Skip to main content
SpringerLink
Log in

Finding Cycles in Nonlinear Autonomous Discrete Dynamical Systems

  • Conference paper
  • First Online:
Harmonic Analysis, Partial Differential Equations, Banach Spaces, and Operator Theory (Volume 2)

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 5))

  • 775 Accesses

Abstract

The goal of this paper is to provide an exposition of recent results of the authors concerning cycle localization and stabilization in nonlinear dynamical systems. Both the general theory and numerical applications to well-known dynamical systems are presented. This paper is a continuation of Dmitrishin et al. (Fejér polynomials and chaos. Springer proceedings in mathematics and statistics, vol 108, pp. 49–75, 2014).

Dedicated to Alexey Solyanik on his 55th birthday

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. J.W. Alexander, Functions which map the interior of the unit disc upon simple regions. Ann. Math. 17, 12–22 (1915–1916)

    Google Scholar 

  2. S. Bielawski, D. Derozier, P. Glorieux, Controlling unstable periodic orbits by a delayed continuous feedback. Phys. Rev. E 49, R971 (1994)

    Article  Google Scholar 

  3. M.E. Bleich, J.E.S. Socolar, Stability of periodic orbits controlled by time-delay feedback. Phys. Lett. A 210, 87 (1996)

    Article  Google Scholar 

  4. S. Boccaletti, C. Grebogi, Y.-C. Lai, H. Mancini, D. Maza, The control of chaos: theory and applications. Phys. Rep. 329, 103–197 (2000)

    Article  MathSciNet  Google Scholar 

  5. T. Dahms, P. Hovel, E. Scholl, Control of unstable steady states by extended time-delayed feedback. Phys. Rev. E 76, 056201 (2007)

    Article  MathSciNet  Google Scholar 

  6. D. Dmitrishin, P. Hagelstein, A. Khamitova, A. Stokolos, On the stability of cycles by delayed feedback control. Linear Multilinear Algebra 64 (8), 1538–1549 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. D. Dmitrishin, A. Khamitova, Methods of harmonic analysis in nonlinear dynamics. C. R. Math. 351 (9–10), 367–370 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Dmitrishin, A. Khamitova, A. Stokolos, Fejér polynomials and chaos, in Special Functions, Partial Differential Equations, and Harmonic Analysis. Springer Proceedings in Mathematics and Statistics, vol. 108 (Springer, Cham, 2014), pp. 49–75

    Google Scholar 

  9. D. Dmitrishin, A. Khamitova, A. Korenovskyi, A. Stokolos, Optimal stabilization of a cycle in nonlinear discrete systems. arXiv:1307.7369 [math.DS]

    Google Scholar 

  10. D. Dmitrishin, A. Khamitova, A. Stokolos, On the generalized linear and non-linear DFC in non-linear dynamics. arXiv:1407.6488 [math.DS]

    Google Scholar 

  11. D. Dmitrishin, A. Khamitova, A. Stokolos, M. Tohaneanu, Complex polynomials and cycles in nonlinear autonomous discrete dynamical systems (in preparation)

    Google Scholar 

  12. J. Hizanidis, R. Aust, E. Scholl: Delay-induced multistability near a globalbifurcation. Int. J. Bifur. Chaos 18, 1759 (2008)

    Article  MATH  Google Scholar 

  13. R.A. Horn, C. Johnson, Matrix analysis. (Cambridge University Press, Cambridge, 1985), p. 561

    Book  MATH  Google Scholar 

  14. P. Hovel, J.E.S. Socolar, Stability domains for time-delay feedback control with latency. Phys. Rev. E 68, 036–206 (2003)

    Article  Google Scholar 

  15. P. Hovel, E. Scholl, Control of unstable steady states by time-delayed feedback methods. Phys. Rev. E 72, 046–203 (2005)

    Article  Google Scholar 

  16. W. Just, H. Benner, E. Scholl, Control of chaos by time–delayed feedback: a survey of theoretical and experimental aspects, in Advances in Solid State Physics, vol. 43, ed. by B. Kramer (Springer, Berlin, 2003), pp. 589–603

