Inverse Shadowing and Weak Inverse Shadowing Property

ABSTRACT

In this paper we show that an -stable diffeomorphism has the weak inverse shadowing property with respect to classes of continuous method and and some of the -stable diffeomorphisms have weak inverse shadowing property with respect to classes . In addition we study relation between minimality and weak inverse shadowing property with respect to class and relation between expansivity and inverse shadowing property with respect to class .

In this paper we show that an -stable diffeomorphism has the weak inverse shadowing property with respect to classes of continuous method and and some of the -stable diffeomorphisms have weak inverse shadowing property with respect to classes . In addition we study relation between minimality and weak inverse shadowing property with respect to class and relation between expansivity and inverse shadowing property with respect to class .

Cite this paper

B. Honary and A. Bahabadi, "Inverse Shadowing and Weak Inverse Shadowing Property,"*Applied Mathematics*, Vol. 3 No. 5, 2012, pp. 478-483. doi: 10.4236/am.2012.35072.

B. Honary and A. Bahabadi, "Inverse Shadowing and Weak Inverse Shadowing Property,"

References

[1] R. Corless and S. Plyugin, “Approximate and Real Trajectories for Generic Dynamical Systems,” Journal of Mathematical Analysis and Applications, Vol. 189, No. 2, 1995, pp. 409-423. doi:10.1006/jmaa.1995.1027

[2] P. Diamond, P. Kloeden, V. Korzyakin and A. Pokrovskii, “Computer Robustness of Semihypebolic Mappings,” Random and computational Dynamics, Vol. 3, 1995, pp. 53-70.

[3] P. Kloeden, J. Ombach and A. Pokroskii, “Continuous and Inverse Shadowing,” Functional Differential Equations, Vol. 6, 1999, pp. 137-153.

[4] K. Lee, “Continuous Inverse Shadowing and Hyperbolic,” Bulletin of the Australian Mathematical Society, Vol. 67, No. 1, 2003, pp. 15-26. doi:10.1017/S0004972700033487

[5] S. Yu. Pilyugin, “Inverse Shadowing by Continuous Methods,” Discrete and Continuous Dynamical Systems, Vol. 8, No. 1, 2002, pp. 29-28. doi:10.3934/dcds.2002.8.29

[6] P. Diamond, Y. Han and K. Lee, “Bishadowing and Hyperbolicity,” International Journal of Bifuractions and Chaos, Vol. 12, 2002, pp. 1779-1788.

[7] T. Choi, S. Kim and K. Lee, “Weak Inverse Shadowing and Genericity,” Bulletin of the Korean Mathematical Society, Vol. 43, No. 1, 2006, pp. 43-52.

[8] B. Honary and A. Zamani Bahabadi, “Weak Strictly Persistence Homeomorphisms and Weak Inverse Shadowing Property and Genericity,” Kyungpook Mathematical Journal, Vol. 49, 2009, pp. 411-418.

[9] R. Gu, Y. Sheng and Z. Xia, “The Average-Shadowing Property and Transitivity for Continuous Flows,” Chaos, Solitons, Fractals, Vol. 23, No. 3, 2005, pp. 989-995.

[10] L.-F. He and Z.-H. Wang, “Distal Flows with the Pseudo Orbit Tracing Property,” Chinese Science Bulletin, Vol. 39, 1994, pp. 1936-1938.

[11] K. Kato, “Pseudo-Orbits and Stabilities Flows,” Memoirs of the Faculty of Science Kochi University Series A Mathematics, Vol. 5, 1984, pp. 45-62.

[12] M. Komouro, “One-Parameter Flows with the PseudoOrbit Tracing Property,” Mathematics and Statistics, Vol. 98, No. 3, 1984, pp. 219-253. doi:10.1007/BF01507750

[13] C. Robinson, “Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,” CRC Press, Boca Raton, 1998.

