臺灣博碩士論文加值系統

分數階系統在過去幾年已有驚人的發展。本論文主要是研究分數階上田振子的渾沌與渾沌同步。在數值分析中，使用Caputo所提出的分數微分定義，並以Adams-Bashforth-Moulton predictor-corrector方法求解。使用相軌跡圖、龐迦萊映射圖和參數分歧圖分析系統對外力之振幅的影響。接著利用電子電路模擬軟體實現分數階上田振子的動力行為。且設計出控制器使系統控制到週期運動。本論文並採用單向耦合和雙向耦合的方法設計同步控制器，使2個等同的上田振子達到渾沌同步行為。

The dynamics of the fractionally-order systems have attracted increasing attentions in recent years. Chaos and synchronization of the fractional-order Ueda oscillator is researched in this study. The fractional-order Ueda oscillator is solved by a Adams-Bashforth-Moulton predictor-corrector method. The phase portraits, the Poincaré map technique and bifurcation are used to study the effect of frequency of external force on the dynamic behaviors of the motion. Then design an electronic circuit to realizate of the fractional-order Ueda oscillator. Further the controll laws are also designed to suppress the chaotic behaviors. Finally, the one-way coupling and two-way coupling method are also designed to make two identical Ueda oscillators to achieve chaos synchronization.

摘要......................................i
Abstract.................................ii
誌謝......................................iii
目錄......................................iv
圖目錄....................................vi
表目錄....................................x
第一章 前言...............................1
1.1 研究動機............................1
1.2 文獻回顧............................2
1.3 研究目的............................3
1.4 研究方法............................3
第二章 分數階上田振子的渾沌分析與電路實現.....4
2.1 問題描述與分數微分定義...............4
2.2 數值方法............................5
2.3 週期外力之振幅對系統動力行為之影響.....7
2.4 電路方法............................15
2.5 電路實現............................19
第三章 分數階上田振子的渾沌控制..............23
3.1 控制方法............................23
3.2 系統控制............................23
第四章 分數階上田振子的渾沌同步..............33
3.1 同步方法............................33
3.2 相同階系統之同步.....................34
3.2 不同階系統之同步.....................46
第五章 結果與討論..........................63
參考文獻..................................64

1. Ueda Y., “Randomly transitional phenomena in the system governed by Duffing’s equation,” J. Stat. Phys. 20, 181-196,1979.
2. Oldham K.B. and Spanier J., The fractional calculus, Academic Press. New York, 1974.
3. Podlubny I., Fractional differential equations, Academic Press, New York, 1999.
4. Miller K.S. and Ross B., An introduction to the fractional calculus and fractional differential equations, Wiley, New York,1993.
5. Lorenz E.N., “Deterministic non-periodic flow,” Journal of Atmospheric Science, Vol. 20, pp.130-141, 1963.
6. Fujisaka H. and Yamada T., “Stability theory of synchronized motion in coupled-oscillator systems,” Progr. Theor. Phys, Vol. 69, pp. 32-47, 1983.
7. Pecora L.M. and Carroll T.L., “Synchronization in chaotic systems,” Phys. Rev. Lett, Vol. 64, pp. 821-824, 1990.
8. Koller R.C., “Application of fractional calculus to the theory of viscoelasticity,” ASME J. Appl. Mech., Vol.51, pp.299-307, 1984.
9. Mainardi F., “Fractional relaxation in anelastic solids,” J. Alloys Comp., Vol.211-1, pp.534-538, 1994.
10. Pritz T., “Analysis of four-parameter fractional derivative model of real solid materials,” J. Sound Vib., Vol.195, pp.103-115, 1996.
11. Papoulia K.D. and Kelly J.M., “Visco-hyperelastic model for filled rubbers used in vibration isolation,” ASME J. Eng. Mater. Technol., Vol.119, pp.292-297, 1997.
12. Koh C.G. and Kelly J.M., “Application of fractional derivatives to seismic analysis of base-isolated models,” Earthquake Eng. Struct. Dyn.; Vol.19, pp.229-241, 1990.
13. Makris N. and Constantinou M.C., “Spring-viscous damper systems of combined seismic and vibration isolation,” Earthquake Eng. Struct. Dyn., Vol.21, pp.649-664, 1992.
14. Shimizu N. and Zhang W., “Fractional calculus approach to dynamic problems of viscoelastic materials,” JSME , Vol.42, pp.825-837, 1999.
15. Rossikhin Y.A. and Shitikova M.V., “Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids,” Appl. Mech. Review, Vol.50, pp.15-67, 1997.
16. Friedrich C., Schiessel H., and Bllumen A., “Constitutive behavior modeling and fractional derivatives,” Advances in the Flow and Rheo. Of Non-Newtonian Fluids-Part A, Siginer DA, Chabra RP, Kee De. Eds., Elsevier, Amsterdam, pp.429-466, 1999.
17. Kempfle S., Schafer I., and Beyer H., “Fractional calculus via functional calculus: theory and applications,” Nonlinear Dyn., Vol.29, pp.99-127, 2002.
18. 葉祖銘, “杜芬方程式具分數微分阻尼之動力分析,” 中華大學機械工程學系碩士論文, 2006.
19. 陳思佑, “分數階陳李系統電路實現與秘密通訊,” 中華大學機械工程學系碩士論文, 2009.
20. Chang C.M. and Chen H.K., "Chaos and hybrid projective synchronization of commensurate and incommensurate fractional-order Chen–Lee systems", Nonlinear Dyn., 2010, doi: 10.1007/s11071-010-9767-6.
21. Caputo M., “Linear models of dissipation whose Q is almost frequency independent-II,” Geophys J. R. Astron Soc., Vol.13, pp.529-539, 1967.
22. Diethelm K., Ford N.J. and Freed A.D., “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dyn., Vol.29, pp.3-22, 2002.
23. Leszczynski J. and Ciesielski M., “A numerical method for solution of ordinary differential equations of fractional order,” arXiv:math.NA/0202276 V1 26 Feb 2002.
24. El-Sayed A.M.A., El-Mesiry A.E.M., and El-Saka H.A.A., “Numerical solution for multi-term fractional (arbitrary) orders differential equations,” Comput., Vol.23, pp.33-54, Math. 2004.
25. Diethelm K. and Ford N.J., “Multi-order fractional differential equations and their numerical solution,” Appl. Math. and Comput., Vol.154, pp.621-640, 2004.
26. Charef A., Sun H.H., Tsao Y.Y., and Onaral B., “Fractal System as Represented by Singularity Function,” IEEE Transactions on Automatic Control, vol. 37, no. 9, pp. 1465-1470, 1992.
27. Wang F.Q. and Liu C.X., “Study on the critical chaotic system with fractional order and circuit experiment,” Acta Physica Sinica vol. 55, no. 8, pp. 3922-3927, 2006.