Images of synchronized chaos: Experiments with circuits

Chaos

Volumn 6, Issue 3, 1996, Pages 262-279

  Rulkov, Nikolai F ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0003255519     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.166174     Document Type: Article

References (46)

1. B. van der Pol, "Forced oscillations in a circuit with non-linear resistance," Philos. Mag. 3, 65-80 (1927).
2. A. A. Andronov and A. A. Witt, "Zur theorie des mitnehmens von van der Pol," Atch. Elektrotech. 16(4), 280-288 (1930) [also see A. A. Andronov, Sobranie Trudov, Moscow, 1956, pp. 51-64 (in Russian)].
3. A. A. Andronov and A. A. Witt, "Zur theorie des mitnehmens von van der Pol," Atch. Elektrotech. 16(4), 280-288 (1930) [also see A. A. Andronov, Sobranie Trudov, Moscow, 1956, pp. 51-64 (in Russian)].
4. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, Berlin, 1974).
5. M. I. Rabinovich and D. I. Trubetskov, Introduction to the Theory of Oscillations and Waves (Kluwer, Amsterdam, 1989).
6. J. A. Glazier and A. Lischaber, "Quasi-periodicity and dynamical systems: An experimentalist's view," IEEE Trans. Circuits Syst. CS-35, 790 (1988).
7. A. V. Gaponov-Grekhov and M. I. Rabinovich, Nonlinearities in Action: Oscillations, Chaos, Order, Fractals (Springer-Verlag, Berlin, 1992), p. 190.
8. V. S. Afraimovich and L.P. Shil'nikov, "Invariant two-dimensional tori, their destruction and stochasticity," in Methods of the Qualitative Theory of Differential Equations (Gorky University Press, Gorky, 1983), pp. 3-26 (in Russian).
9. S. Ostlund, D. Rand, J. Sethna, and E. Siggia, "Universal properties of the transition from quasi-periodicity to chaos in dissipative systems," Physica D 8, 303-342 (1983).
10. H. Fujisaka and T. Yamada, "Stability theory of synchronized motions in coupled oscillator systems," Prog. Theor. Phys. 69, 32-46 (1984).
11. V. S. Afraimovich, N. N. Verichev, and M. I. Rabinovich, "Stochastic synchronization of oscillations in dissipative systems," Izv. VUZ. Radiofiz. RPQAEC 29, 795-803 (1986).
12. A. S. Pikovsky, "On the interaction of strange attractors," Z. Phys. B Condensed Matter 55, 149-154 (1984).
13. H. G. Schuster, S. Martin, and W. Martiessen, "New method for determining the largest Liapunov exponent of simple nonlinear systems," Phys. Rev. A 33, 3547-5349 (1986).
14. L. M. Pecora and T. L. Carroll, "Synchronization in chaotic systems," Phys. Rev. Lett. 64, 821-824 (1991).
15. T. L. Carroll and L. M. Pecora, "Synchronizing Chaotic Systems," IEEE Trans. Circuits Syst. CS-38, 453-456 (1991).
16. N. N. Verichev, "Mutual synchronization of chaotic oscillations in the Lorenz systems," in Methods of the Qualitative Theory of Differential Equations (Gorky University Press, Gorky, 1986), pp. 47-57 (in Russian).
17. N. F. Rulkov, A. R. Volkovskii, A. Rodriguez-Lozano, E. Del Rio, and M. G. Velarde, "Mutual synchronization of chaotic self-oscillators with dissipative coupling," Int. J. Bifurc. Chaos Appl. Sci. Eng. 2, 669-676 (1992).
18. C. W. Wu and L. O. Chua, "A unified framework for synchronization and control of dynamical systems," Int. J. Bifurc. Chaos Appl. Sci. Eng. 4, 979-998 (1994).
19. V. Afraimovich and H. M. Rodridues, "Uniform ultimate boundedness and synchronization for nonautonomous equations," preprint CDSN94-202, Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, 1994, p. 39.
20. P. Badola, V. R. Kumar, and B. D. Bulkarni, "Effects of coupling nonlinear systems with complex dynamics," Phys. Lett. A 155, 365-372 (1991).
21. J. F. Heagy, T. L. Carroll, and L. M. Pecora, "Synchronous chaos in coupled oscillator systems," Phys. Rev. E 50, 1874-1885 (1994).
22. A. S. Pikovsky and P. Grassberger, "Symmetry breaking bifurcation for coupled chaotic attractors," J. Phys. A Math. Gen. 24, 4587-4597 (1991).
