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

For citation:

Blekhman I. I., Landa P. S. Conjugate resonances in nonlinear systems under biharmonical action. Vibro-induced bifurcations. Izvestiya VUZ. Applied Nonlinear Dynamics, 2002, vol. 10, iss. 1, pp. 44-51. DOI: 10.18500/0869-6632-2002-10-1-44-51

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Language: 
Russian
Heading: 
Bifurcations in Dynamical Systems
Article type: 
Article
UDC: 
531.3+534
DOI: 
10.18500/0869-6632-2002-10-1-44-51

Conjugate resonances in nonlinear systems under biharmonical action. Vibro-induced bifurcations

Autors: 
Blekhman Ilya Israelivich, Institute of Problems of Mechanical Engineering of Russian Academy of Science
Landa Polina Solomonovna, Lomonosov Moscow State University
Abstract: 

Using a bistable oscillator described by a Duffing equation as an example, we consider resonances caused by a biharmonical external force with essentially different frequencies. We show that these resonances are conjugate; they appear when either the low or high frequency changes. The resonances take place also as the amplitude of the high-frequency action varies. Besides we show that the high-frequency action induces the bifurcation of the change in the number of stable steady states in the system; so the seeming resonance in an overdamped oscillator is caused just this bifurcation.

Key words: 
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Acknowledgments: 
This work was partially supported by the Russian Foundation for Basic Research (Grant 01-01-00227) and within the framework of the Integrated Programme 19 of the Presidium of the Russian Academy of Sciences (Project 1.4).
Reference: 
  1. Landa PS, McClintock PVE. Vibrational resonance. J. Phys. A: Math. Gen. 2000;33(45):L433-L438. DOI: 10.1088/0305-4470/33/45/103.
  2. Anishchenko VS, Neiman AB, Moss F, Shimansky-Geier L. Stochastic resonance: noise-enhanced order. Phys. Usp. 1999;42(1):7-36. DOI: 10.1070/PU1999v042n01ABEH000444.
  3. Landa PS. Regular and Chaotic Oscillations. Berlin, Heidelberg: Springer-Verlag; 2001. 397 p. DOI: 10.1007/978-3-540-45252-2.
  4. Blekhman II. Vibrational Mechanics: Nonlinear Dynamic Effects, General Approach, Applications. World Scientific; 2000. 536 p. DOI: 10.1142/4116.
  5. Malkin IG. Some Problems of the Theory of Nonlinear Oscillations. Moscow: Gostekhizdat; 1956. 492 p. (in Russian).
  6. Kolovsky M.Z. On the influence of high-frequency disturbances on resonant oscillations in a nonlinear system. In: Dynamics and Strength of Machines. Proceedings of the Leningrad Polytechnic Institute. No. 226. Moscow, Leningrad: Mashgiz; 1963. P. 7 (in Russian).
  7. Nayfeh AH, Mook DT. Nonlinear Oscillations. New York: Wiley-Interscience; 1979. 720 p. DOI: 10.1002/9783527617586.
Received: 
10.03.2002
Accepted: 
15.04.2002
Available online: 
13.12.2023
Published: 
31.07.2002
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