ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Talagaev Y. V., Tarakanov A. F. Multiparametrical analysis based on Melnikov criterion and optimal chaos suppression in periodically driven dynamic systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 2011, vol. 19, iss. 4, pp. 77-90. DOI: 10.18500/0869-6632-2011-19-4-77-90

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
PDF icon 2011no4p077.pdf
(downloads: 405)
Language: 
Russian
Heading: 
Deterministic Chaos
Article type: 
Article
UDC: 
517.923:517.977.5
DOI: 
10.18500/0869-6632-2011-19-4-77-90

Multiparametrical analysis based on Melnikov criterion and optimal chaos suppression in periodically driven dynamic systems

Autors: 
Talagaev Yury Viktorovich, Balashov branch of the Saratov State University. N.G. Chernyshevsky
Tarakanov Andrej Fedorovich, Borisoglebsk State Pedagogical Institute (BSPI)
Abstract: 

The results that illustrate the fruitfulness of the idea of optimal parametric correction for the analysis and optimization of the class of periodically driven chaotic systems are presented. Two problems that reveal the peculiarities of suppression of chaotic dynamics and present the method of regulation of the behavior of dissipative nonlinear oscillator were solved with the help of Melnikov criterion. The analytical results were compared to the solution of double-criteria problem that uses the conditions of Pontryagin maximum principle to find optimal parametric perturbations. The efficiency evaluations of various forms of parametric perturbations on the system found with the help of two independent methods correspond to each other. 

Key words: 
Multiparametrical analysis
nonlinear oscillators
optimal chaos suppression
Melnikov criterion
maximum principle
Reference: 
  1. Zhang H, Liu D, Wang Z. Controlling Chaos: Suppression, Synchronization and Chaotification. Series: Communications and Control Engineering. London: Springer; 2009. 344 p. DOI: /10.1007/978-1-84882-523-9.
  2. Sanjuan MAF, Grebogi C, editors. Recent Progress in Controlling Chaos. Singapore: World Scientific; 2010. 440 p. DOI: 10.1142/7563.
  3. Rega G, Lenci S, Thompson JMT. Controlling chaos: The OGY Method, Its Use in Mechanics, and an Alternative Unified Framework for Control of Non-regular Dynamics. In: Thiel M, et al. Nonlinear Dynamics and Chaos Advances and Perspectives. Berlin: Springer–Verlag; 2010. P. 211–269. DOI: 10.1007/978-3-642-04629-2_11.
  4. Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys. Rev. Lett. 1990;64(11):1196–1199. DOI: 10.1103/PhysRevLett.64.1196.
  5. Warncke J, Bauer M, Martienssen W. Multiparameter control of high-dimensional chaotic systems. Europhys. Lett. 1994;25(5):323–356. DOI: 10.1209/0295-5075/25/5/002.
  6. Barreto E, Grebogi C. Multiparameter control of chaos. Phys. Rev. E. 1995;54(4):3553–3557. DOI: 10.1103/PhysRevE.52.3553.
  7. Pyragas K, Pyragas V. Delayed feedback control techniques. In: Sanjuan MAF, Grebogi C, editors. Recent Progress in Controlling Chaos. Singapore: World Scientific; 2010. P. 103.
  8. Lima R, Pettini M. Suppression of chaos by resonant parametric perturbation. Phys. Rev. A. 1990;41(2):726–733. DOI: 10.1103/PhysRevA.41.726.
  9. Braiman Y, Goldhirsch I. Taming chaotic dynamics with weak periodic perturbations. Phys. Rev. Lett. 1991;66(20):2545–2548. DOI: 10.1103/physrevlett.66.2545.
  10. Kivshar YS, Rodelsperger F, Benner H. Suppression of chaos by nonresonant parametric perturbation. Phys. Rev. E. 1994;49(1):319–324. DOI: 10.1103/PhysRevE.49.319.
  11. Belhaq M, Houssni M. Quasi-periodic oscillations, chaos and suppression of chaos in a nonlinear oscillator driven by parametric and external excitations. Nonlinear Dyn. 1999;18(1):1–24. DOI: 10.1023/A:1008315706651.
  12. Schwalger T, Dzhanoev A, Loskutov A. May chaos always be suppressed by parametric perturbations? Chaos. 2006;16(2):023109. DOI: 10.1063/1.2195787.
  13. Guckenheimer J, Holmes F. Nonlinear oscillations, dynamical systems and bifurcations of vector fields. NY: Springer; 1983. 462 p. DOI: 10.1007/978-1-4612-1140-2.
  14. Loskutov A. Dynamical chaos: systems of classical mechanics. Phys. Usp. 2007;50(9):939–964. DOI: 10.1070/PU2007v050n09ABEH006341.
  15. Kuznetsov SP. Dynamic Chaos (Lecture Course). Moscow: Fizmatlit; 2001. 296 p. (in Russian).
  16. Chacon R. Suppression of chaos by selective resonant parametric perturbation. Phys. Rev. E. 1995;51(1):761–764. DOI: 10.1103/PhysRevE.51.761.
  17. Qu Z, Hu G, Yang G, Qin G. Phase effect in taming nonautonomous chaos by weak harmonic perturbations. Phys. Rev. Lett. 1995;74(10):1736–1739. DOI: 10.1103/PhysRevLett.74.1736.
  18. Zambrano S, Allaria E, Brugioni S, Leyva I, Meucci R, Sanjuan MAF, Arecchi FT. Numerical and experimental exploration of phase control of chaos. Chaos. 2006;16(1):013111. DOI: 10.1063/1.2161437.
  19. Talagaev JV, Tarakanov AF. Optimal chaos suppression and transition processes in сorrected multiparametrical oscillatory systems. Izvestiya VUZ. Applied Nonlinear Dynamics. 2008;16(5):99–114 (in Russian). DOI: 10.18500/0869-6632-2008-16-5-99-114.
  20. Gorelik V, Talagaev Y, Tarakanov А. Optimal processes of chaotic uncertainty correction. Proc. 18th IEEE International Conference on Control Applications, Part of 2009 IEEE Multi-conference on Systems and Control. July 8–10, 2009, Saint Petersburg, Russia. IEEE; 2009. P. 878–883. DOI: 10.1109/CCA.2009.5281155.
  21. Afanasyev VN, Kolmanovsky VB, Nosov VR. Mathematical Theory of Design of Control Systems. Third Edition. Moscow: Vysshaya Shkola; 2003. 615 p. (in Russian).
Received: 
16.02.2011
Accepted: 
12.07.2011
Published: 
30.09.2011
Short text (in English):
PDF icon a2011no4p077.pdf
(downloads: 111)
  • 1865 reads