    Chapter  Google Scholar 

  17. W. Just, T. Bernard, M. Ostheimer, E. Reibold, H. Benner, Mechanism of time-delayed feedback control. Phys. Rev. Lett. 78, 203 (1997). Time-Delayed Feedback Control 63

    Google Scholar 

  18. W. Just, B. Fiedler, V. Flunkert, M. Georgi, P. Hovel, E. Scholl, Beyond odd number limitation: a bifurcation analysis of time-delayed feedback control. Phys. Rev. E 76, 026–210 (2007)

    Article  MathSciNet  Google Scholar 

  19. Ö. Morgül, Further stability results for a generalization of delayed feedback control. Nonlinear Dyn. 70 (2), 1255–1262 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. I.D. Murray, Mathematical Biology. An Introduction, 3rd edn. (Springer, New York, 2002), p. 551

    MATH  Google Scholar 

  21. E. Scholl, P. Hovel, V. Flunkert, M.A. Dahlem, H. Nakajima, On analytical properties of delayed feedback control of chaos. Phys. Lett. A 232, 207 (1997)

    Article  MathSciNet  Google Scholar 

  22. E. Scholl, H. G. Schuster (eds.), Handbook of Chaos Control (Wiley- VCH, Weinheim, 2008). Second completely revised and enlarged edition

    Google Scholar 

  23. J.E.S. Socolar, D.W. Sukow, D.J. Gauthier, Stabilizing unstable periodic orbits in fast dynamical systems. Phys. Rev. E 50, 32–45 (1994)

    Article  Google Scholar 

  24. J.E.S. Socolar, D.J. Gauthier, Analysis and comparison of multiple-delay schemes for controlling unstable fixed points of discrete maps. Phys. Rev. E 57, 65–89 (1998)

    Google Scholar 

  25. Y.-P. Tian, Z. Jiandong, Full characterization on limitation of generalized delayed feedback control for discrete-time systems. Phisica D 198, 248–257 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. d.S.M. Vieira, A.J. Lichtenberg, Controlling chaos using nonlinear feedback with delay. Phys. Rev. E 54, 1200–1207 (1996)

    Google Scholar 

  27. S. Yanchuk, M. Wolfrum, P. Hovel, E. Scholl, Control of unstable steady states by long delay feedback. Phys. Rev. E 74, 026201 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Odessa National Polytechnic University, 1 Shevchenko Ave., Odessa, 65044, Ukraine

    Dmitriy Dmitrishin

  2. Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, 30458, USA

    Anna Khamitova & Alexander M. Stokolos

  3. University of Kentucky, Lexington, KY, 40506, USA

    Mihai Tohaneanu

Authors
  1. Dmitriy Dmitrishin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anna Khamitova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexander M. Stokolos
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mihai Tohaneanu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dmitriy Dmitrishin .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico, USA

    María Cristina Pereyra

  2. Department of Mathematics, Venezuelan Institute for Scientific Research, Caracas, Venezuela

    Stefania Marcantognini

  3. Department of Mathematical Sciences, Georgia Southern University, Statesboro, Georgia, USA

    Alexander M. Stokolos

  4. Department of Mathematical and Actuarial Sciences, Roosevelt University Chicago, Chicago, Illinois, USA

    Wilfredo Urbina

Rights and permissions

Reprints and permissions

Copyright information

© 2017 The Author(s) and the Association for Women in Mathematics

About this paper

Cite this paper

Dmitrishin, D., Khamitova, A., Stokolos, A.M., Tohaneanu, M. (2017). Finding Cycles in Nonlinear Autonomous Discrete Dynamical Systems. In: Pereyra, M., Marcantognini, S., Stokolos, A., Urbina, W. (eds) Harmonic Analysis, Partial Differential Equations, Banach Spaces, and Operator Theory (Volume 2). Association for Women in Mathematics Series, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-51593-9_8

Download citation

Publish with us

Policies and ethics

Search

Navigation