在外加振盪磁場中阻尼磁針的非線性動力學分析

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姓名

許庭瑋(Ting-wei Hsu) 查詢紙本館藏

畢業系所

生物物理研究所

論文名稱

在外加振盪磁場中阻尼磁針的非線性動力學分析
(Numerical and theoretical analysis of the nonlinear dynamics of a damped compass under external oscillatory magnetic field)

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摘要(中)

我們研究了振盪磁場中的阻尼磁針系統，振盪磁場由垂直交錯的兩部分合成，一部分是方向大小固定的磁場B1（可以是地球磁場），另一部分是以正弦形式、振幅為B2的外加振盪磁場。磁針的擺動是複雜的非線性振盪，它會在外磁場振幅增加時，經由週期倍增路徑變成混沌的振盪。系統的運動方程式具有「角度反轉，同時時間平移二分之奇數倍個磁場振盪週期」的對稱性。因為這個對稱性，磁針振盪運動有對稱的週期和混沌吸引子共存。我們用數值方法解微分方程組和畫出相空間圖，檢查當系統的參數改變時，吸引子的性質如何隨參數改變，例如：成對對稱的吸引子是如何出現和合併。有趣的是，我們不只發現了具有相同週期的吸引子成對對稱地出現。在某些參數區間，彼此不對稱的兩個單數週期吸引子也可以共存，以及一對成對對稱的週期二吸引子和一個混沌吸引子的共存也被我們發現。

摘要(英)

We consider a magnetic dipole (compass needle) under a constant magnetic (Earth′s) field and an external sinusoidally oscillating magnetic field (of magnitude B2) that is perpendicular to the former. The angular motion displays complex nonlinear oscillations and undergoes a period-doubling route to chaos. The equation of motion of the system possesses a special symmetry when angle inversion together with time translation of half of the driving period is applied. Due to this symmetry, coexistence of attractors, including symmetric periodic states and symmetric chaotic strange attractors, occurs. The properties of these attractors, such as how the symmetric attractor pairs appear and merge, as revealed by numerical solution of the differential equations and phase portraits, are examined in detail as the parameters of the system change. Interestingly, it is found that in addition to the coexistence of symmetric limit cycle attractor pair (both having the same period state), two different odd-periodic states not related by symmetry, can coexist. In addition, a pair of symmetric period-2 limit cycles and a chaotic attractor can coexist in certain parameter regimes.

關鍵字(中)

★ 非線性動力學
★ 非線性振盪
★ 吸引子共存
★ 對稱性

關鍵字(英)

★ Nonlinear dynamics
★ Nonlinear oscillation
★ Attractors coexisting
★ Symmetry

論文目次

摘要 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
目錄 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
圖目錄 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
表目錄 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
符號說明 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
一、緒論 . . . . . . . . . . . . . . . . . . . . . . . . 1
1-1 研究動機 . . . . . . . . . . . . . . . . . . . . . . 2
1-2 理論背景 . . . . . . . . . . . . . . . . . . . . . . 2
1-2-1 非線性動力系統 . . . . . . . . . . . . . . . . . . 2
1-2-2 相空間 . . . . . . . . . . . . . . . . . . . . . . . 3
1-2-3 吸引子 . . . . . . . . . . . . . . . . . . . . . . . 3
1-2-4 Bifurcation（分岔、分歧） . . . . . . . . . . . . 4
1-2-5 Poincaré 映射 . . . . . . . . . . . . . . . . . . . 10
1-2-6 吸引子共存 . . . . . . . . . . . . . . . . . . . . 11
二、文獻探討 . . . . . . . . . . . . . . . . . . . . . . 13
2-1 吸引子共存 . . . . . . . . . . . . . . . . . . . . 13
2-2 磁針、單擺系統 . . . . . . . . . . . . . . . . . . 19
三、研究方法 . . . . . . . . . . . . . . . . . . . . . . 23
3-1 外加振盪磁場中的阻尼磁針系統 . . . . . . . . . 23
3-2 理論分析方法 . . . . . . . . . . . . . . . . . . . 23
3-2-1 分析系統的運動方程式 . . . . . . . . . . . . . . 23
3-2-2 分析系統的對稱性 . . . . . . . . . . . . . . . . 24
3-3 數值分析方法 . . . . . . . . . . . . . . . . . . . 24
3-3-1 使用四階 Runge-Kutta 法及 Euler 法數值模擬
系統 . . . . . . . . . . . . . . . . . . . . . . . . 24
3-3-2 使用 Poincaré 截面和時間平均法幫助判斷磁針
振盪的週期 . . . . . . . . . . . . . . . . . . . . 25
四、研究結果 . . . . . . . . . . . . . . . . . . . . . . 29
4-1 磁針振盪吸引子隨 b2 的變化 . . . . . . . . . . . 29
4-2 系統對稱性和磁針振盪吸引子存在情形的分析 . 42
4-2-1 只有成對對稱吸引子的共存 . . . . . . . . . . . 42
4-2-2 只有自我對稱吸引子的共存 . . . . . . . . . . . 42
4-2-3 成對對稱吸引子和自我對稱吸引子的共存 . . . . 43
4-2-4 系統不允許偶數週期的自我對稱吸引子 . . . . . 43
4-3 Basins of attraction 的性質 . . . . . . . . . . . . 44
4-3-1 自我相似的結構 . . . . . . . . . . . . . . . . . . 44
4-3-2 與系統對稱性的關係 . . . . . . . . . . . . . . . 45
4-3-3 隨參數 b2 的變化 . . . . . . . . . . . . . . . . . 47
五、討論、結論與未來展望 . . . . . . . . . . . . . . 49
5-1 討論 . . . . . . . . . . . . . . . . . . . . . . . . 49
5-1-1 與磁針、單擺系統文獻的比較 . . . . . . . . . . 49
5-1-2 與其他吸引子共存系統文獻的比較 . . . . . . . . 50
5-2 結論 . . . . . . . . . . . . . . . . . . . . . . . . 50
5-3 未來展望 . . . . . . . . . . . . . . . . . . . . . . 51
參考文獻 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
附錄一 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

