Skip to main content
Log in

Control of multistability in hidden attractors

The European Physical Journal Special Topics Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

Hidden attractors have a basin of attraction which is not connected with unstable equilibrium. Certain systems with hidden attractor show multistability for a range of parameter. Multistability or coexistence of different attractors in nonlinear systems often creates inconvenience and therefore, needs to be avoided to obtain a desired specific output from the system. We discuss the control of multistability in the hidden attractor through the scheme of linear augmentation, that can drive the multistable system to a monostable state. With the proper choice of control parameters a shift from multistability to monostability can be achieved. This transition from multiple attractors to a single attractor is confirmed by calculating the basin size as a measure. When a nonlinear system with hidden attractors is coupled with a linear system, two important transitions are observed with the increase of coupling strength: transition from multistability to monostability and then stabilization of newly created equilibrium point via suppression of oscillations.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. E. Ott, Chaos is Dynamical System (Cambridge University Press, Cambridge, 1993)

  2. S.H. Strogatz, Nonlinear Dynamics and Chaos (Westview Press, New York, 1994)

  3. A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear sciences (Cambridge University Press, Cambridge, 2003)

  4. N.A. Gubar’, J. Appl. Math. Mech. 25, 1519 (1961)

    Article  MathSciNet  Google Scholar 

  5. S. Timoshenko, Vibration Problems in Engineering (Van Nostran NY, 1928)

  6. A. Andronov, A. Vitt, S. Khaikin, Nonlinear Oscillations (Pergamon, Oxford, 1966)

  7. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems (Interscience, NY, 1950)

  8. L.O. Chua, G.N. Lin, IEEE Trans. Circ. Syst. 37, 885 (1990)

    Article  MathSciNet  Google Scholar 

  9. N.V. Kuznetsov, G.A. Leonov, V.I. Vagaitsev, IFAC Proc. 4, 29 (2010)

    Google Scholar 

  10. G.A. Leonov, V.I. Vagaitsev, N.V. Kuznetsov, Phys. Lett. A 375, 2230 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  11. G.A. Leonov, N.V. Kuznetsov, V.I. Vagaitsev, Physica D 241, 1482 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  12. V.I. Arnold, Experimental Mathematics (Fazis, Moscow 2005) [in Russian]

  13. G.A. Leonov, Appl. Math. Mech. 74, 24 (2010)

    Article  MathSciNet  Google Scholar 

  14. N.V. Kuznetsov, O.A. Kuznetsova, G.A. Leonov, Differ. Equ. Dyn. Syst. 21, 29 (2013)

    Article  MathSciNet  Google Scholar 

  15. G.A. Leonov, N.V. Kuznetsov, Int. J. Bifurcat. Chaos 23, 1330002 (2013)

    Article  MathSciNet  Google Scholar 

  16. S. Wang, G. Chen, Nonlinear Dyn. 71, 249 (2013)

    Google Scholar 

  17. M.A. Aizerman, Lecture on Automatic Control Theory (Fizmatgiz, Moscow, 1958) [in Russian]

  18. R.E. Kalman, Trans. ASME 79, 553 (1957)

    MathSciNet  Google Scholar 

  19. A.M. Letov, Stability of Nonlinear Control Systems (GTTI, Moscow, 1955) [in Russian]

  20. M.A. Aizerman, F.R. Gantmakher, Absolute Stability of Control Systems (Akad. Nauk SSSR, Moscow, 1963) [in Russian]

  21. S.L. Lefschetz, Stability of Nonlinear Control Systems (Academic, New York, London, 1965)

  22. F. Atteneave, Sci. Am. 225, 63 (1971)

    Article  Google Scholar 

  23. F.T. Arecchi, F. Lisi, Phys. Rev. Lett. 49, 94 (1982)

    Article  ADS  Google Scholar 

  24. F.T. Arecchi, R. Meucci, G. Puccioni, J. Tredicce, Phys. Rev. Lett. 49, 1982 (1982)

    Google Scholar 

  25. U. Feudel Int. J. Bifurc. Chaos 18, 1607 (2008)

    Article  MathSciNet  Google Scholar 

  26. A. Chudzik, P. Perlikowski, A. Stefanski, T. Kapitaniak, Int. J. Bifurc. Chaos 21, 1907 (2011)

    Article  MathSciNet  Google Scholar 

  27. S.L.T. de Souza, A.M. Batista, I.L. Caldas, R.L. Viana, T. Kapitaniak, Chaos, Solitons Fractals 32, 758 (2007)

    Article  ADS  Google Scholar 

  28. B. Blazejczyk-Okolewska, T. Kapitaniak, Chaos, Solitons Fractals 9, 1439 (1998)

    Article  ADS  Google Scholar 

  29. T. Kapitaniak, Phys. Rev. E 47, 1408 (1993)

    Article  ADS  Google Scholar 

  30. T. Kapitaniak, J. Sound Vibration 102, 440 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  31. C. Grebogi, S.W. McDonald, E. Ott, J.A. Yorke, Phys. Lett. A 99, 415 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  32. S.W. McDonald, C. Grebogi, E. Ott, J.A. Yorke, Physica D 17, 125 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  33. Y.-C. Lai, R.L. Winslow, Phys. Rev. Lett. 72, 1640 (1994)

    Article  ADS  Google Scholar 

  34. U. Chaudhuri, A. Prasad, Phys. Lett. A 378, 713 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  35. A.N. Pisarchik, U. Feudel, Physics Reports 540, 167 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  36. P.R. Sharma, A. Sharma, M.D. Shrimali, A. Prasad, Phys. Rev. E 83, 067201 (2011)

    Article  ADS  Google Scholar 

  37. P.R. Sharma, M.D. Shrimali, A. Prasad, U. Feudel, Phys. Lett. A 377, 2329 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  38. P.R. Sharma, A. Singh, A. Prasad, M.D. Shrimali, Eur. Phys. J. Special Topics 223, 1531 (2014)

    Article  ADS  Google Scholar 

  39. P.R. Sharma, M.D. Shrimali, A. Prasad, N.V. Kuznetsov, G.A. Leonov, Int. J. Bifur. Chaos 25, 1550061 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. Prasad.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sharma, P.R., Shrimali, M.D., Prasad, A. et al. Control of multistability in hidden attractors. Eur. Phys. J. Spec. Top. 224, 1485–1491 (2015). https://doi.org/10.1140/epjst/e2015-02474-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjst/e2015-02474-y

Navigation