對於漏斗型若斯勒吸子觸發動力學及同步化的研究

Investigation of the spiking dynamics and synchronization of the funnel-type Rössler attractors

鄭安良(An-Liang Cheng)

指導教授 ： 陳義裕

國立臺灣大學/理學院/物理學研究所/博士(2018年)

https://doi.org/10.6342/NTU201800290

摘要

我們研究了漏斗型若斯勒吸子的觸發動力學以及同步穩定性。首先，我們研究了漏斗型若斯勒吸子連續觸發的動力學。利用合適的平均方法以及連接公式，我們可以把原本相當困難分析的系統所展現的時間連續混沌行為，化簡為只有四個參數的遞迴關係式。這個方法的優點是可以幫助我們觀察尖峰高度和兩個尖峰期間如何隨時間演化，以及當我們改變系統參數時的變化。我們也研究了兩個耦合漏斗型若斯勒吸子的同步穩定性。對於耦合漏斗型若斯勒吸子的同步穩定性，有時候可以近似若斯勒吸子的軌道為在平面上向外螺旋的運動來有效的描述。我們證明這個近似方法只對於研究耦合螺旋型若斯勒吸子的同步穩定性有效。但是當處理耦合的漏斗型若斯勒吸子時，我們必須要考慮到連續觸發行為所造成的貢獻。利用適當的時間權重平均方法，我們可以重建出耦合漏斗型若斯勒吸子的同步穩定性並得到和原本數值解符合的結果。我們也分析研究了分離向量如何隨時間演化，並證明了當執行時間權重平均方法時，如何選取合適的初始分離向量對於重建同步穩定性的重要性。

關鍵字

若斯勒吸子 ； 螺旋型若斯勒吸子 ； 漏斗型若斯勒吸子 ； 連接公式 ； 混沌同步化 ； 李亞普諾夫指數

並列摘要

The spiking dynamics and synchronous stability of the funnel-type Rössler attractor are studied. First, we investigate the dynamics of the consecutive triggering behavior in a funnel-type Rössler attractor. Using a suitable averaging method and connection formulas, we reduce the much more difficult time continuous chaotic behavior of the original system into a set of recursion relations involving only four parameters. This approach has the merits of helping one see more easily how the height of the peaks and the peak-to-peak durations behave and vary as one tunes the system parameters. We also study the synchronous stability of the coupled funnel-type Rössler attractors. The study of the synchronous stability of two coupled Rössler attractors sometimes
can be effectively described by approximating the trajectory on the attractor as an outward planar spiral. We show that this is true only when one is dealing with
the spiral-type attractor. But when the equally important funnel-type attractor is encountered, a properly constructed time-weighted average must be used to yield a prediction that agrees well with the original numerical results. We also show analytically how the separation vector evolves in time, and demonstrate why this study matters when one tries to perform the time-weighted average.

並列關鍵字

Rössler attractor ； Spiral-type Rössler attractor ； Funnel-type Rössler attractor ； Connection formula ； Synchronization of chaos ； Lyapunov exponent

參考文獻

[24] A. L. Cheng, Y. Y. Chen, Chaos: An Interdisciplinary Journal of Nonlinear Science, 27, 073117 (2017).

[46] L. Huang, Q. Chen, Y. C. Lai, L. M. Pecora, Phys. Rev. E, 80, 036204 (2009).

[7] L. Chen, C. Qiu, H. Huang, G. Qi, J. Wang, Phys. Lett. A, 375, 149–154 (2010).

[48] A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, Physica D, 16, 285-317 (1985).

[17] J. Kurths, M. C. Romano, M. Thiel, G. V. Osipov, M. V. Ivanchenko, I. Z. Kiss, J. L. Hudson, Nonlinear Dyn., 44, 135–149 (2006).