1063-7850/01/2706- $21.00 © 2001 MAIK “Nauka/Interperiodica” 0476 Technical Physics Letters, Vol. 27, No. 6, 2001, pp. 476–479. Translated from Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, Vol. 27, No. 11, 2001, pp. 78–85. Original Russian Text Copyright © 2001 by Shabunin, Demidov, Astakhov, Anishchenko. The term “synchronization of chaos” refers to a number of physical phenomena such as the suppression of chaotic oscillations under the action of external peri- odic factors [1], the transition to completely identical oscillations in coupled oscillators (totally synchronized chaos) [2, 3] or to the oscillations identical to within a delay time in the subsystems (lag-synchronization) [4], the base frequency locking in the spectrum of chaotic oscillations [5], the instantaneous phase locking [6], and a deterministic relationship (including the case of delay-time x 1 (t ) = f [x 2 (t – τ)] between oscillations in the subsystems (generalized synchronization) [7]. A somewhat ambiguous terminology reflects the variety of manifestations of the phenomenon of oscillation matching due to the interaction of oscillators. For better understanding the interplay between var- ious types of the chaos synchronization, it would be helpful to introduce a characteristic capable of measur- ing the degree of matching in the behavior of oscillators and quantitatively describing the degree of chaos syn- chronization in the system studied. It must be noted that a system can escape from the regime of chaos synchro- nization as a result of some change in the system parameters, this process proceeding gradually via a sequence of intermediate stages. For example, the breakage of a totally synchronized chaos may be accompanied by intermittent behavior, whereby the coupled oscillators exhibit synchronized oscillations for some time and chaotic oscillations otherwise [8, 9]. In this case, the task of quantitatively describing the escape from the synchronized chaos regime is also encountered. We believe that the necessary requirements to a quantity measuring the degree of chaos synchroniza- tion are as follows: (i) clear physical meaning facilitat- ing interpretation of the results; (ii) universality, mak- ing the measure applicable to various types of consistent behavior of the interacting subsystems; (iii) indepen- dence of a particular type of the dynamic system, mak- ing it possible to determine the degree of synchroniza- tion using the time patterns of oscillations in the sub- systems; (iv) robustness, implying that small perturbations (changes in the oscillation regime, noise, distortions) would not significantly alter the value of the degree of synchronization. With an allowance of these requirements, we sug- gest that such a measure of the consistent behavior can be represented by the amount of information provided by knowledge of the state of one oscillator for deter- mining the state of another oscillator. If the state of one oscillator uniquely determines that of another, we may speak of the total synchronization (the degree of syn- chronization is unity). In the contrary, if the state of one oscillator does not influence the state of another, we may speak of the absence of synchronization (zero degree of synchronization). Using the language of mathematics, a quantitative measure of such informa- tion is offered by a difference between the total and conditional entropies calculated for all realizations of the oscillators [10]: (1) The so defined information is zero when the condi- tional entropy is equal to the total (unconditional) entropy, that is, when the state of the second oscillator (y) does not affect the variable (x) distribution of the first oscillator; the information is maximum (S x ) if the state of the second oscillator uniquely determines that of the first oscillator (S x |y = 0). By normalizing the information to the unconditional entropy, we obtain a function acquiring a zero value in the case of com- pletely independent behavior of oscillators and increas- ing to unity for the fully consistent behavior of the sub- systems. I S x S xy . – = The Volume of Information as a Measure of the Chaos Synchronization A. V. Shabunin, V. V. Demidov, V. V. Astakhov, and V. S. Anishchenko Saratov State University, Saratov, Russia e-mail: valya@chaos.ssu.runnet.ru Received December 4, 2000 Abstract—A characteristic is suggested for evaluation of the degree of synchronization of the chaotic oscilla- tions in a system of two coupled oscillators. The proposed value is tested by application to the case of two uni- directionally coupled logistic maps. It is shown that this characteristic is stable with respect to a low noise and a nonlinear distortion of the signal. © 2001 MAIK “Nauka/Interperiodica”.