Skip to main content
Log in

State space reconstruction applied to a multiparameter chaos control method

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

The idea of the chaos control is the stabilization of unstable periodic orbits (UPOs) embedded in chaotic attractors. The OGY method achieves system stabilization by using small perturbations promoted in the neighborhood of the desired orbit when the trajectory crosses a Poincaré section. A generalization of this method considers multiple actuations of parameters and sections, known as semi-continuous multiparameter method. This paper investigates the state space reconstruction applied to this general method, allowing chaotic behavior control of systems with non-observable states using multiple control parameters from time series analysis, avoiding the use of governing equations. As an application of the proposed multiparameter general formulation it is presented an uncoupled approach where the control parameters do not influence the system dynamics when they are not active. This method is applied to control chaos in a nonlinear pendulum using delay coordinates to perform state space reconstruction. Results show that the proposed procedure can be applied together with delay coordinates providing UPO stabilization.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Andrievskii BR, Fradkov AL (2004) Control of chaos: methods and applications. II. Applications. Autom Remote Control 65(4):505–533

    Article  MATH  MathSciNet  Google Scholar 

  2. Arecchi FT, Boccaletti S, Ciofini M, Meucci R (1998) The control of chaos: theoretical schemes and experimental realizations. Int J Bifurc Chaos 8(8):1643–1655

    Article  MATH  Google Scholar 

  3. Auerbach D, Cvitanovic P, Eckmann J-P, Gunaratne G, Procaccia I (1987) Exploring chaotic motion through periodic orbits. Phys Rev Lett 58(23):2387–2389

    Article  ADS  MathSciNet  Google Scholar 

  4. Barreto E, Grebogi C (1995) Multiparameter control of chaos. Phys Rev E 54(4):3553–3557

    Article  ADS  Google Scholar 

  5. Bessa WM, de Paula AS, Savi MA (2009) Chaos control using an adaptive fuzzy sliding mode controller with application to a nonlinear pendulum. Chaos Solitons Fractals 42(2):784–791

    Article  MATH  ADS  Google Scholar 

  6. Bessa WM, de Paula AS, Savi MA (2012) Sliding mode control with adaptive fuzzy dead-zone compensation for uncertain chaotic systems. Nonlinear Dyn 70(3):1989–2001

    Article  Google Scholar 

  7. Bessa WM, de Paula AS, Savi MA (2013) Adaptive fuzzy sliding mode control of a chaotic pendulum with noisy signals. ZAMM J Appl Math Mech. doi:10.1002/zamm.201200214

    Google Scholar 

  8. de Korte RJ, Schouten JC, van den Bleek CMV (1995) Experimental control of a chaotic pendulum with unknown dynamics using delay coordinates. Phys Rev E 52(4):3358–3365

    Article  ADS  Google Scholar 

  9. De Paula AS, Savi MA (2008) A multiparameter chaos control method applied to maps. Braz J Phys 38(4):537–543

    Article  ADS  Google Scholar 

  10. De Paula AS, Savi MA (2009) A multiparameter chaos control method based on OGY approach. Chaos Solitons Fractals 40(3):1376–1390

    Article  MATH  ADS  Google Scholar 

  11. De Paula AS, Savi MA (2009) Controlling chaos in a nonlinear pendulum using an extended time-delayed feedback control method. Chaos Solitons Fractals 42(5):2981–2988

    Article  MATH  ADS  Google Scholar 

  12. De Paula AS, Savi MA (2011) Comparative analysis of chaos control methods: a mechanical system case study. Int J Non-Linear Mech 46(8):1076–1089

    Article  Google Scholar 

  13. De Paula AS, Savi MA, Pereira-Pinto FHI (2006) Chaos and transient chaos in an experimental nonlinear pendulum. J Sound Vib 294(3):585–595

    Article  ADS  Google Scholar 

  14. De Paula AS, Savi MA, Wiercigroch M, Pavlovskaia E (2012) Bifurcation control of a parametric pendulum. Int J Bifurc Chaos 22(5):1–14, Article 1250111

  15. Dressler U, Nitsche G (1992) Controlling chaos using time delay coordinates. Phys Rev Lett 68(1):1–4

    Article  ADS  MathSciNet  Google Scholar 

  16. Ferreira BB, de Paula AS, Savi MA (2011) Chaos control applied to heart rhythm dynamics. Chaos Solitons Fractals 44(8):587–599

    Article  ADS  Google Scholar 

  17. Fradkov AL, Evans RJ, Andrievsky BR (2006) Control of chaos: methods and applications in mechanics. Phylos Trans R Soc 364:2279–2307

