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## A new method for selecting arbitrary Poincare section

Zhang Shi, Wang Pan, Zhang Rui-Hao, Chen Hong
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• #### 摘要

针对用庞加莱截面分析非线性系统时较难选择合适截面的问题, 提出了一种投影时域法. 该方法可以在时域上直观地判断所选平面是否合适并实时准确地调整平面的方向与位置. 文中给出了投影时域法的完整定义并从理论上详细分析了该方法的原理; 同时研究了在时域上判断所选平面是否是合适的庞加莱截面的条件以及如何调整所选平面的方向与位置; 最后, 利用投影时域法对三种典型的三维或四维非线性系统进行了仿真实验, 实验结果证明了该方法的有效性和实用性.

#### Abstract

Poincare section is an important method for analyzing nonlinear systems. Choosing a suitable plane as the Poincare section is the key to using the Poincare section to analyze a nonlinear system. At present, it is still a difficult problem to select a suitable Poincare section when analyzing a nonlinear system. This is caused by two reasons. On the one hand, the classical method for selecting a partial Poincare section only applies to analyze a part of the nonlinear system orbit, whether the selected plane is a suitable Poincare section is affected by the different initial points. On the other hand, according to the actual situation, different researchers have different needs for Poincare section. In order to solve this problem, a new method named Projection Time Domain method is put forward in this paper. This method can help us not only directly reflect the intersection between the nonlinear system orbit and the selected plane, but also accurately adjust the direction and position of the selected plane in real time. It can be used to quickly find a plane which fully intersects the nonlinear system orbit or an arbitrary plane as a Poincare section. In this paper, the complete definition of Projection Time Domain method is given firstly. Then, the principle of Projection Time Domain method is theoretically analyzed in detail. At the same time, the rules for determining whether the selected plane is a suitable Poincare section in the time domain are also studied. Finally, it is introduced how to quantify the direction and position of the selected plane in the phase space. The simulation experiments are conducted with three typical three-dimensional and four-dimensional nonlinear systems by using this new method. The experimental results consistent with the theoretical analysis, which demonstrate the effectiveness and practicability of this method.

#### 作者及机构信息

###### 通信作者: 陈红, chenhongdeepred@163.com
• 基金项目: 国家级-国家自然科技基金(61471158)

#### 施引文献

• 图 1  轨道在平面上的投影

Fig. 1.  The projection of a trajectory onto a plane.

图 2  $F({l_1}(t), {l_2}(t))$${l_3}(t)$的波形

Fig. 2.  Waveform of $F({l_1}(t), {l_2}(t))$ and ${l_3}(t)$.

图 3  涡卷的相轨与波形　(a)相轨; (b)波形

Fig. 3.  Phase trajectoy and waveform of vortex: (a) Phase trajectoy; (b) waveform.

图 4  平面一的实验结果　(a)时域波形; (b)相图

Fig. 4.  Experimental results of the first plane: (a) Waveform; (b) phase diagram.

图 5  平面二的实验结果　(a)时域波形; (b)相图

Fig. 5.  Experimental results of the second plane: (a) Waveform;(b) phase diagram.

图 6  平面三的实验结果　(a)时域波形; (b)相图

Fig. 6.  Experimental results of the third plane: (a) Waveform; (b) phase diagram.

图 7  平面一的实验结果　(a) C = 0时的波形; (b) C = 0时的相图; (c) C = 1.5时的波形; (d) C = 1.5时的相图; (e) C = –1.5时的波形; (f) C = –1.5时的相图

Fig. 7.  Experimental results of the first plane: (a) Waveform while C = 0; (b) phase diagram while C = 0; (c) waveform while C = 1.5; (d) phase diagram while C = 1.5; (e) waveform while C = –1.5; (f) phase diagram while C = –1.5.

图 8  平面二的实验结果　(a)波形; (b)相图

Fig. 8.  Experimental results of the second plane: (a) Waveform; (b) phase diagram.

图 9  $w = 0$时的实验结果　(a)波形; (b)相图

Fig. 9.  Experimental results while $w = 0$: (a) Waveform; (b) phase diagram.

图 10  $z = 0$时的实验结果　(a)波形; (b)相图

Fig. 10.  Experimental results while $z = 0$: (a) Waveform; (b) phase diagram.

图 11  $y = 0$时的实验结果　(a)波形; (b)相图

Fig. 11.  Experimental results while $y = 0$: (a) Waveform; (b) phase diagram.

图 12  $x = 0$时的实验结果　(a)波形; (b)相图

Fig. 12.  Experimental results while $x = 0$: (a) Waveform; (b) phase diagram.