi: 10.3934/dcds.2016105
[14]	Geonho Lee, Sangdong Kim, Young-Sam Kwon. Large time behavior for the full compressible magnetohydrodynamic flows. Communications on Pure and Applied Analysis, 2012, 11 (3) : 959-971. doi: 10.3934/cpaa.2012.11.959
[15]	Shan Ma, Chunyou Sun. Long-time behavior for a class of weighted equations with degeneracy. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1889-1902. doi: 10.3934/dcds.2020098
[16]	Toshikazu Kuniya, Mimmo Iannelli. $R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission. Mathematical Biosciences & Engineering, 2014, 11 (4) : 929-945. doi: 10.3934/mbe.2014.11.929
[17]	Young-Pil Choi, Cristina Pignotti. Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays. Networks and Heterogeneous Media, 2019, 14 (4) : 789-804. doi: 10.3934/nhm.2019032
[18]	Grant Cairns, Barry Jessup, Marcel Nicolau. Topologically transitive homeomorphisms of quotients of tori. Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 291-300. doi: 10.3934/dcds.1999.5.291
[19]	Katrin Gelfert. Non-hyperbolic behavior of geodesic flows of rank 1 surfaces. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 521-551. doi: 10.3934/dcds.2019022
[20]	Ciprian G. Gal, Maurizio Grasselli. Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 1-39. doi: 10.3934/dcds.2010.28.1

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Bin Yu
on Google Scholar
Bin Yu

[Back to Top]