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V.M. Doroshenko, Yu.P. Emelianova, A.P. Kuznetsov, J.V. Sedova
A METHOD OF LYAPUNOV CHARTS:
ILLUSTRATIONS TO THE THEORY OF COUPLED SELF-OSCILLATING SYSTEMS
We demonstrate scopes of the Lyapunov charts method for identification of various regimes in nonlinear systems. We show that Lyapunov charts reveal the sophisticated and complex structure of the parameter space. Illustrations are given for the ensembles of coupled self-oscillating elements of Van der Pol type oscillator and also for the coupled phase oscillators. The considered approach is common for a variety ofphysical systems
Lyapunov exponents, Lyapunov chart, quasiperiodic oscillations, invariant tori,
Amold resonance web
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1.
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= ~^--m sin + sin 0^ sin(0 + j).
()
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- , 1 = 0 ;
- , 2 = 0 ;
- , 1 = 2.
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2 , 2- [2].
3 . . - ї.
4 , , [17, 18]. [16] , . , [14, 19], , .
(8) 2 - 2 . , 2 = 0.36 , . 8. , . 7.
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0 ! 71
. 7. (8);
3 = 0.2 , = 0.4
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5
vn w+v +mZ sin(v, v ) (9)
i=1
yn - n- , Wn - .
(9) . 0 = y2 1, = V2 .. -1 = - Wj. .
0 1 + 0 + m[2 sin 0 + sin +sin(j+a) + sin(j + a + b) sin(0 + )
sin(0 + + a) sin(0 + + a + b)],
2 1 +j + m[2sin + sin 0 + sin a sin(0 + j) sin(j + a) +
+ sin(a + b) sin(+a + b)], (1Q)
a 3 2 +a + m[2sin a + sin + sin b + sin(0 + ) sin(+a)
sin(a + b) sin(0 + +a)],
b 4 3 +b + m[2 sin b + sin a + sin(+a) sin( + a + b) sin(a + b) +
+ sin(0 + + a) sin(0 + + a + b)].
(10), (4,m) , . 9. . , , . . . , 1 =w2 w1 , , . 1 = - W1 :
1 = 0, 1 = 24. . 9.
(m ^ 0.05) . . - . , . 10. , . , , [14, 19]. . , , , , . , . 10 , .
. 9 m ^ 0.15 . , , . 10.
, . 10. , , . , . , .
. 9. ;
2 = 0.1, 3 = 0.45, 4 = 1
. 10. . 9
. . . , . , . , . . , -, , ї. -, , , .. .
. , . [20], [21], [22]. [23-25]. . . , , [26]. , , [27]. , , , ї. , .
-1726.2014.2. .. ( 14-02-31064).

1. Benettin G. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them. Part I: Theory. Part II: Numerical application / G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn // Meccanica. Vol. 15. 1980. P. 9-30.
2. .. / .. . .: , 2006. 356 .
3. .. / .. . .: , 1980. 359 .
4. . . / . , . , . . .: , 2003. 496 .
5. Broer H. Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems / H. Broer, C. Simo, R. Vitolo // Regular and Chaotic Dynamics. Vol. 16. 2011. Issue 12. P. 154-184.
6. Kuznetsov A.P. Properties of synchronization in the systems of non-identical coupled van der Pol and van der Pol-Duffing oscillators. Broadband synchronization / A.P. Kuznetsov, Ju.P. Roman // Physica
D. Vol. 238. 2009. Issue 16. P. 1499-1506.
7. Emelianova Yu.P. Synchronization and multi-frequency oscillations in the low-dimensional chain of the self-oscillators / Yu.P. Emelianova, A.P. Kuznetsov, I.R. Sataev, L.V. Turukina // Physica D. Vol. 244. 2013. Issue 1. P. 36-49.
8. .. / .. , .. , .. . 2- . .: , 2005. 292 .
9. Phase model analysis of two lasers injected field / A.I. Khibnik, Y. Braimanc, T.A.B. Kennedyd, K. Wiesenfeld // Physica D. Vol. 111. 1998. Issue 1-4. P. 295-310.
10. .. / .. // . 2003. . 33. 4. . 283-306.
11. .. - / .. , .. // . . 2004. . 12. 5. . 16-31.
12. Kuznetsov A.P. Dynamical systems of different classes as models of the kicked nonlinear oscillator / A.P. Kuznetsov, L.V. Turukina, E. Mosekilde // International Journal of Bifurcation and Chaos. Vol. 11. 2001. Issue 4. P. 1065-1078.
13. .. / .. . .-: , 2004. 288 .
14. Baesens . Simple resonance regions of torus diffeomorphisms / C. Baesens, J. Guckenheimer, S. Kim // Patterns and dynamics in reactive media. Vol. 37. 1991. P. 1-9.
15. .. , / .. . : , 2005. 424 .
16. Broer H. The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: the Arnol'd resonance web / H. Broer, C. Simo, R. Vitolo // Bulletin of the Belgian Mathematical Society. Vol. 15. 2008. Issue 5. P. 769-787.
17. Froeschle . Analysis of the chaotic behavior of orbits diffusing along the Arnold web / C. Froeschle,
E. Lega, M. Guzzo // Celestial Mechanics and Dynamical Astronomy. Vol. 95. 2006. Issue 1-4. P. 141-153.
18. Guzzo M. First numerical evidence of global Arnold diffusion in quasi--integrable systems / M. Guzzo, E. Lega, C. Froeschle // http://arxiv.org/abs/nlin/0407059.
19. / .. , .. , .. , .. // . . 2012. . 20. 2. . 112-137.
20. Hoff A. Bifurcation structures and transient chaos in a four-dimensional Chua model / A. Hoff, D.T. da Silva, C. Manchein, H.A. Albuquerque // Physics Letters A. Vol. 378. 2014. P. 171-177.
21. -. . 1, 2 / .. -, .. , .. , .. , .. // . 2013. 2 (70). . 1. C. 12-28.
22. / .. , .. , .. , .. // . . 44. 2014. 1. . 17-22.
23. .. - - / .. , .., .. // . . 8. 2012. 3. . 473-482.
24. .. / .. , .. // . .9. 2013. 3. . 409-419.
25. .. - / .. , .. , .. // . 2012. 4 (68). . 72-76.
26. / .. , .. , .. , .. // . 2013. 1 (69). C. 33-39.
27. Linsay P.S. Three-frequency quasiperiodicity, phase locking and the onset of chaos / P.S. Linsay, A.W. Cumming // Physica D. Vol. 40. 1989. P. 196-217.
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Valentina M. Doroshenko -
Undergraduate
Department of Nonlinear Processes,
Yuri Gagarin State Technical University of Saratov
Yulia P. Emelianova -
Ph. D., Associate Professor Department of Instrument Engineering,
Yuri Gagarin State Technical University of Saratov
Alexander P. Kuznetsov -Dr. Sc., Professor,
Leading Researcher: Kotel'nikov Institute of Radio Engineering and Electronics of the Russian Academy of Sciences (Saratov Branch)
-
- , . .
Yulia V. Sedova -Ph.D., Senior Researcher:
Kotel'nikov Institute of Radio Engineering and Electronics of the Russian Academy of Sciences (Saratov Branch)
15.01.14, 11.03.14