时标正弦动力学方程稳定性与分岔分析

胡文; 赵广浩; 张弓; 张景乔; 刘贤龙

doi:10.7498/aps.61.170505

摘要

本文研究时标上正弦动力学方程的平衡点稳定性和分岔现象. 研究表明随时标参数的变化,正弦动力学方程展现出完全不同的解, 会产生n倍周期分岔和平衡点分裂等特有现象.同时,不增加系统参数, 仅改变时标的复杂性就能扩展动力学方程处于混沌状态的参数空间, 这为时标上动力学方程在混沌加密和雷达波形设计等领域的应用提供了潜在的优势.

关键词:

A time scale is a nonempty closed subset of the real numbers R. Recently, the dynamic equations on time scale have received much attention, which have the generalized forms of differential and differential dynamic equations. In this paper, we study the stabilities of fixed points and bifurcations of the sine dynamic equations on time scale. The results show that the solutions of the sine dynamic equations become different with the time scale parameter changing. And n-period-doubling bifurcations and splits of fixed points are observed. Moreover, the chaotic parameter spaces of the dynamic equations are expanded by the increase of complexity of time scale but without increasing the system parameter, thus providing a potential advantage for chaos encryption, radar waveform design and other application areas.

Keywords:

作者及机构信息

1.
南京航空航天大学电子信息工程学院, 南京 210016;

2.
南京长江电子信息产业集团有限公司, 南京 210037

基金项目: 国家自然科学基金(批准号: 61071163, 61001151); 南京航空航天大学科研启动金, 南京航空航天大学基本科研业务费专项科研项目(批准号: NS2010096) 和航空科学基金(批准号: 2009ZC52038, 2008ZC52026)资助的课题.

Authors and contacts

1.
College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China;

2.
Nanjing Changjiang Electronics Group CO., LTD, Nanjing 210037, China

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 61071163, 61001151), the Nanjing University of Aeronautics and Astronautics, Scientific Research Fund, the NUAA Research Funding (Grant No. NS2010096), and the Aeronautical Science Foundations of China (Grant Nos. 2009ZC52038, 2008ZC52026).

参考文献

[1]	Schell M, Fraser S, Kapral R 1983 Physical Review A 28 373
[2]	Chirikov B V 1979 Physics reports 52 263
[3]	Huberman B A 1980 Appl. Phys. Lett. 37 750
[4]	Winfree A T 2001 The geometry of biological time(New York: Springer) p101
[5]	Larger L, Udaltsov V S, Poinsot S 2005 Journal of optical technology 72 378
[6]	Jie X, Ke-Ping L, Dani Le F 2010 Chinese Physics Letters 27 020504
[7]	Tarasov V E, Edelman M 2010 Chaos: An Interdisciplinary Journal of Nonlinear Science 20 023127
[8]	Arnol'd V I 1961 Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 25 21
[9]	Feigenbaum M J, Kadanoff L P, Shenker S J 1982 Physica D: Nonlinear Phenomena5 370
[10]	Ruelle D, Takens F 1971 Commun. Math. Phys. 20 167
[11]	Glass L, Perez R 1982 Physical Review Letters 48 1772
[12]	Schell M, Fraser S, Kapral R 1982 Physical Review A 26 504
[13]	Fraser S, Kapral R 1982 Physical Review A 25 3223
[14]	Chang S J, Wortis M, Wright J A 1981 Physical Review A 24 2669
[15]	Xu J, Long K P, Fournier-Prunaret D 2010 Chinese Phys. Lett. 27 080506
[16]	Lalescu C C 2010 Arxiv preprint arXiv 1011 6552
[17]	Afsar O, Tirnakli U 2010 Physical Review E 82 046210
[18]	Nayak C R, Gupte N 2010 Arxiv preprint arXiv 1011 5492
[19]	Santhiah M, Philominathan P 2010 Pramana 75 403
[20]	Xu J, Charg P, Fournier-Prunaret D 2010 SCIENCE CHINA Information Sciences 53 129
[21]	Agarwal R, Bohner M, O'regan D 2002 Journal of Computational and Applied Mathematics 141 1
[22]	Agarwal D 2000 Applied Mathematics Letters 13 7
[23]	Peterson A C, Raffoul Y N 2005 Advances in Difference Equations 2005 133
[24]	Hoffacker J, Tisdell C C 2005 Computers & Mathematics with Applications 49 1327
[25]	Li T, Han Z, Sun S 2009 Electronic Journal of Qualitative Theory of Differential Equations 60 1
[26]	Grace S R, Agarwal R P, Kaymak Alan B 2010 Journal of Applied Mathematics and Computing 32 205
[27]	Alvarez G, Li S 2006 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 16 2129
[28]	Willsey M S, Cuoco K M, Oppenheim A V 2011 Aerospace and Electronic Systems, IEEE Transactions on, Cambridge, July 2011, p1974

施引文献

[1]	Schell M, Fraser S, Kapral R 1983 Physical Review A 28 373
[2]	Chirikov B V 1979 Physics reports 52 263
[3]	Huberman B A 1980 Appl. Phys. Lett. 37 750
[4]	Winfree A T 2001 The geometry of biological time(New York: Springer) p101
[5]	Larger L, Udaltsov V S, Poinsot S 2005 Journal of optical technology 72 378
[6]	Jie X, Ke-Ping L, Dani Le F 2010 Chinese Physics Letters 27 020504
[7]	Tarasov V E, Edelman M 2010 Chaos: An Interdisciplinary Journal of Nonlinear Science 20 023127
[8]	Arnol'd V I 1961 Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 25 21
[9]	Feigenbaum M J, Kadanoff L P, Shenker S J 1982 Physica D: Nonlinear Phenomena5 370
[10]	Ruelle D, Takens F 1971 Commun. Math. Phys. 20 167
[11]	Glass L, Perez R 1982 Physical Review Letters 48 1772
[12]	Schell M, Fraser S, Kapral R 1982 Physical Review A 26 504
[13]	Fraser S, Kapral R 1982 Physical Review A 25 3223
[14]	Chang S J, Wortis M, Wright J A 1981 Physical Review A 24 2669
[15]	Xu J, Long K P, Fournier-Prunaret D 2010 Chinese Phys. Lett. 27 080506
[16]	Lalescu C C 2010 Arxiv preprint arXiv 1011 6552
[17]	Afsar O, Tirnakli U 2010 Physical Review E 82 046210
[18]	Nayak C R, Gupte N 2010 Arxiv preprint arXiv 1011 5492
[19]	Santhiah M, Philominathan P 2010 Pramana 75 403
[20]	Xu J, Charg P, Fournier-Prunaret D 2010 SCIENCE CHINA Information Sciences 53 129
[21]	Agarwal R, Bohner M, O'regan D 2002 Journal of Computational and Applied Mathematics 141 1
[22]	Agarwal D 2000 Applied Mathematics Letters 13 7
[23]	Peterson A C, Raffoul Y N 2005 Advances in Difference Equations 2005 133
[24]	Hoffacker J, Tisdell C C 2005 Computers & Mathematics with Applications 49 1327
[25]	Li T, Han Z, Sun S 2009 Electronic Journal of Qualitative Theory of Differential Equations 60 1
[26]	Grace S R, Agarwal R P, Kaymak Alan B 2010 Journal of Applied Mathematics and Computing 32 205
[27]	Alvarez G, Li S 2006 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 16 2129
[28]	Willsey M S, Cuoco K M, Oppenheim A V 2011 Aerospace and Electronic Systems, IEEE Transactions on, Cambridge, July 2011, p1974

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