參考文獻

[1] S. H. Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, Westview press, 1994.
[2] F. Moon and G.-X. Li, “Fractal basin boundaries and homoclinic orbits for periodic motion in a two-well potential”, Physical Review Letters, volume 55(14), p. 1439, 1985.
[3] J. Sprott, “Simplest chaotic flows with involutional symmetries”, International Journal of Bifurcation and Chaos, volume 24(01), p. 1450009, 2014.
[4] M. P. Dafilis et al., “Dynamical crises, multistability and the influence of the duration of immunity in a seasonally-forced model of disease transmission”, Theoretical Biology and Medical Modelling, volume 11(1), p. 43, 2014.
[5] F. Moon, J. Cusumano, and P. Holmes, “Evidence for homoclinic orbits as a precursor to chaos in a magnetic pendulum”, Physica D: Nonlinear Phenomena, volume 24(1), pp. 383–390, 1987.
[6] J. Jeong and S.-Y. Kim, “Bifurcations in a horizontally driven pendulum”, Journal of the Korean Physical Society, volume 35(5), pp. 393–398, 1999.
[7] R. M. May et al., “Simple mathematical models with very complicated dynamics”, Nature, volume 261(5560), pp. 459–467, 1976.
[8] M. J. Feigenbaum, “Quantitative universality for a class of nonlinear transformations”, Journal of statistical physics, volume 19(1), pp. 25–52, 1978.
[9] A. N. Pisarchik and U. Feudel, “Control of multistability”, Physics Reports, volume 540(4), pp. 167–218, 2014.
[10] J. E. Ferrell and E. M. Machleder, “The biochemical basis of an all-or-none cell fate switch in xenopus oocytes”, Science, volume 280(5365), pp. 895–898, 1998.
[11] C. P. Bagowski and J. E. Ferrell, “Bistability in the jnk cascade”, Current Biology, volume 11(15), pp. 1176–1182, 2001.
[12] U. S. Bhalla, P. T. Ram, and R. Iyengar, “Map kinase phosphatase as a locus of flexibility in a mitogen-activated protein kinase signaling network”, Science, volume 297(5583), pp. 1018–1023, 2002.
[13] F. R. Cross et al., “Testing a mathematical model of the yeast cell cycle”, Molecular Biology of the Cell, volume 13(1), pp. 52–70, 2002.
[14] J. R. Pomerening, E. D. Sontag, and J. E. Ferrell, “Building a cell cycle oscillator: hysteresis and bistability in the activation of cdc2”, Nature Cell Biology, volume 5(4), pp. 346–351, 2003.
[15] C. P. Bagowski et al., “The jnk cascade as a biochemical switch in mammalian cells: ultrasensitive and all-or-none responses”, Current Biology, volume 13(4), pp. 315–320, 2003.
[16] W. Sha et al., “Hysteresis drives cell-cycle transitions in xenopus laevis egg extracts”, Proceedings of the National Academy of Sciences, volume 100(3), pp. 975–980, 2003.
[17] B. M. Slepchenko and M. Terasaki, “Bio-switches: what makes them robust?”, Current opinion in genetics & development, volume 14(4), pp. 428–434, 2004.
[18] D. Angeli, J. E. Ferrell, and E. D. Sontag, “Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems”, Proceedings of the National Academy of Sciences, volume 101(7), pp. 1822–1827, 2004.
[19] J. D. Chung and G. Stephanopoulos, “On physiological multiplicity and population heterogeneity of biological systems”, Chemical Engineering Science, volume 51(9), pp. 1509–1521, 1996.
