| Generalized synchronization (GS) is one of the most interesting types of the synchronous chaotic system behavior [Rulkov N.F., Sushchik M.M., Tsimring L.S., Abarbanel H.D.I. Phys. Rev. E. 51 (2) (1995) 980-994]. It is observed predominantly in unidirectionally coupled chaotic dynamical systems, the drive and response ones, and means the presence of the functional relation between the drive and response system states. GS is well studied at present both in systems with a small number of degrees of freedom and spatially extended media. In particular, the effective methods for the GS regime detection [Abarbanel H.D.I., Rulkov N.F., Sushchik M.M. Phys. Rev. E. 53 (5) (1996) 4528-4535; Pyragas K. Phys. Rev. E. 56 (5) (1997) 5183-5188] and mechanisms of the synchronous regime arising [Hramov A.E., Koronovskii A.A. Phys. Rev. E. 71 (6) (2005) 067201; Hramov A.E., Koronovskii A.A., Popov P.V. Phys. Rev. E. 72 (3) (2005) 037201; Hramov A.E., Koronovskii A.A., Moskalenko O.I. Europhysics Letters. 72 (6) (2005) 901-907] have been revealed. Intermittent behavior near the boundary of the GS regime onset has been discovered [Hramov A.E., Koronovskii A.A. Europhysics Letters. 70 (2) (2005) 169-175]. Effect of noise on the GS regime arising has been studied [Guan S., Lai Y.-C., Lai C.-H. Phys. Rev. E. 73 (2006) 046210; Moskalenko O.I., Hramov A.E., Koronovskii A.A., Ovchinnikov A.A. Phys. Rev. E, 2009 (submitted)]. GS is known to be used in a number of practical applications. In particular, there are known methods for secure information transmission [Murali K., Lakshmanan M. Phys. Lett. A. 241 (1998) 303-310; Terry J.R., VanWiggeren G.D. Chaos, Solitons and Fractals. 12 (2001) 145-152; Koronovskii A.A., Moskalenko O.I., Popov P.V., Hramov A.E. BRAS: Physics. 72 (1) (2008) 131-135] possessing several advantages in comparison with the other ones based on the complete or phase synchronization. At the same time, the method proposed by us previously [Koronovskii A.A., Moskalenko O.I., Popov P.V., Hramov A.E. BRAS: Physics. 72 (1) (2008) 131-135] surpasses all of them, i.e. it is easily realizable, confidential enough and possess a remarkable stability to noise in the communication channel. But the technical realization of such communication scheme demands the generators of the transmitter and receiver having some specific peculiarities, connected, first of all, with the simplicity of their practical realization and possibility of realization of the stable to noise GS regime in them. One of the simplest models of chaotic generators is radiotechnical generator with delayed feedback described in [Ponomarenko V.I., Prokhorov M.D. Phys. Rev. E. 66 (2002) 026215]. Specific peculiarity of time-delayed systems is known to be a complicated theoretical analysis connected, first of all, with the infinite dimension of their phase space. At the same time, it is easy enough to realize the delay in practice in microwave or optical band and therefore to carry out the experimental study of time-delayed systems. Time-delayed systems are well studied at present both theoretically and experimentally [Franceschini G., Bose S., Scholl E. Phys. Rev. E. 60 (5) (1999) 5426-5434; Bleich M.E., Hochheiser D., Moloney J.V., Socolar J.E.S. Phys. Rev. E. 55 (3) (1997) 2119-2126; Bezruchko B.P., Karavaev A.S., Ponomarenko V.I., Prokhorov M.D. Phys. Rev. E. 64 (1) (2001) 056216-1-056216-6; Prokhorov M.D., Ponomarenko V.I. Phys. Rev. E. 72 (2005) 016210]. At the same time, such investigations refer to their autonomous dynamics, whereas non-autonomous behavior of time-delayed systems, in particular, the GS regime, has not been studied well. As an exception one can refer to the paper [Ghosh D., Ray A., Chowdhury A.R. Modern Physics Letters B. 22 (19) (2008) 1867-1878] where relation between the GS and time-scale synchronization in two different time-delayed systems with modulated delay time has been studied. Obviously, the use of the GS regime for practical purposes demands additional investigations to be carried out. At the same time, investigation of the GS regime in time-delayed systems could be considered as an independent problem having a large fundamental significance. In present report we have investigated the generalized synchronization regime in radiotechnical generators with delayed feedback and analyze the possibility of their application as basic elements of secure communication scheme on the basis of GS in the presence of noise proposed by us previously [Koronovskii A.A., Moskalenko O.I., Hramov A.E. Method of hidden information transfer. Patent for invention No 2349044. 10.03.2009]. We have revealed the mechanisms of the GS regime arising in such systems for different values of the control parameter mismatch both in the absence and presence of noise. We have shown both theoretically and experimentally that the threshold value of the synchronous regime arising changes sharply with the variation of the control parameter of the drive system. Moreover, the threshold of the GS regime does not almost depend on the intensity of noise if the noise intensity is comparable with the drive signal one. Revealed peculiarities make possible to use radiotechnical generators with delayed feedback as a transmitting and receiving generators in the method for secure transmission of information on the basis of GS in the presence of noise. We have also discovered one more interesting effect connected with the behavior of the boundary of the GS regime in unidirectionally coupled radiotechnical generators with time delay. We have shown that the threshold value of the GS regime onset becomes decreasing if the power of noise is greater than the signal one. This fact has a large fundamental significance; in particular, it confirms that the noise could play a constructive role at the GS regime over. |
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