Synchronizing hyperchaos with a single variable A. Tamas ˇ evic ˇ ius and A. C ˇ enys Semiconductor Physics Institute, LT 2600 Vilnius, Lithuania Received 1 July 1996 Chaos synchronism is investigated in hyperchaotic systems. Regarding the possible applications in secure communications, the synchronization of the hyperchaotic systems via only one dynamical variable is demon- strated. As an illustration three examples including two hyperchaotic electronic circuits and the Ro ¨ssler hy- perchaotic equations are considered. S1063-651X9712501-6 PACS numbers: 05.45.+b, 89.70.+c The most intriguing feature of the chaotic synchronization is its potential possibility to be applied in secure communi- cations 1–4. However, it has been recently realized 5 that masking signals by means of comparatively simple chaos with only one positive Lyapunov exponent does not ensure high level of security. In some cases decoding can be per- formed using common signal processing methods. The straightforward way to overcome this shortcoming is to em- ploy more complex hyperchaotic signals. It was, however, commonly believed that synchronization of the hyperchaotic systems can not be achieved by a single variable coupling 6. One could think that for hyperchaotic systems it is nec- essary to transmit as much variables as there are positive Lyapunov exponents. Very recently Peng et al. demonstrated 7, that this assumption is incorrect and hyperchaotic sys- tems can be synchronized with a single transmitted signal. Their idea is to transmit a scalar signal constructed in the form of the linear combination of the original variables. Given a hyperchaotic system dx  / dt =F   x   , 1 where x   R m is an m -dimensional state vector x  = x 1 , x 2 ,..., x m  , one can construct a complex signal u ( t ) =K  x  =K 1 x 1 ( t ) +K 2 x 2 ( t ) +••• +K m x m ( t ). The trans- mitted signal u ( t ) is then applied to all the variables of the response system with another weight vector B  dx  r / dt =F   x  r  +B   u -K  x  r  . 2 By the proper adjustment of both, vector K  and vector B  , the synchronization can be achieved with only one scalar trans- mitted signal u ( t ). From the practical point of view, however, the above method can lead to some inconvenience. To implement the method one needs to have direct access to all or at least two variables in the transmitter as well as in the receiver system. This may appear to be rather complicated in the real situa- tions. In the present paper we argue that the problem can be solved by the proper selection of the original hyperchaotic system. We give two examples of hyperchaotic electronic circuits, which can be immediately synchronized with a single variable. For the first hyperchaotic circuit suggested by Matsumoto, Chua, and Kobayashi 8 synchronization is achieved by the proper choice of the single transmitted vari- able. This situation often appears also for the usual chaotic systems with one positive Lyapunov exponent, when syn- chronization is sensitive to the choice of the synchronizing variable. The second hyperchaotic circuit 9 can be synchro- nized using any variable. In addition, we suggest a modification to the method of Peng et al.’s. The modified technique enables one to employ the original single variable approach even in the case when the original hyperchaotic system, like the Ro ¨ ssler equations, cannot be synchronized immediately with a single variable. The basic idea is to transform the variables of the original system. Considering for simplicity only linear transformation   =Cx  , 3 where C is an arbitrarily chosen matrix, we construct an ‘‘improved’’ system d   / dt =      . 4 All the essential features, like the Lyapunov exponents, di- mensions, etc., of the new system remain unchanged except for the synchronization properties. With the proper choice of the matrix C one can expect to achieve the synchronization in the new hyperchaotic system with the single variable transmitted and applied to only one variable of the response system. As compared with the method of Peng et al. this corresponds to only one nonzero component B i in the vector B  . There is no general algorithm for choosing the matrix C, but some hints can be suggested. One could try to con- struct the new vector   in such a way that only one equation would have an ‘‘unstable’’ positive diagonal element. There is hope to synchronize hyperchaotic systems via this single ‘‘unstable’’ variable. We demonstrate the perfor- mance of this approach for the hyperchaotic Ro ¨ ssler system. To make sure that the synchronization is robust in a spe- cific hyperchaotic system we estimate the conditional Lyapunov exponents introduced by Pecora and Carroll 1. The largest conditional Lyapunov exponent plotted against the scalar coefficient B i provides the synchronization thresh- old. Example 1. Let us consider the electronic circuit of Mat- sumoto, Chua, and Kobayashi et al. 8 characterized by two positive Lyapunov exponents,  1 =0.24 and  2 =0.06. The dynamics of the circuit is described by 8 PHYSICAL REVIEW E JANUARY 1997 VOLUME 55, NUMBER 1 55 1063-651X/97/551/2973/$10.00 297 © 1997 The American Physical Society