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Introducing sine-squared potential，the particle motion equation in the crystalline undulator field is reduced to the generalized pendulum equation with a dampping and a force terms in the classical mechanics frame in the dipole approximation. The properties of the phase plane are ananysed for a non-peturbated system by means of Jacobian elliptic function and the elliptic integral，and the solution of the equation and the period of the particle motion for this system are expressed exactly. The global bifurcation and a chaotic behaviour with the Smale horseshoe for the 3 kinds of orbits in a phase plane are analysed by Melnikov method. The critical condition of the system entering into a bifurcation or a chaoc is found. The result shows that critical condition is related to the parameters of the system，by suitably regulating the parameters of the system，the bifurcation or the chaos can be avoided or controlled in principle.
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Keywords:
- crystalline undulator radiation /
- Melnikov method /
- bifurcation /
- chaos
[1] ［1］Kumakhov M A 1976 Phys. Lett. A 57 17
[2] ［2］Shao M Z，Luo S Y 1986 Chin. Laser 13 607 (in Chinese) ［邵明珠、罗诗裕 1986 中国激光 13 607］
[3] ［3］Korol A，Solovyov A V，Greiner W 1998 J. Phys. G 24 L45
[4] ［4］Korol A，Solovyov A V，Greiner W 1999 Int. J. Mod. Phys. E 8 49
[5] ［5］Gevorging L A，Hovsepyan L A 2007 J. Contemp. Phys. 42 87
[6] ［6］Luo S Y，Shao M Z 2005 High Energ. Phys. Nucl. Phys. 29 193 (in Chinese) ［罗诗裕、邵明珠 2005 高能物理与核物理 29 193
[7] ［7］Korol A V，Solovyov A V，Greiner W 2008 Nucl. Inst. Method. Phys. Rese. B 266 1173
[8] ［8］Wiedemann H 2006 Advance Radiation Sources and Application (Natherlands: Springer) p165
[9] ［9］Tabrizi M，Korol A V，Solovyov A V 2007 Phys. Rev. Lett. 98 164801
[10] ］Korol A V，Solovyov A V，Greiner W 2004 Int. J. Mod. Phys. E 13 867
[11] ］Wiedemann H 2006 Advance Radiation Sources and Application (Natherlands: Springer) p191
[12] ］Luo S Y，Shao M Z 2009 Chin. Laser 36 1378 (in Chinese) ［罗诗裕、邵明珠 2009 中国激光 36 1378］
[13] ］Luo S Y，Shao M Z 2003 Chin. J. Semiconductors 24 485 (in Chinese) ［罗诗裕、邵明珠 2003 半导体学报 24 485 ］
[14] ］Luo S Y，Shao M Z 1984 Chin. Phys. 4 670
[15] ］Luo S Y，Shao M Z 2005 Chin. J. Semiconductors 26 294 (in Chinese) ［罗诗裕、邵明珠 2005 半导体学报 26 294］
[16] ］Luo S Y，Shao M Z 2006 Acta Phys. Sin. 55 1324 (in Chinese) ［罗诗裕、邵明珠 2006 物理学报 55 1324］
[17] ］Deng C L，Shao M Z，Luo S Y 2006 Acta Phys. Sin. 55 1721 (inChinese) ［邓成良、邵明珠、罗诗裕 2006 物理学报 55 1721］
[18] ］Shao M Z，Luo S Y 2005 Acta Phys. Sin. 54 4092 (in Chinese) ［邵明珠、罗诗裕 2005 物理学报 54 4092］
[19] ］Luo S Y，Shao M Z 2004 Acta Phys. Sin. 53 1157 (in Chinese) ［罗诗裕、邵明珠 2004 物理学报 53 1157］
[20] ］Luo S Y，Shao M Z 2004 Acta Phys. Sin. 53 1940 (in Chinese) ［罗诗裕、邵明珠 2004 物理学报 53 1940］
[21] ］Shao M Z，Luo S Y 2007 Acta Phys. Sin. 56 3407 (in Chinese) ［邵明珠、罗诗裕 2007 物理学报 56 3407］
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[1] ［1］Kumakhov M A 1976 Phys. Lett. A 57 17
[2] ［2］Shao M Z，Luo S Y 1986 Chin. Laser 13 607 (in Chinese) ［邵明珠、罗诗裕 1986 中国激光 13 607］
[3] ［3］Korol A，Solovyov A V，Greiner W 1998 J. Phys. G 24 L45
[4] ［4］Korol A，Solovyov A V，Greiner W 1999 Int. J. Mod. Phys. E 8 49
[5] ［5］Gevorging L A，Hovsepyan L A 2007 J. Contemp. Phys. 42 87
[6] ［6］Luo S Y，Shao M Z 2005 High Energ. Phys. Nucl. Phys. 29 193 (in Chinese) ［罗诗裕、邵明珠 2005 高能物理与核物理 29 193
[7] ［7］Korol A V，Solovyov A V，Greiner W 2008 Nucl. Inst. Method. Phys. Rese. B 266 1173
[8] ［8］Wiedemann H 2006 Advance Radiation Sources and Application (Natherlands: Springer) p165
[9] ［9］Tabrizi M，Korol A V，Solovyov A V 2007 Phys. Rev. Lett. 98 164801
[10] ］Korol A V，Solovyov A V，Greiner W 2004 Int. J. Mod. Phys. E 13 867
[11] ］Wiedemann H 2006 Advance Radiation Sources and Application (Natherlands: Springer) p191
[12] ］Luo S Y，Shao M Z 2009 Chin. Laser 36 1378 (in Chinese) ［罗诗裕、邵明珠 2009 中国激光 36 1378］
[13] ］Luo S Y，Shao M Z 2003 Chin. J. Semiconductors 24 485 (in Chinese) ［罗诗裕、邵明珠 2003 半导体学报 24 485 ］
[14] ］Luo S Y，Shao M Z 1984 Chin. Phys. 4 670
[15] ］Luo S Y，Shao M Z 2005 Chin. J. Semiconductors 26 294 (in Chinese) ［罗诗裕、邵明珠 2005 半导体学报 26 294］
[16] ］Luo S Y，Shao M Z 2006 Acta Phys. Sin. 55 1324 (in Chinese) ［罗诗裕、邵明珠 2006 物理学报 55 1324］
[17] ］Deng C L，Shao M Z，Luo S Y 2006 Acta Phys. Sin. 55 1721 (inChinese) ［邓成良、邵明珠、罗诗裕 2006 物理学报 55 1721］
[18] ］Shao M Z，Luo S Y 2005 Acta Phys. Sin. 54 4092 (in Chinese) ［邵明珠、罗诗裕 2005 物理学报 54 4092］
[19] ］Luo S Y，Shao M Z 2004 Acta Phys. Sin. 53 1157 (in Chinese) ［罗诗裕、邵明珠 2004 物理学报 53 1157］
[20] ］Luo S Y，Shao M Z 2004 Acta Phys. Sin. 53 1940 (in Chinese) ［罗诗裕、邵明珠 2004 物理学报 53 1940］
[21] ］Shao M Z，Luo S Y 2007 Acta Phys. Sin. 56 3407 (in Chinese) ［邵明珠、罗诗裕 2007 物理学报 56 3407］
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