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Kiveleva K. G., Frajman L. A. The bifurcation analysis of nonautonomous pendulum equation from the theory of phase-locked loop. Izvestiya VUZ. Applied Nonlinear Dynamics, 1994, vol. 2, iss. 2, pp. 27-35.

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Language: 
Russian
Heading: 
Bifurcations in Dynamical Systems
Article type: 
Article
UDC: 
531.01

The bifurcation analysis of nonautonomous pendulum equation from the theory of phase-locked loop

Autors: 
Kiveleva Klara Georgievna, Institute of Applied Mathematics and Cybernetics. Nizhny Novgorod state University
Frajman Ljudmila Alekseevna, Institute of Applied Mathematics and Cybernetics. Nizhny Novgorod state University
Abstract: 

The bifurcation analysis of periodic solutions and homoclynic structures in nonautonomous system of differential equations which models phase—locked loop (PLL) is carried out with the qualitative numerical method using computer modelling. The chao tization of dynamics of the system is investigated. The bifurcation diagrams on the parameter plane with regions corresponding to different qualitative dynamics are presented. The achieved results are interpreted applying to PLL.

Key words: 
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Acknowledgments: 
The authors are grateful to V.N. Belykh for his attention and assistance in the work. The work was carried out with financial support from the Russian Foundation for Basic Research (project 93-013-16253).
Reference: 
  1. Belyustina LN, Belykh VN. Qualitative study of the dynamic system on the cylinder. Differential Equations. 1973;9(3):403-415.
  2. Stoker JJ. Nonlinear Vibrations in Mechanical and Electrical Systems. N.Y.: Wiley; 1992. 296 p.
  3. Belykh VN, Pedersen NF, Soerensen ON. Shunted—Josephson—junction model. II.The nonautonomous case. Phys. Rev. B. 1977;16(11):4860-4871. DOI: 10.1103/PhysRevB.16.4860.
  4. Belyustina LN, Belykh VN. On the global structure of the phase space splitting of one non-autonomous system. Differential Equations. 1973;9(4):595-608.
  5. Belyustina LN, Belykh VN. Homoclinic structures generated by the simplest model of phase self-adjustment. In: Shakhgildyan VV, Belyustina LN, editors. Phase Synchronization. М.: Radio i Svyaz; 1975. P. 97.
  6. Belyustina LN, Belykh VN. On a nonautonomous phase system of equations with a small parameter, which contains invariant tori and crude homoclinic curves. Radiophys. Quantum Electron. 1972;15(7):793-800. DOI: 10.1007/BF01031989.
  7. Kiveleva KG, Fraiman LА. Finding fixed points of the point mapping of the plane into the plane. In: Algorithms and Programs. М.: All-Russian Scientific and Technical Information Center; 1979. № 3 (29).
  8. Kiveleva KG, Fraiman LА. Finding the characteristic numbers of fixed points and critical directions of separatistic invariant curves of the point mapping of the plane into the plane generated by solutions of the non-autonomous periodic table of the second order. In: Algorithms and Programs. М.: All-Russian Scientific and Technical Information Center; 1979. № 3 (29).
  9. Belyustina LN, Ezhevskaya NА. A program for calculating the coordinates of fixed points of point mapping of a plane to a plane based on analogue of the secant method. In: Algorithms and Programs. М.: All-Russian Scientific and Technical Information Center; 1979. № 3 (29).
  10. Fraiman LА. Algorithms for qualitative and numerical study of some mathematical models of phase synchronisation systems. In: Theoretical Electrical Engineering: Republican Scientific and Technical Collection. Lviv: Lviv University Publishing; 1986. Vol. 41. P. 30.
  11. Fraiman LА. Study of bifurcations of the non-autonomous equation of phase synchronisation by the qualitative-numerical method. In: Neymark YuI, editor. Dynamics of Systems. Numerical Methods of Researching Dynamical Systems: Interuniversity Thematic Collection of Scientific Works. Gorky: Gorky University Publishing; 1982. P. 127.
  12. Arnold VI. Ordinary Differential Equation. Berlin: Springer; 1992. 338 p.
Received: 
29.04.1994
Accepted: 
12.07.1994
Published: 
08.08.1994
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