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The generalized synchronization （GS） of two different unidirectional coupled Lorenz systems is studied. According to the method of auxiliary-system, by using the theories of stability and the boundary of the responsed system, a sufficient criterion is rigorously proven. Furthermore, based on the modified system approach, GS is classified into three types, the first type,the second type and the third type of GS when the modified system has an asymptotically stable equilibrium of zero solution, asymptotically stable equilibrium of non-zero solution, asymptotically stable limit cycles, respectively. Moreover, using the Routh-Hurwitz theorem to analyze the stability of equilibrium of the modified system, the existence of the first type and the second type of GS are strictly theoretically proved. Numerical simulations show the effectiveness of the method.
[1] ［1]Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821
[2] ［2]Rulkov N F, Sushchik M M, Tsimring L S 1995 Phy. Rev. E 51 980
[3] ［3]Abarbanel H D I, Rulkov N F, Sushchik M M 1996 Phys.Rev. E 53 4528
[4] ［4]Hramov A E , Koronoviskii A A 2005 Phys.Rev. E 71 067201
[5] ［5]Min F H, Wang Z Q 2005 Acta Phy. Sin. 54 4026（in Chinese）［闵富红、王执铨 2005 54 4026 ］
[6] ［6]Chen G R, Lü J H 2003 The Dynamics Analysis, Control and Synchronization of Lorenz Family （Beijing: Science Press） （in Chinese）［陈关荣、吕金虎2003 Lorenz系统族的动力学分析、控制与同步 （北京：科学出版社）］
[7] ［7]Tao C H 2006 Phys. Lett. A 348 201
[8] ［8]Yu Y G, Zhang S C 2004 Chaos, Solitons and Fractals 22 189
[9] ［9]Wu X F, Chen G R, Cai J P 2007 Physica D 229 52
[10] ［10]Li F, Hu A H, Xu Z Y 2006 Acta Phys. Sin. 55 590 （in Chinese）［李芳、胡爱花、徐振源 2006 55 590］
[11] ［11]Guo L X, Xu Z Y 2008 Chaos 18 033134
[12] ［12]Guo B L 1995 Nonlinear Evolution Equations （1st ed） （Shanghai: Shanghai Scientific and Technological Education Publishing House） p208 （in Chinese）［郭柏灵1995 非线性演化方程（上海:上海科技教育出版社） 第208页］
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[1] ［1]Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821
[2] ［2]Rulkov N F, Sushchik M M, Tsimring L S 1995 Phy. Rev. E 51 980
[3] ［3]Abarbanel H D I, Rulkov N F, Sushchik M M 1996 Phys.Rev. E 53 4528
[4] ［4]Hramov A E , Koronoviskii A A 2005 Phys.Rev. E 71 067201
[5] ［5]Min F H, Wang Z Q 2005 Acta Phy. Sin. 54 4026（in Chinese）［闵富红、王执铨 2005 54 4026 ］
[6] ［6]Chen G R, Lü J H 2003 The Dynamics Analysis, Control and Synchronization of Lorenz Family （Beijing: Science Press） （in Chinese）［陈关荣、吕金虎2003 Lorenz系统族的动力学分析、控制与同步 （北京：科学出版社）］
[7] ［7]Tao C H 2006 Phys. Lett. A 348 201
[8] ［8]Yu Y G, Zhang S C 2004 Chaos, Solitons and Fractals 22 189
[9] ［9]Wu X F, Chen G R, Cai J P 2007 Physica D 229 52
[10] ［10]Li F, Hu A H, Xu Z Y 2006 Acta Phys. Sin. 55 590 （in Chinese）［李芳、胡爱花、徐振源 2006 55 590］
[11] ［11]Guo L X, Xu Z Y 2008 Chaos 18 033134
[12] ［12]Guo B L 1995 Nonlinear Evolution Equations （1st ed） （Shanghai: Shanghai Scientific and Technological Education Publishing House） p208 （in Chinese）［郭柏灵1995 非线性演化方程（上海:上海科技教育出版社） 第208页］
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