[14] S. Yu. Pilyugin, “Introduction to Structurally Stable Systems of Differential Equations,” Birkhauser Verlag, Boston, 1992. doi:10.1007/978-3-0348-8643-7

[15] I. P. Malta, “Hyperbolic Birkhoff Centers,” Transactions of the American Mathematical Society, Vol. 262, 1980, pp. 181-193. doi:10.1090/S0002-9947-1980-0583851-4

[16] O. B. Plaenevskaya, “Weak Shadowing for Two-Dimensional Diffeomorphism,” Vestnik St. Petersburg University: Mathematics, Vol. 31, 1998, pp. 49-56.

[1] R. Corless and S. Plyugin, “Approximate and Real Trajectories for Generic Dynamical Systems,” Journal of Mathematical Analysis and Applications, Vol. 189, No. 2, 1995, pp. 409-423. doi:10.1006/jmaa.1995.1027

[2] P. Diamond, P. Kloeden, V. Korzyakin and A. Pokrovskii, “Computer Robustness of Semihypebolic Mappings,” Random and computational Dynamics, Vol. 3, 1995, pp. 53-70.

[3] P. Kloeden, J. Ombach and A. Pokroskii, “Continuous and Inverse Shadowing,” Functional Differential Equations, Vol. 6, 1999, pp. 137-153.

[4] K. Lee, “Continuous Inverse Shadowing and Hyperbolic,” Bulletin of the Australian Mathematical Society, Vol. 67, No. 1, 2003, pp. 15-26. doi:10.1017/S0004972700033487

[5] S. Yu. Pilyugin, “Inverse Shadowing by Continuous Methods,” Discrete and Continuous Dynamical Systems, Vol. 8, No. 1, 2002, pp. 29-28. doi:10.3934/dcds.2002.8.29

[6] P. Diamond, Y. Han and K. Lee, “Bishadowing and Hyperbolicity,” International Journal of Bifuractions and Chaos, Vol. 12, 2002, pp. 1779-1788.

[7] T. Choi, S. Kim and K. Lee, “Weak Inverse Shadowing and Genericity,” Bulletin of the Korean Mathematical Society, Vol. 43, No. 1, 2006, pp. 43-52.

[8] B. Honary and A. Zamani Bahabadi, “Weak Strictly Persistence Homeomorphisms and Weak Inverse Shadowing Property and Genericity,” Kyungpook Mathematical Journal, Vol. 49, 2009, pp. 411-418.

[9] R. Gu, Y. Sheng and Z. Xia, “The Average-Shadowing Property and Transitivity for Continuous Flows,” Chaos, Solitons, Fractals, Vol. 23, No. 3, 2005, pp. 989-995.

[10] L.-F. He and Z.-H. Wang, “Distal Flows with the Pseudo Orbit Tracing Property,” Chinese Science Bulletin, Vol. 39, 1994, pp. 1936-1938.

[11] K. Kato, “Pseudo-Orbits and Stabilities Flows,” Memoirs of the Faculty of Science Kochi University Series A Mathematics, Vol. 5, 1984, pp. 45-62.

[12] M. Komouro, “One-Parameter Flows with the PseudoOrbit Tracing Property,” Mathematics and Statistics, Vol. 98, No. 3, 1984, pp. 219-253. doi:10.1007/BF01507750

[13] C. Robinson, “Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,” CRC Press, Boca Raton, 1998.

[14] S. Yu. Pilyugin, “Introduction to Structurally Stable Systems of Differential Equations,” Birkhauser Verlag, Boston, 1992. doi:10.1007/978-3-0348-8643-7

[15] I. P. Malta, “Hyperbolic Birkhoff Centers,” Transactions of the American Mathematical Society, Vol. 262, 1980, pp. 181-193. doi:10.1090/S0002-9947-1980-0583851-4

[16] O. B. Plaenevskaya, “Weak Shadowing for Two-Dimensional Diffeomorphism,” Vestnik St. Petersburg University: Mathematics, Vol. 31, 1998, pp. 49-56.