Periodic shadowing of vector fields

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July 2016, 36(7): 3623-3638. doi: 10.3934/dcds.2016.36.3623

Periodic shadowing of vector fields

Jifeng Chu ^1, , Zhaosheng Feng ^2, and Ming Li ^3,

1.	Department of Mathematics, Hohai University, Nanjing 211100, China
2.	School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, TX 78539
3.	School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

Received March 2015 Revised December 2015 Published March 2016

Abstract
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A vector field has the periodic shadowing property if for any $\varepsilon>0$ there is $d>0$ such that, for any periodic $d$-pseudo orbit $g$ there exists a periodic orbit or a singularity in which $g$ is $\varepsilon$-shadowed. In this paper, we show that a vector field is in the $C^1$ interior of the set of vector fields satisfying the periodic shadowing property if and only if it is $\Omega$-stable. More precisely, we prove that the $C^1$ interior of the set of vector fields satisfying the orbital periodic shadowing property is a subset of the set of $\Omega$-stable vector fields.

Keywords: homoclinic connection, diffeomorphism, $\Omega$-stability., periodic shadowing property, Vector fields, hyperbolic singularity.

Mathematics Subject Classification: Primary: 37C50, 37C20; Secondary: 37J4.

Citation: Jifeng Chu, Zhaosheng Feng, Ming Li. Periodic shadowing of vector fields. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3623-3638. doi: 10.3934/dcds.2016.36.3623

References:

[1]	D. V. Anosov, On a class of invariant sets of smooth dynamical systems, Proc. 5th Int. Conf. on Nonlin. Oscill., Kiev, 2 (1970), 39-45.
[2]	C. Bonatti, M. Li and D. Yang, A robustly chain transitive attractor with singularities of different indices, J. Inst. Math. Jussieu, 12 (2013), 449-501. doi: 10.1017/S1474748012000710.
[3]	R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., Vol. 470, Springer, Berlin, 1975.
[4]	S. Gan, M. Li and S. B. Tikhomirov, Oriented shadowing property and $\Omega$-stability for vector fields, J. Dynam. Differential Equations, 28 (2016), 225-237. doi: 10.1007/s10884-014-9399-5.
[5]	S. Gan and L. Wen, Nonsingular star flows satisfy Axiom A and the nocycle condition, Invent. Math., 164 (2006), 279-315. doi: 10.1007/s00222-005-0479-3.
[6]	K. Lee and K. Sakai, Structural stability of vector fields with shadowing, J. Differential Equations, 232 (2007), 303-313. doi: 10.1016/j.jde.2006.08.012.
[7]	A. Morimoto, The Method of Pseudo-Orbit Tracing and Stability of Dynamical Systems, Sem. Note, Vol. 39, Tokyo Univ., 1979.
[8]	A. V. Osipov, S. Yu. Pilyugin and S. B. Tikhomirov, Periodic Shadowing and $\Omega$-stability, Regul. Chaotic Dyn., 15 (2010), 404-417. doi: 10.1134/S1560354710020255.
[9]	K. J. Palmer, Shadowing in Dynamical Systems: Theory and Applications, Kluwer, 2000. doi: 10.1007/978-1-4757-3210-8.
[10]	K. J. Palmer, S. Yu. Pilyugin and S. B. Tikhomirov, Lipschitz shadowing and structural stability of flows, J. Differential Equations, 252 (2012), 1723-1747. doi: 10.1016/j.jde.2011.07.026.
[11]	S. Yu. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Math., Vol. 1706, Springer, 1999.
[12]	S. Yu. Pilyugin, Variational shadowing, Discrete Contin. Dyn. Syst. B, 14 (2010), 733-737. doi: 10.3934/dcdsb.2010.14.733.
[13]	S. Yu. Pilyugin and S. B. Tikhomirov, Vector fields with the oriented shadowing property, J. Differential Equations, 248 (2010), 1345-1375. doi: 10.1016/j.jde.2009.09.024.
[14]	S. Yu. Pilyugin and S. B. Tikhomirov, Lipschitz shadowing implies structural stability, Nonlinearity, 23 (2010), 2509-2515. doi: 10.1088/0951-7715/23/10/009.
[15]	C. Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math., 7 (1977), 425-437. doi: 10.1216/RMJ-1977-7-3-425.
[16]	K. Sakai, Pseudo orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math., 31 (1994), 373-386.
[17]	K. Sawada, Extended $f$-orbits are approximated by orbits, Nagoya Math. J., 79 (1980), 33-45.
[18]	L. Wen, A uniform $C^1$ connecting lemma, Discrete Contin. Dyn. Syst., 8 (2002), 257-265. doi: 10.3934/dcds.2002.8.257.
[19]	L. Wen and Z. Xia, $C^1$ connecting lemmas, Trans. Amer. Math. Soc., 352 (2000), 5213-5230. doi: 10.1090/S0002-9947-00-02553-8.
[20]	S. Zhu, S. Gan and L. Wen, Indices of singularities of robustly transitive sets, Discrete Cont. Dyn. Syst., 21 (2008), 945-957. doi: 10.3934/dcds.2008.21.945.

