环域反馈下复Ginzburg-Landau系统的螺旋波动力学行为

doi:10.19596/j.cnki.1001-246x.8360

摘要/Abstract

摘要：

针对自激振荡系统的复金兹伯格-朗道(Complex Ginzbury-Landau, 简称CGL)方程, 研究圆形环域与方形环域两种反馈控制下的螺旋波动力学。结果表明: 圆形环域反馈控制下, 螺旋波波头通常经过一段过渡漂移后进入圆形吸引子, 圆形吸引子的半径以及反馈刚启动时波头的漂移方向随环域参数呈周期性变化, 过渡漂移阶段波头轨道的平缓程度与复反馈信号模的时间函数中钟形部分的陡度有关, 且反馈增益的正负与大小也会影响受控螺旋波的动力学行为。方形环域反馈控制下的螺旋波波头的吸引子更为丰富, 主要包括方形吸引子、小的极限环吸引子、菱形吸引子以及点吸引子, 点吸引子通常位于方形环域的两条对角线上, 且波头运动随环域控制参数呈现规律性变换。

关键词: 螺旋波, Ginzburg-Landau系统, 反馈控制, 吸引子

Abstract:

We studied dynamics of spiral waves subjected to a feedback control, where the feedback signal is obtained with integral in an annular region.As the annular region is circular, the spiral tip usually enters a circular attractor after a transient drift, and radius of the attractor varies periodically with parameters of the annular region.With the change of these parameters, drift direction of the spiral tip at the beginning of feedback also has a periodic change.In the phase of transient drift, modulus function of the complex feedback signal is bell-shaped, and the steepness of the bell is related to the smoothness of the drift trajectory.The sign and value of the feedback gain affect the dynamical behavior of the controlled spiral.As the annular region is square, final attractors of the spiral tip are more abundant, including mainly square attractor, limit circle attractor with small amplitude, diamond attractor and point attractor, where the point attractors are usually on the diagonal lines of the square.Here, trajectory of the spiral tip has a regular change with control parameters of the annual region as well.

Key words: spiral wave, Ginzburg-Landau system, feedback control, attractor

陈绍英, 王雪丽, 高志梅, 袁国勇. 环域反馈下复Ginzburg-Landau系统的螺旋波动力学行为[J]. 计算物理, 2022, 39(1): 118-126.

Shaoying CHEN, Xueli WANG, Zhimei GAO, Guoyong YUAN. Dynamics of Spiral Waves in Complex Ginzburg-Landau Systems Subjected to Feedback Derived from an Annular Domain[J]. Chinese Journal of Computational Physics, 2022, 39(1): 118-126.

图/表 11

图1 环域内圆半径不同时的螺旋波快照与波头运动轨道(a) rfb1=20；(b) rfb1=40；(c) rfb1=60 (两个粉色圆分别为测量环域的内外边界，黑色圆点描述反馈启动前波头的位置，黑色线为反馈启动后波头的运动轨迹。)

Fig.1 Spiral snapshots and spiral tip tractories (a) rfb1=20; (b) rfb1=40; (c) rfb1=60 (The two pink circles are inner and outer boundaries of the annular measuring region. The black dot describes location of the spiral tip before the feedback is switched on. The black line is the tip trajectory after the feedback is applied.)

图2 (a) 四种不同rfb1下波头的运动轨道; (b)圆形吸引子半径rtip随环域内半径rfb1的变化

Fig.2 (a) Tip trajectories at four rfb1; (b) Attractor radius rtip as a function of inner radius rfb1 of a annular measuring region under the control of the feedback signal

图3 (a) 五种不同h下波头的运动轨道；(b)圆形吸引子半径rtip随环域宽度h的变化

Fig.3 (a) Tip trajectories at five h; (b) Variation of rtip with h under the control of a feedback signal

图4 正负增益下螺旋波波头的运动(a) rfb1=28; (b) rfb1=30; (c) rfb1=40; (d) rfb1=42 (黑色轨道或点对应正增益kfb=0.02，红色轨道或点对应负增益kfb=-0.02。)

Fig.4 Spiral tip trajectiories under positive and negative gains (a) rfb1=28; (b) rfb1=30; (c) rfb1=40; (d) rfb1=42 (The black trajectories and point attractors are obtained under kfb=0.02, and the case of kfb=-0.02 is illustrated by red color.)

图5 正反馈增益大小对螺旋波动力学的影响

Fig.5 Effect of positive feedback gain on spiral dynamics

图6 负反馈增益对螺旋波动力学的影响(h=30, rfb1=42) (a) kfb=-0.015; (b) kfb=-0.02; (c) kfb=-0.025; (d) kfb=-0.035

Fig.6 Effect of negative feedback gain on spiral dynamics(h=30 and rfb1=42) (a) kfb=-0.015; (b) kfb=-0.02; (c) kfb=-0.025; (d) kfb=-0.035

图7 过渡漂移阶段波头的轨道与相应的复反馈信号模时间函数

Fig.7 Transient drift tractories and modular functions of the complex feedback signal

图8 环域参数dfb1不同取值对应的波头轨道

Fig.8 Trajectories of the spiral tip at different dfb1

图9 环域参数l不同取值对应的波头轨道(a)2 ≤l≤20; (b) 21≤l≤46

Fig.9 Trajectories of the spiral tip at different l (a) 2≤l≤20; (b) 21≤l≤46

图10 正反馈增益kfb对过渡漂移轨道的影响

Fig.10 Effect of kfb on drift trajectiory in the transient phase

图11 负反馈增益对螺旋波动力学的影响(dfb1=20, l=5) (a) kfb=-0.015; (b) kfb=-0.02; (c) kfb=-0.0225; (d) kfb=-0.025

Fig.11 Effect of negative feedback gain on spiral dynamics (dfb1=20, l=5) (a) kfb=-0.015; (b) kfb=-0.02; (c) kfb=-0.0225; (d) kfb=-0.025

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