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By introducing periodically alternate current source as well as suitable values for the parameters to ensure that there exists order gap between the natural frequency and the exited frequency, a two-time scale namely, a fast-slow coupled non-smooth generalized Chua’s circuit model is established. Based on the corresponding generalized autonomous system, the stabilities of the equilibrium points in different regions are investigated, from which the critical conditions related to different types of bifurcation forms are obtained. At the same time, combining the theory of Clarke derivative, different types of non-conventional bifurcation models which may occur when the trajectory passes across the non-smooth boundaries are explored. Furthermore, with the combination of the generalized phase portraits, two typical periodic bursting phenomena namely, the Fold/Fold and Fold/Hopf periodic bursters, and their associated bifurcation mechanisms are analysed in detail.
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Keywords:
- non-smooth /
- generalized Chua’ /
- s circuit /
- two time scales
[1] Shooshtari A, Pasha Zanoosi A 2010 Appl. Math. Model. 34 1918
[2] Haselbacher A, Najjar F M, Massa L, Moser R D 2010 J. Comput. Phys. 229 325
[3] Merkin J H, Taylor A F 2012 Physica D 241 1336
[4] Ernesto P, Dulce M, Soledad M, Jose M G, Santiago L, Julian J G 2006 Neurosci. Lett. 394 152
[5] Jia Z D, Leimkuhler B 2003 Future Generation Comput. Syst. 19 415
[6] Knoll D A, Chacon L, Margolin L, Mousseau V 2003 J. Comput. Phys. 185 583
[7] Rinberg A, Taylor A L, Mdarder E 2013 Plos Computat. Biol. 9 e1002857
[8] Strizhak P E, Kawczynski A L 1995 J. Phys. Chem. 99 10830
[9] Ji Y, Bi Q S 2010 Phys. Lett. A 374 1434
[10] Izhikevich E M 2000 Int. J. Bifur. Chaos 10 1171
[11] Chua L O, Lin G N 1990 IEEE Trans. Circ. Syst. 37 885
[12] Zhai D Q, Liu C X, Liu Y, Xu Z 2010 Acta Phys. Sin. 59 816 (in Chinese) [翟笃庆, 刘崇新, 刘尧, 许喆 2010 物理学报 59 816]
[13] Chen Z Y, Zhang X F, Bi Q S 2008 Nonlin. Anal.: Real World Appl. 9 1158
[14] Stouboulos I N, Miliou A N, Valaristos A P 2007 Chaos Solition. Fract. 33 1256
[15] Koliopanos C L, Kyprianidis I M, Stouboulos I N 2003 Chaos Solition. Fract. 16 173
[16] Yang Z M, Zhang J, Ma Y J, Bai Y L, Ma S Q 2010 Acta Phys. Sin. 59 3007 (in Chinese) [杨志民, 张洁, 马永杰, 摆玉龙, 马胜前 2010 物理学报 59 3007]
[17] Binazadeh T, Shafiei M H 2013 Commun. Nonlinear Sci. Numer. Simulat. 18 1071
[18] Ji Y, Bi Q S 2012 Acta Phys. Sin. 61 010202 (in Chinese) [季颖, 毕勤胜 2012 物理学报 61 010202]
[19] Zhang Y, Bi Q S 2011 Chin. Phys. B 20 010504-1
[20] Li X H, Bi Q S 2012 Acta Phys. Sin. 61 020504 (in Chinese) [李向红, 毕勤胜 2012 物理学报 61 020504]
[21] Zhang Z D, Li Y Y, Bi Q S 2013 Phys. Lett. A 377 975
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[1] Shooshtari A, Pasha Zanoosi A 2010 Appl. Math. Model. 34 1918
[2] Haselbacher A, Najjar F M, Massa L, Moser R D 2010 J. Comput. Phys. 229 325
[3] Merkin J H, Taylor A F 2012 Physica D 241 1336
[4] Ernesto P, Dulce M, Soledad M, Jose M G, Santiago L, Julian J G 2006 Neurosci. Lett. 394 152
[5] Jia Z D, Leimkuhler B 2003 Future Generation Comput. Syst. 19 415
[6] Knoll D A, Chacon L, Margolin L, Mousseau V 2003 J. Comput. Phys. 185 583
[7] Rinberg A, Taylor A L, Mdarder E 2013 Plos Computat. Biol. 9 e1002857
[8] Strizhak P E, Kawczynski A L 1995 J. Phys. Chem. 99 10830
[9] Ji Y, Bi Q S 2010 Phys. Lett. A 374 1434
[10] Izhikevich E M 2000 Int. J. Bifur. Chaos 10 1171
[11] Chua L O, Lin G N 1990 IEEE Trans. Circ. Syst. 37 885
[12] Zhai D Q, Liu C X, Liu Y, Xu Z 2010 Acta Phys. Sin. 59 816 (in Chinese) [翟笃庆, 刘崇新, 刘尧, 许喆 2010 物理学报 59 816]
[13] Chen Z Y, Zhang X F, Bi Q S 2008 Nonlin. Anal.: Real World Appl. 9 1158
[14] Stouboulos I N, Miliou A N, Valaristos A P 2007 Chaos Solition. Fract. 33 1256
[15] Koliopanos C L, Kyprianidis I M, Stouboulos I N 2003 Chaos Solition. Fract. 16 173
[16] Yang Z M, Zhang J, Ma Y J, Bai Y L, Ma S Q 2010 Acta Phys. Sin. 59 3007 (in Chinese) [杨志民, 张洁, 马永杰, 摆玉龙, 马胜前 2010 物理学报 59 3007]
[17] Binazadeh T, Shafiei M H 2013 Commun. Nonlinear Sci. Numer. Simulat. 18 1071
[18] Ji Y, Bi Q S 2012 Acta Phys. Sin. 61 010202 (in Chinese) [季颖, 毕勤胜 2012 物理学报 61 010202]
[19] Zhang Y, Bi Q S 2011 Chin. Phys. B 20 010504-1
[20] Li X H, Bi Q S 2012 Acta Phys. Sin. 61 020504 (in Chinese) [李向红, 毕勤胜 2012 物理学报 61 020504]
[21] Zhang Z D, Li Y Y, Bi Q S 2013 Phys. Lett. A 377 975
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