For citation:
Balanov A. G., Vadivasova T. E., Postnov D. E., Sosnovtseva O. V. Bifurcation of chaotic synchronization in Rossler system with harmonic exitation. Izvestiya VUZ. Applied Nonlinear Dynamics, 1997, vol. 5, iss. 5, pp. 31-43. DOI: 10.18500/0869-6632-1997-5-5-31-43
Bifurcation of chaotic synchronization in Rossler system with harmonic exitation
Synchronization of chaotic oscillations in nonautonomous Rossler system from the point of view its universality for regular and chaotic regimes is investigated. Comparative analysis of different methods of the boundary of chaotic synchronization diagnostics, namely, the locking of basic frequencies. the stabilisation of difference о instant phases of oscillations, and the levelling of mean time of returning to а secant surface, shows the equivalence of the mentioned approaches for spiral chaos. A set of results from numerical simulation allows us to suggest and substantiate hypothesis that there is a bifurcation of chaotic regime as an analogue of a tangent bifurcation for a limit cycle. Analysing the spectrum of Lyapunov exponents two types of chaos outside synchronization region which differ by number of zero Lyapunov exponents are found.
- Huygens С. Horologium Oscilatorium. Paris: Apud F. Muguet; 1673. 161 p.
- Van der Pol B. Theory of the amplitude of free and forced triod vibrations. Radio Rev. 1920;1:701-710.
- Gaponov VI. Two connected generators with soft excitation. Tech. Phys. 1936;6(5):801.
- Teodorchik КF. To the theory of synchronization of relaxation self-oscillations. Soviet Phys. Doklady. 1943;40(2):63-66.
- Khokhlov RV. To the theory of capture at low amplitude of external force. Soviet Phys. Doklady. 1954;97(3):411-414.
- Andronov АА, Vitt АА. On the mathematical theory of capture. In: Collection of Works АА Andronov. М.:Academy of Sciences USSR Publishing; 1956. P. 70-84.
- Minakova II, Teodorchik КF. On the theory of synchronization of auto-oscillations of any form. Soviet Phys. Doklady. 1956;106(4):658-660.
- Hayashi C. Nonlinear Oscillations in Physical Systems. N.Y.: McGraw-Hill; 1964. 392 p.
- Blekhman II. Synchronization of Dynamic Systems. М.: Nauka; 1971. 894 p.
- Blekhman II. Synchronization in Nature and Technology. М.: Nauka; 1981. 351 p.
- Landa PS, Lariontsev ЕG. Beating and synchronization modes of counter waves in a rotating ring gas laser. Radio Engineering and Electronic Physics. 1970;15(6):1214.
- Romanovskii YuМ. O mutual synchronization of many auto-oscillation systems connected through a common environment. Radiophys. Quantum Electron. 1972;15(5):718-724.
- Demyanchenko AG. Synchronization of Harmonic Oscillation Generators. М.: Energiya; 1976. 240 p.
- Sskupoy VF, Kopylov VP. About synchronization of the FM autogenerator. Radio Engineering and Electronic Physics. 1979;24(7):1374-1379.
- Landa PS. Autooscillations in Systems with a Finite Number of Degrees of Freedom. М.: Nauka; 1980. 359 p.
- Kuznetsov YuА, Landa PS, Olchovoy АF. Amplitude threshold of synchronization as a measure of chaos in stochastic auto-oscilling systems. Soviet Phys. Doklady. 1985;281(2):1164-1169.
- Landa PS, Rendel YuS, Sher VА. Synchronization of oscillations in the Lorentz system. Radiophys. Quantum Electron. 1989;32(9):1172-1174.
- Dykman G, Landa Р, Neimark Y. Synchronising of chaotic oscillations by external forse. Chaos, Solitons & Fractals. 1992;1(4):339–353. DOI: 10.1016/0960-0779(91)90025-5.
- Afraimovich VS, Verichev NN, Rabinovich МI. Stochastic synchronization of oscillation in dissipative systems. Radiophys. Quantum Electron. 1986;29(9):795-803. DOI: 10.1007/BF01034476.
- Volkovskii AR, Rulkov NF. Experimental study of bifurcations on the threshold of stochastic synchronization. Tech. Phys. Lett. 1989;15(7):5-10.
- Ресоrа L, Carroll Т. Synchronization in chaotic systems. Phys. Rev. Lett. 1990;64(8):821-824. DOI: 10.1103/PhysRevLett.64.821.