[20] J. E. Ferrell, “Self-perpetuating states in signal transduction: positive feedback, double-negative feedback and bistability”, Current Opinion in Cell Biology, volume 14(2), pp. 140–148, 2002.
[21] J. Hertz, A. Krogh, and R. G. Palmer, Introduction to the theory of neural computation, volume 1, Addison-Wesley, New York, 1991.
[22] C. Canavier et al., “Nonlinear dynamics in a model neuron provide a novel mechanism for transient synaptic inputs to produce long-term alterations of postsynaptic activity”, Journal of Neurophysiology, volume 69(6), pp. 2252–2257, 1993.
[23] J. Braun and M. Mattia, “Attractors and noise: twin drivers of decisions and multistability”, NeuroImage, volume 52(3), pp. 740–751, 2010.
[24] M. Laurent and N. Kellershohn, “Multistability: a major means of differentiation and evolution in biological systems”, Trends in Biochemical Sciences, volume 24(11), pp. 418–422, 1999.
[25] J. P. Newman and R. J. Butera, “Mechanism, dynamics, and biological existence of multistability in a large class of bursting neurons”, Chaos: An Interdisciplinary Journal of Nonlinear Science, volume 20(2), p. 023118, 2010.
[26] S. Sridhar et al., “Suppression of cardiac alternans by alternatingperiod-feedback stimulations”, Physical Review E, volume 87(4), p. 042712, 2013.
[27] V. Croquette and C. Poitou, “Cascade of period doubling bifurcations and large stochasticity in the motions of a compass”, Journal de Physique Lettres, volume 42(24), pp. 537–539, 1981.
[28] F. Moon and P. J. Holmes, “A magnetoelastic strange attractor”, Journal of Sound and Vibration, volume 65(2), pp. 275–296, 1979.
[29] U. Feudel, “Complex dynamics in multistable systems”, International Journal of Bifurcation and Chaos, volume 18(06), pp. 1607–1626, 2008.
[30] A. Pisarchik, “Controlling the multistability of nonlinear systems with coexisting attractors”, Physical Review E, volume 64(4), p. 046203, 2001.
[31] B. Martínez-Zérega and A. Pisarchik, “Efficiency of the control of coexisting attractors by harmonic modulation applied in different ways”, Physics Letters A, volume 340(1), pp. 212–219, 2005.
[32] T. U. Singh, A. Nandi, and R. Ramaswamy, “Coexisting attractors in periodically modulated logistic maps”, Physical Review E, volume 77(6), p. 066217, 2008.
[33] J. C. Sprott, X. Wang, and G. Chen, “Coexistence of point, periodic and strange attractors”, International Journal of Bifurcation and Chaos, volume 23(05), p. 1350093, 2013.
[34] E. N. Lorenz, “Deterministic nonperiodic flow”, Journal of the Atmospheric Sciences, volume 20(2), pp. 130–141, 1963.
[35] N. Shuichi, “Constant temperature molecular dynamics methods”, Progress of Theoretical Physics Supplement, volume 103, pp. 1–46, 1991.
[36] W. G. Hoover, “Remark on “some simple chaotic flows””, Physical Review E, volume 51(1), p. 759, 1995.
[37] V. Chizhevsky, “Multistability in dynamical systems induced by weak periodic perturbations”, Physical Review E, volume 64(3), p. 036223, 2001.
[38] A. Pisarchik and B. Goswami, “Annihilation of one of the coexisting attractors in a bistable system”, Physical Review Letters, volume 84(7), p. 1423, 2000.

指導教授

黎璧賢、陳志強(Pik-yin Lai Chi-keung Chan)

審核日期

2016-1-28

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