Chaotic Properties on Time Varying Map and Its Set Valued Extension

Affiliation(s)

Department of Mathematics, Zakir Hussain College, University of Delhi, Delhi, India.

Department of Mathematics and Statistics, Ramjas College, University of Delhi, Delhi, India.

Department of Mathematics, Zakir Hussain College, University of Delhi, Delhi, India.

Department of Mathematics and Statistics, Ramjas College, University of Delhi, Delhi, India.

ABSTRACT

Every autonomous dynamical system （*X*, *f*）induces a set-valued dynamical system on the space of compact subsets of *X*. In this paper we have investigated some chaotic relations between a nonautonomous dynamical system and its set valued extension.

Every autonomous dynamical system （

Cite this paper

A. Khan and P. Kumar, "Chaotic Properties on Time Varying Map and Its Set Valued Extension,"*Advances in Pure Mathematics*, Vol. 3 No. 3, 2013, pp. 359-364. doi: 10.4236/apm.2013.33051.

A. Khan and P. Kumar, "Chaotic Properties on Time Varying Map and Its Set Valued Extension,"

References

[1] P. Touhey, “Yet Another Definition of Chaos,” American Mathematical Monthly, Vol. 104, No. 5, 1997, pp. 411-414. doi:10.2307/2974734

[2] M. Vellekoop and R. Berglund, “On Intervals, Transitivity = Chaos,” American Mathematical Monthly, Vol. 101, No. 4, 1994, 353-355. doi:10.2307/2975629

[3] J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, “On Devaney’s Definition of Chaos,” American Mathematical Monthly, Vol. 99, No. 4, 1992, pp. 332-334. doi:10.2307/2324899

[4] S. N. Elaydi, “Discrete Chaos,” Chapman & Hall/CRC, Boca Raton, 2000.

[5] R. L. Devaney, “An Introduction to Chaotic Dynamical Systems,” 2nd Edition, Addision-Welsey, New York, 1989.

[6] C. Tian and G. Chen, “Chaos of a Sequence of Maps in a Metric Space,” Chaos, Solitons and Fractals, Vol. 28, No. 4, 2006, pp. 1067-1075. doi:10.1016/j.chaos.2005.08.127

[7] Y. M. Shi and G. R. Chen, “Chaos of Time-Varying Discrete Dynamical Systems,” Journal of Difference Equations and Applications, Vol. 15, No. 5, 2009, pp. 429-449. doi:10.1080/10236190802020879

[8] Y. M. Shi, “Chaos in Nonautonomous Discrete Dynamical Systems Approached by Their Subsystems,” RFDP of Higher Education of China, Beijing, 2012.

[9] P. Sharma and A. Nagar, “Topological Dynamics on Hyperspaces,” Applied General Topology, Vol. 11, No. 1, 2010, pp. 1-19.

[10] H. Roman-Flores and Y. Chalco-Cano, “Robinsons Chaos in Set-Valued Discrete Systems,” Chaos, Solitons and Fractals, Vol. 25, No. 1, 2005, pp. 33-42.

[11] J. Banks, “Chaos for Induced Hyperspace Maps,” Chaos, Solitons and Fractals, Vol. 25, No. 3, 2005, pp. 681-685. doi:10.1016/j.chaos.2004.11.089

[12] H. Roman-Flores, “A Note on Transitivity in Set Valued Discrete Systems,” Chaos, Solution and Fractals, Vol. 17, No. 1, 2003, pp. 99-104. doi:10.1016/S0960-0779(02)00406-X

[13] R. B. Gu and W. J. Guo, “On Mixing Properties in Set Valued Discrete System,” Chaos, Solitons and Fractals, Vol. 28, No. 3, 2006, pp. 747-754. doi:10.1016/j.chaos.2005.04.004

[1] P. Touhey, “Yet Another Definition of Chaos,” American Mathematical Monthly, Vol. 104, No. 5, 1997, pp. 411-414. doi:10.2307/2974734

[2] M. Vellekoop and R. Berglund, “On Intervals, Transitivity = Chaos,” American Mathematical Monthly, Vol. 101, No. 4, 1994, 353-355. doi:10.2307/2975629

[3] J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, “On Devaney’s Definition of Chaos,” American Mathematical Monthly, Vol. 99, No. 4, 1992, pp. 332-334. doi:10.2307/2324899

[4] S. N. Elaydi, “Discrete Chaos,” Chapman & Hall/CRC, Boca Raton, 2000.

[5] R. L. Devaney, “An Introduction to Chaotic Dynamical Systems,” 2nd Edition, Addision-Welsey, New York, 1989.

[6] C. Tian and G. Chen, “Chaos of a Sequence of Maps in a Metric Space,” Chaos, Solitons and Fractals, Vol. 28, No. 4, 2006, pp. 1067-1075. doi:10.1016/j.chaos.2005.08.127

[7] Y. M. Shi and G. R. Chen, “Chaos of Time-Varying Discrete Dynamical Systems,” Journal of Difference Equations and Applications, Vol. 15, No. 5, 2009, pp. 429-449. doi:10.1080/10236190802020879

[8] Y. M. Shi, “Chaos in Nonautonomous Discrete Dynamical Systems Approached by Their Subsystems,” RFDP of Higher Education of China, Beijing, 2012.

[9] P. Sharma and A. Nagar, “Topological Dynamics on Hyperspaces,” Applied General Topology, Vol. 11, No. 1, 2010, pp. 1-19.

[10] H. Roman-Flores and Y. Chalco-Cano, “Robinsons Chaos in Set-Valued Discrete Systems,” Chaos, Solitons and Fractals, Vol. 25, No. 1, 2005, pp. 33-42.

[11] J. Banks, “Chaos for Induced Hyperspace Maps,” Chaos, Solitons and Fractals, Vol. 25, No. 3, 2005, pp. 681-685. doi:10.1016/j.chaos.2004.11.089

[12] H. Roman-Flores, “A Note on Transitivity in Set Valued Discrete Systems,” Chaos, Solution and Fractals, Vol. 17, No. 1, 2003, pp. 99-104. doi:10.1016/S0960-0779(02)00406-X

[13] R. B. Gu and W. J. Guo, “On Mixing Properties in Set Valued Discrete System,” Chaos, Solitons and Fractals, Vol. 28, No. 3, 2006, pp. 747-754. doi:10.1016/j.chaos.2005.04.004