- Chua L, Itoh M, Kocarev L, Eckert K. Chaos synchronization in Chua’s circuit. In: Madan RN, editor. Chua’s Circuits: A Paradigma for Chaos. Singapore: World Scientific; 1993. P. 309-324. DOI: 10.1142/9789812798855_0013.
- Abarbanel HDI, Rabinovich MI, Selverston А, Bazhenov MV, Huerta R, Sushchik MM, Rubchinskii LL. Synchronisation in neural networks. Phys. Usp. 1996;39(4):337-362. DOI: 10.1070/pu1996v039n04abeh000141.
- Anishchenko VS, Postnov DE. Effect of the basic frequency locking of chaotic auto-oscillations. Synchronization of strange attractors. Tech. Phys. Lett. 1988;14(6):569-573.
- Anishchenko VS, Vadivasova TE, Postnov DE, Safonova MA. Synchronization оf chaos. Int. J. Bifurc. Chaos. 1992;2(3): 633-644. DOI: 10.1142/S0218127492000756.
- Anishchenko VS, Vadivasova TE, Postnov DE. Synchronization оf chaos. In: Proc. First International Conference on Applyed Synergetic and Synergetic Engeneering. 21-23 June,1994, Erlangen, Germany. P.200.
- Rosenblum M, Picovsky A, Kurths J. Phase synchronization of chaotic oscillators. Phys. Rev. Lett. 1996;76(11):1804-1807. DOI: 10.1103/PhysRevLett.76.1804.
- Rosenblum MG, Pikovsky AS, Kurths J. Phase synchronization in driven and coupled chaotic oscillators. IЕЕЕ Trans. Circuits Syst. I. 1997;44(10):874-881. DOI: 10.1109/81.633876.
- Shalfeev VD, Osipov GV. Chaotic phase synchronization of coupled PLLS. In: Рrос. of 5—th International Specialist Workshop оп Nonlincar Dynamics of Electronic Systems. 26—27 June, 1997, Moscow, Russia. P. 139-144.
- Pikovsky А, Osipov G, Rosenblum M, Zaks M, Kurths Y. Attractor—repeller collision and eyelet intermittency at the transition to phase synchronization. Phys.Rev.Lett. 1997;79(1):47-50. DOI: 10.1103/PhysRevLett.79.47.
- Anishchenko VS, Aranson NS, Postnov DE, Rabinovich MI. Spatial synchronization and bifurcation of chaos development in the chain of connected generators. Soviet Phys. Doklady. 1986;286(5):1120-1124.
- Osipov GV, Pikovsky AS, Rosenblum MG, Kurths J. Phase synchronization effects in а lattice of nonidentical Rossler oscillators. Phys. Rev. Е. 1997;55(3):2353-2361. DOI: 10.1103/PhysRevE.55.2353.
- Astakhov VV, Bezruchko BP, Gulyaev YuV, Seleznev EP. Multi-stable modes in Feigenbaum systems with dissipative communication. Tech. Phys. Lett. 1989;15(3):60-65.
- Anishchenko VS, Vadivasova TE, Astakhov VV, Sosnovtseva OV, Wu CW, Chua LО. Dynamics оf two coupled Chua’s circuits. Int. J. Bifurc. Chaos. 1995;5(6):1677-1699. DOI: 10.1142/S0218127495001241.
- Anishchenko VS, Silchenko АN, Khovanov IА. Mutual synchronization of switching processes in coupled Lorenz systems. Tech. Phys. Lett. 1997;23:300-302. DOI: 10.1134/1.1261855.
- Neiman AB. Synchronizationlike phenomena in coupled stochastic bistable systems. Phys. Rev. Е. 1994;49(4):3484-3487. DOI: 10.1103/physreve.49.3484.
- Shulgin BV, Neiman AB, Anishchenko VS. Mean switching frequency locking in stochastic bistable systems driven by periodic force. Phys. Rev. Lett. 1995;75:4157-4160. DOI: 10.1103/PhysRevLett.75.4157.
- Khovanov IA, Anishchenko VS. The mechanism of stochastic resonance in a system with intermittence of the “chaos-chaos” type. Tech. Phys. Lett. 1996;22(20):75-81.
- Anishchenko VS. Complex Oscillations in Simple Systems. М.: Nauka; 1990. 312 p. (in Russian).
- Heagy JF, Hammel SM. The birth оf strange nonchaotic attractors. Physica D. 1994;70(1-2):140-153. DOI: 10.1016/0167-2789(94)90061-2.
- Bendat JS, Piersol А. Random Data: Analysis and Measurement Procedures. N.Y.: Wiley; 1971. 407 p.
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