15 January 1996 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA PHYSICS LETTERS A ELSEVIER Physics Letters A 210 (1996) 279-282 Steady state locking in coupled chaotic systems Andrzej Stefanski, Tomasz Kapitaniak zyxwvutsrqponmlkjihgfedcbaZYXWV Division of Dynamics, Technical University of Lodz, Stefanowskiego I/ 15, 90-924 Lodz, Poland Received 8 June 1995; revised manuscript received 29 September 1995; accepted for publication 21 October 1995 Communicated by CR. Doering zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK Abstract Two Lorenz systems working in different chaotic ranges can be stabilized simultaneously in the same steady state by coupling them through the negative feedback mechanism. This kind of locking is robust as it can be realized for a wide range of parameters. PACS: 05.45. + b Recently it has been demonstrated that two identical chaotic systems k = fix> and j = fl y) (x, y E Iw”, n 2 3) coupled with each other can be synchronized [l-6]. Coupling of homochaotic systems (i.e. systems given by the same set of ODES but with different values of the system parameters) can lead to the practical synchronization (i.e. x r y, but I] x - y (I G E where E is a vector of small parameters) [7-91. In such coupled systems we can also observe a significant change of the chaotic behavior of one or both systems [11,12]. This so-called “controlling chaos by chaos” procedure has some potential importance in a range of contexts varying from mechanical and electrical systems [7], where control of the system is the objective, to geophysical systems, like the atmosphere or oceans, where improvement in basic understanding and prediction is the main motivation [ 11,121. Other phenomena are possible as the coupled systems become a new augmented system, which has its own dynamics. In Ref. [lo] it was shown that two Lorenz systems working in different chaotic ranges can be stabilized simultaneously in different periodic orbits by coupling them through certain system parameters. In what follows we introduce another phenomenon characteristic for such systems. We consider two monochaotic systems i=f(q, x) (1) and Y=f(as* Y) (2) (x, YE [w”, Q, 2 E [w, n 2 3) coupled with each other by negative feedback. The augmented system is as follows, ’ k=f(a,, x) +qy-x), j,=f(azv Y) +&(x-y), (3) 0375.9601/96/$12.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0375-9601(95)00878-O