show all references

References:

[1]	D. V. Anosov, On a class of invariant sets of smooth dynamical systems, Proc. 5th Int. Conf. on Nonlin. Oscill., Kiev, 2 (1970), 39-45.
[2]	C. Bonatti, M. Li and D. Yang, A robustly chain transitive attractor with singularities of different indices, J. Inst. Math. Jussieu, 12 (2013), 449-501. doi: 10.1017/S1474748012000710.
[3]	R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math., Vol. 470, Springer, Berlin, 1975.
[4]	S. Gan, M. Li and S. B. Tikhomirov, Oriented shadowing property and $\Omega$-stability for vector fields, J. Dynam. Differential Equations, 28 (2016), 225-237. doi: 10.1007/s10884-014-9399-5.
[5]	S. Gan and L. Wen, Nonsingular star flows satisfy Axiom A and the nocycle condition, Invent. Math., 164 (2006), 279-315. doi: 10.1007/s00222-005-0479-3.
[6]	K. Lee and K. Sakai, Structural stability of vector fields with shadowing, J. Differential Equations, 232 (2007), 303-313. doi: 10.1016/j.jde.2006.08.012.
[7]	A. Morimoto, The Method of Pseudo-Orbit Tracing and Stability of Dynamical Systems, Sem. Note, Vol. 39, Tokyo Univ., 1979.
[8]	A. V. Osipov, S. Yu. Pilyugin and S. B. Tikhomirov, Periodic Shadowing and $\Omega$-stability, Regul. Chaotic Dyn., 15 (2010), 404-417. doi: 10.1134/S1560354710020255.
[9]	K. J. Palmer, Shadowing in Dynamical Systems: Theory and Applications, Kluwer, 2000. doi: 10.1007/978-1-4757-3210-8.
[10]	K. J. Palmer, S. Yu. Pilyugin and S. B. Tikhomirov, Lipschitz shadowing and structural stability of flows, J. Differential Equations, 252 (2012), 1723-1747. doi: 10.1016/j.jde.2011.07.026.
[11]	S. Yu. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Math., Vol. 1706, Springer, 1999.
[12]	S. Yu. Pilyugin, Variational shadowing, Discrete Contin. Dyn. Syst. B, 14 (2010), 733-737. doi: 10.3934/dcdsb.2010.14.733.
[13]	S. Yu. Pilyugin and S. B. Tikhomirov, Vector fields with the oriented shadowing property, J. Differential Equations, 248 (2010), 1345-1375. doi: 10.1016/j.jde.2009.09.024.
[14]	S. Yu. Pilyugin and S. B. Tikhomirov, Lipschitz shadowing implies structural stability, Nonlinearity, 23 (2010), 2509-2515. doi: 10.1088/0951-7715/23/10/009.
[15]	C. Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math., 7 (1977), 425-437. doi: 10.1216/RMJ-1977-7-3-425.
[16]	K. Sakai, Pseudo orbit tracing property and strong transversality of diffeomorphisms on closed manifolds, Osaka J. Math., 31 (1994), 373-386.
[17]	K. Sawada, Extended $f$-orbits are approximated by orbits, Nagoya Math. J., 79 (1980), 33-45.
[18]	L. Wen, A uniform $C^1$ connecting lemma, Discrete Contin. Dyn. Syst., 8 (2002), 257-265. doi: 10.3934/dcds.2002.8.257.
[19]	L. Wen and Z. Xia, $C^1$ connecting lemmas, Trans. Amer. Math. Soc., 352 (2000), 5213-5230. doi: 10.1090/S0002-9947-00-02553-8.
[20]	S. Zhu, S. Gan and L. Wen, Indices of singularities of robustly transitive sets, Discrete Cont. Dyn. Syst., 21 (2008), 945-957. doi: 10.3934/dcds.2008.21.945.

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