American Institute of Mathematical Sciences

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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

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 & Continuous Dynamical Systems - A, 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., 2 (1970), 39. Google Scholar

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R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Lecture Notes in Math., (1975). Google Scholar

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S. Gan, M. Li and S. B. Tikhomirov, Oriented shadowing property and $\Omega$-stability for vector fields,, J. Dynam. Differential Equations, 28 (2016), 225. doi: 10.1007/s10884-014-9399-5. Google Scholar

[5]

S. Gan and L. Wen, Nonsingular star flows satisfy Axiom A and the nocycle condition,, Invent. Math., 164 (2006), 279. doi: 10.1007/s00222-005-0479-3. Google Scholar

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K. Lee and K. Sakai, Structural stability of vector fields with shadowing,, J. Differential Equations, 232 (2007), 303. doi: 10.1016/j.jde.2006.08.012. Google Scholar

[7]

A. Morimoto, The Method of Pseudo-Orbit Tracing and Stability of Dynamical Systems,, Sem. Note, 39 (1979). Google Scholar

[8]

A. V. Osipov, S. Yu. Pilyugin and S. B. Tikhomirov, Periodic Shadowing and $\Omega$-stability,, Regul. Chaotic Dyn., 15 (2010), 404. doi: 10.1134/S1560354710020255. Google Scholar

[9]

K. J. Palmer, Shadowing in Dynamical Systems: Theory and Applications,, Kluwer, (2000). doi: 10.1007/978-1-4757-3210-8. Google Scholar

[10]

K. J. Palmer, S. Yu. Pilyugin and S. B. Tikhomirov, Lipschitz shadowing and structural stability of flows,, J. Differential Equations, 252 (2012), 1723. doi: 10.1016/j.jde.2011.07.026. Google Scholar

[11]

S. Yu. Pilyugin, Shadowing in Dynamical Systems,, Lecture Notes in Math., 1706 (1999). Google Scholar

[12]

S. Yu. Pilyugin, Variational shadowing,, Discrete Contin. Dyn. Syst. B, 14 (2010), 733. doi: 10.3934/dcdsb.2010.14.733. Google Scholar

[13]

S. Yu. Pilyugin and S. B. Tikhomirov, Vector fields with the oriented shadowing property,, J. Differential Equations, 248 (2010), 1345. doi: 10.1016/j.jde.2009.09.024. Google Scholar

[14]

S. Yu. Pilyugin and S. B. Tikhomirov, Lipschitz shadowing implies structural stability,, Nonlinearity, 23 (2010), 2509. doi: 10.1088/0951-7715/23/10/009. Google Scholar

[15]

C. Robinson, Stability theorems and hyperbolicity in dynamical systems,, Rocky Mountain J. Math., 7 (1977), 425. doi: 10.1216/RMJ-1977-7-3-425. Google Scholar

[16]

K. Sakai, Pseudo orbit tracing property and strong transversality of diffeomorphisms on closed manifolds,, Osaka J. Math., 31 (1994), 373. Google Scholar

[17]

K. Sawada, Extended $f$-orbits are approximated by orbits,, Nagoya Math. J., 79 (1980), 33. Google Scholar

[18]

L. Wen, A uniform $C^1$ connecting lemma,, Discrete Contin. Dyn. Syst., 8 (2002), 257. doi: 10.3934/dcds.2002.8.257. Google Scholar

[19]

L. Wen and Z. Xia, $C^1$ connecting lemmas,, Trans. Amer. Math. Soc., 352 (2000), 5213. doi: 10.1090/S0002-9947-00-02553-8. Google Scholar

[20]

S. Zhu, S. Gan and L. Wen, Indices of singularities of robustly transitive sets,, Discrete Cont. Dyn. Syst., 21 (2008), 945. doi: 10.3934/dcds.2008.21.945. Google Scholar

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., 2 (1970), 39. Google Scholar

[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. doi: 10.1017/S1474748012000710. Google Scholar

[3]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Lecture Notes in Math., (1975). Google Scholar

[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. doi: 10.1007/s10884-014-9399-5. Google Scholar

[5]

S. Gan and L. Wen, Nonsingular star flows satisfy Axiom A and the nocycle condition,, Invent. Math., 164 (2006), 279. doi: 10.1007/s00222-005-0479-3. Google Scholar

[6]

K. Lee and K. Sakai, Structural stability of vector fields with shadowing,, J. Differential Equations, 232 (2007), 303. doi: 10.1016/j.jde.2006.08.012. Google Scholar

[7]

A. Morimoto, The Method of Pseudo-Orbit Tracing and Stability of Dynamical Systems,, Sem. Note, 39 (1979). Google Scholar

[8]

A. V. Osipov, S. Yu. Pilyugin and S. B. Tikhomirov, Periodic Shadowing and $\Omega$-stability,, Regul. Chaotic Dyn., 15 (2010), 404. doi: 10.1134/S1560354710020255. Google Scholar

[9]

K. J. Palmer, Shadowing in Dynamical Systems: Theory and Applications,, Kluwer, (2000). doi: 10.1007/978-1-4757-3210-8. Google Scholar

[10]

K. J. Palmer, S. Yu. Pilyugin and S. B. Tikhomirov, Lipschitz shadowing and structural stability of flows,, J. Differential Equations, 252 (2012), 1723. doi: 10.1016/j.jde.2011.07.026. Google Scholar

[11]

S. Yu. Pilyugin, Shadowing in Dynamical Systems,, Lecture Notes in Math., 1706 (1999). Google Scholar

[12]

S. Yu. Pilyugin, Variational shadowing,, Discrete Contin. Dyn. Syst. B, 14 (2010), 733. doi: 10.3934/dcdsb.2010.14.733. Google Scholar

[13]

S. Yu. Pilyugin and S. B. Tikhomirov, Vector fields with the oriented shadowing property,, J. Differential Equations, 248 (2010), 1345. doi: 10.1016/j.jde.2009.09.024. Google Scholar

[14]

S. Yu. Pilyugin and S. B. Tikhomirov, Lipschitz shadowing implies structural stability,, Nonlinearity, 23 (2010), 2509. doi: 10.1088/0951-7715/23/10/009. Google Scholar

[15]

C. Robinson, Stability theorems and hyperbolicity in dynamical systems,, Rocky Mountain J. Math., 7 (1977), 425. doi: 10.1216/RMJ-1977-7-3-425. Google Scholar

[16]

K. Sakai, Pseudo orbit tracing property and strong transversality of diffeomorphisms on closed manifolds,, Osaka J. Math., 31 (1994), 373. Google Scholar

[17]

K. Sawada, Extended $f$-orbits are approximated by orbits,, Nagoya Math. J., 79 (1980), 33. Google Scholar

[18]

L. Wen, A uniform $C^1$ connecting lemma,, Discrete Contin. Dyn. Syst., 8 (2002), 257. doi: 10.3934/dcds.2002.8.257. Google Scholar

[19]

L. Wen and Z. Xia, $C^1$ connecting lemmas,, Trans. Amer. Math. Soc., 352 (2000), 5213. doi: 10.1090/S0002-9947-00-02553-8. Google Scholar

[20]

S. Zhu, S. Gan and L. Wen, Indices of singularities of robustly transitive sets,, Discrete Cont. Dyn. Syst., 21 (2008), 945. doi: 10.3934/dcds.2008.21.945. Google Scholar

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