In order to obtain asymptotical synchronization, we combine negative feedback control and dislocated negative feedback control with partial replacement to the nonlinear terms of the response system, a coupling version that was less explored. All these unidirectional coupling schemes are applied between Lorenz systems where we consider some values for the control parameters that lead to chaotic behavior. The sufficient conditions for global stable synchronization are ob- tained from a different approach of the Lyapunov direct method for the transversal system. In one of the coupling we apply a result based on the classification of the symmetric matrix AT +A as negative definite, where A is characterizing the transversal system. In the other couplings presented here, the sufficient conditions are based on the derivative in- crease of an appropriate Lyapunov function. In fact, the effectiveness of the coupling between systems with equal dimension follows from the Ros´ario D. Laureano, Diana A. Mendes and Manuel Alberto M. Ferreira analysis of the synchronization error, e(t), and, if the system variables can be bounded by positive constants, then the derivative of an appro- priate Lyapunov function can be increased as required by the Lyapunov direct method. Mathematics Subject Classification: 34D06, 34D23

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asymptotical chaos synchronization, negative feedback control, chaotic Lorenz system, unidirectional coupling, global stability, Lyapunov direct method

• Mathematics - Natural Sciences