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. Fraser AM, Swinney HL (1986) Independent coordinates for strange attractors from mutual information. Phys Rev A 33:1134–1140

    Article  MATH  ADS  MathSciNet  Google Scholar 

  19. Grebogi C, Lai Y-C (1997) Controlling chaotic dynamical systems. Syst Control Lett 31:307–312

    Article  MATH  MathSciNet  Google Scholar 

  20. Hübinger B, Doerner R, Martienssen W, Herdering M, Pitka R, Dressler U (1994) Controlling chaos experimentally in systems exhibiting large effective Lyapunov exponents. Phys Rev E 50(2):932–948

    Article  ADS  Google Scholar 

  21. Kapitaniak T (1992) Controlling chaotic oscillators without feedback. Chaos Solitons Fractals 2(5):512–530

    Article  ADS  MathSciNet  Google Scholar 

  22. Kapitaniak T (2005) Controlling chaos: theoretical and practical methods in non-linear dynamics. Academic Press Inc, San Diego

    Google Scholar 

  23. Ogorzalek M (1994) Chaos control: how to avoid chaos or take advantage of it. J Franklin Inst 331B(6):681–704

    Article  MATH  MathSciNet  Google Scholar 

  24. Otani M, Jones AJ (1997) Guiding chaotic orbits Research Report—Imperial College of Science Technology and Medicine, London

  25. Ott E, Grebogi C, Yorke JA (1990) Controlling chaos. Phys Rev Lett 64(11):1196–1199

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. Pereira-Pinto FHI, Ferreira AM, Savi MA (2004) Chaos control in a nonlinear pendulum using a semi-continuous method. Chaos Solitons Fractals 22(3):653–668

    Article  MATH  ADS  Google Scholar 

  27. Pereira-Pinto FHI, Ferreira AM, Savi MA (2005) State space reconstruction using extended state observers to control chaos in a nonlinear pendulum. Int J Bifurc Chaos 15(12):4051–4063

    Article  MATH  Google Scholar 

  28. Pyragas K (1992) Continuous control of chaos by self-controlling feedback. Phys Lett A 170:421–428

    Article  ADS  Google Scholar 

  29. Pyragas K (2006) Delayed feedback control of chaos. Phylos Trans R Soc 364:2309–2334

    Article  MATH  ADS  MathSciNet  Google Scholar 

  30. Rhodes C, Morari M (1997) False-nearest-neighbors algorithm and noise-corrupted time series. Phys Rev E 55(5):6162–6170

    Article  ADS  Google Scholar 

  31. Ritz T, Schweinsberg ASZ, Dressler U, Doerner R, Hübinger B, Martienssen W (1997) Chaos control with adjustable control times. Chaos Solitons Fractals 8(9):1559–1576

    Article  ADS  Google Scholar 

  32. Savi MA (2006) Nonlinear dynamics and chaos, Editora E-papers (in portuguese)

  33. Shinbrot T, Grebogi C, Ott E, Yorke JA (1993) Using small perturbations to control chaos. Nature 363:411–417

    Article  ADS  Google Scholar 

  34. So P, Ott E (1995) Controlling chaos using time delay coordinates via stabilization of periodic orbits. Phys Rev E 51(4):2955–2962

    Article  ADS  MathSciNet  Google Scholar 

  35. Socolar JES, Sukow DW, Gauthier DJ (1994) Stabilizing unstable periodic orbits in fast dynamical systems. Phys Rev E 50(4):3245–3248

    Article  ADS  Google Scholar 

  36. Yanchuk S, Kapitaniak T (2001) Symmetry-increasing bifurctaion as a predictor of a chaos-hyperchaos transition in coupled systems. Phys Rev E 64(056235):1–5

    Google Scholar 

  37. Yanchuk S, Kapitaniak T (2001) Chaos-hyperchaos transition in coupled Rossler systems. Phys Lett A 290:139–144

    Article  MATH  ADS  Google Scholar 

Download references

Acknowledgments

The authors would like to acknowledge the support of the Brazilian Research Agencies CNPq, CAPES and FAPERJ and through the INCT-EIE (National Institute of Science and Technology—Smart Structures in Engineering) the CNPq and FAPEMIG. The Air Force Office of Scientific Research (AFOSR) is also acknowledged.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Marcelo Amorim Savi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

de Paula, A.S., Savi, M.A. State space reconstruction applied to a multiparameter chaos control method. Meccanica 50, 207–216 (2015). https://doi.org/10.1007/s11012-014-0066-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-014-0066-z

Keywords

Navigation