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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.
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Keywords:
- sine map /
- bifurcation /
- chaos /
- dynamic equations on time scales
[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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[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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