23. P. Ashwin, J. Buescu, and I. Stewart, "Bubbling of attractors and synchronization of chaotic oscillators," Phys. Lett. A 193, 126-139 (1994).
24. E. Ott and J. C. Sommerer, "Blowout bifurcations: The occurrence of riddled basins and on-off intermittency," Phys. Lett. A 188, 39-47 (1994).
25. R. Brown, N. F. Rulkov, and N. B. Tufillaro, "Synchronization of chaotic systems: The effects of additive noise and drift in the dynamics of the driving," Phys. Rev. E 50, 4488-4508 (1994).
26. J. F. Heagy, T. L. Carroll, and L. M. Pecora, "Desynchronization by periodic orbits," Phys. Rev. E 52, R1253-R1256 (1995).
27. K. Pyragas, "Continuous control of chaos by self-controlling feedback," Phys. Lett. A 170, 421-428 (1992).
28. N. F. Rulkov, L. S. Tsimring, and H. D. I. Abarbanel, "Tracking unstable orbits in chaos using dissipative feedback control," Phys. Rev. E 50, 314-324 (1995).
29. A. Kittel, K. Pyragas, and R. Richter, "Prerecorded history of a system as an experimental tool to control chaos," Phys. Rev. E 50, 262-268 (1994).
30. N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, and H. D. I. Abarbanel, "Generalized synchronization of chaos in directionally coupled chaotic systems," Phys. Rev. E 51, 980 (1995).
31. H. D. I. Abarbanel, N. F. Rulkov and M. M. Sushchik, "Generalized synchronization of chaos: The auxiliary system approach," Phys. Rev. E 53, 4528-4535 (1996).
32. G. I. Dykman, P. S. Landa, and Y. I. Neymark, "Synchronizing the chaotic oscillations by external force," Chaos Solitons Fractals 1, 339-353 (1991).
33. V. S. Anishchenko, Dynamical Chaos in Physical Systems: Experimental Investigation of Self-Oscillating Circuits (Teubner-Verlag, Leipzig, 1989).
34. V. V. Alekseev and A. Yu. Loskutov, "Control of a system with a strange attractor through periodic parametric action," Sov. Phys. Dokl. 32, 270-271 (1987).
35. R. Lima and M. Pettini, "Suppression of chaos by resonant parametric perturbations," Phys. Rev. A 41, 726-733 (1990).
36. Y. Braiman and I. Goldhirsch, "Taming chaotic dynamics with weak periodic perturbations," Phys. Rev. Lett. 66, 2545-2548 (1991).
37. Y. S. Tang, A. I. Mees, and L. O. Chua, "Synchronization and chaos (triggered oscillator model)," IEEE Trans. Circuits Sys. CS-30, 620-626 (1983).
38. T. Allen, "Complicated, but rational, phase-locking responses of an ideal time-base oscillator," IEEE Trans. Circuits Sys. CS-30, 627-632 (1983).
39. T. L. Carroll, J. Heagy, and L. M. Pecora, "Synchronization and desyn-chronization in pulse coupled relaxation oscillators," Phys. Lett. A 186, 225-229 (1994).
40. L. Glass and M. C. Mackey, From Clocks to Chaos. The Rhythms of Life (Princeton University Press, Princeton, NJ, 1988).
41. R. E. Mirollo and S. H. Strogatz, "Synchronization of pulse-coupled biological oscillators," SIAM J. Appl. Math. 50, 1645-1662 (1990).
42. S. V. Kiyashko, A. S. Pikovsky, and M. I. Rabinovich, "Self-exciting oscillator for the radio-frequency range with stochastic behavior," Radio Eng. Electron. Phys. 25, 74-79 (1980).
43. P. A. Bernhardt, "The autonomous chaotic relaxation oscillator: An electrical analogue to the dripping faucet," Physica D 52, 489-527 (1991).
44. J. M. Perez, Z. Yu, J. M. Kovalskii, G. Albert, C. L. Littler, and X. N. Song, "Synchronization of chaos in coupled tunnel diode relaxation oscillators," in Proceedings of the 1st Experimental Chaos Conference (World Scientific, Singapore, 1991), p. 327.
45. N. F. Rulkov and A. R. Volkovskii, "Threshold synchronization of chaotic relaxation oscillations," Phys. Lett. A 179, 332-336 (1993).
46. A. S. Dmitriev, V. Ya. Kislov, and A. O. Starkov, "Experimental study of the formation and interaction of strange attractors in a self-excited ring oscillator," Sov. Phys. Tech. Phys. 30, 1439-1440 (1985).