基于现场可编程逻辑门阵列的磁控忆阻电路对称动力学行为分析

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摘要

将含绝对值项的磁控忆阻器引入改进型蔡氏电路, 构建新型磁控忆阻混沌电路, 通过分岔图与Lyapunov指数谱创新性地观察到系统的对称分岔行为, 揭示系统双参数平面内运动状态分布的对称性. 同时, 基于忆阻电路参数-初值平面的系统运动分布图, 分析对称吸引域内系统的多稳态特性, 相图的绘制进一步证明电路多稳态现象的存在性. 此外, 应用现场可编程逻辑门阵列完成电路实验, 在数字示波器上捕捉实验结果, 证明所构磁控忆阻电路的物理可实现性.

关键词:

Abstract

The lack of the relationship between flux and charge has been made up for by the memristor which is suitable to constructing chaotic circuits as a nonlinear element. Commonly, the memristor-based chaotic systems are constructed by introducing the model of memristor into various classical nonlinear circuits, and more special and abundant dynamic behaviors are existent in these memristive systems. With the deepening of research, several novel nonlinear phenomena of memristor circuits have been found, such as hidden attractors, self-excited attractors and anti-monotonic characteristic. Meanwhile, multistability of a memristor-based circuit explained by the coexistence of multiple attractors with different topological structures is a typical phenomenon in a nonlinear system, and it is also one of the hotspots in this field. In addition, the chaotic sequences generated by the memristive circuits are used as additional signals for information transmission or image encryption. Therefore, the study of modeling memristor systems and analyzing various nonlinear behaviors is of certain valuable. In this paper, a four-dimensional flux-controlled memeristive circuit is constructed by introducing an active memristor with absolute value into an improved Chua’s circuit, and the special dynamic behaviors are observed. Through the bifurcation diagrams and Lyapunov exponent spectra, the symmetric bifurcations are shown, and the symmetric system states in parameter mappings are found. Besides, the distribution maps of memristive circuit are used to analyze the multistability in a symmetrical attraction domain, and the corresponding phase diagrams are depicted to confirm the existence of multistability. Furthermore, the circuit experiments of the flux-controlled memeristive circuit are implemented by the field programmable gate array simulation, and the experimental results are obtained on a digital oscilloscope, which proves the physical implementability of the memristor-based system.

Keywords:

作者及机构信息

南京师范大学电气与自动化工程学院，南京　210023

通信作者: 闵富红, minfuhong@njnu.edu.cn

基金项目: 国家自然科学基金(批准号: 61871230)资助的课题.

Authors and contacts

School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210023, China

Corresponding author: Min Fu-Hong, minfuhong@njnu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61871230).

文章全文 : translate this paragraph

参考文献

[1]	Chua L O 1971 IEEE Trans. Circ. Theory 18 507 Google Scholar
[2]	林毅, 刘文波, 沈骞 2018 物理学报 67 230502 Google Scholar Lin Y, Liu W B, Shen Q 2018 Acta Phys. Sin. 67 230502 Google Scholar
[3]	Olumodeji O A, Gottardi M 2017 Integration 58 438 Google Scholar
[4]	Joglekar Y N, Wolf S J 2009 Eur. J. Phys. 30 661 Google Scholar
[5]	Guo M, Gao Z H, Xue Y B, Dou G, Li Y X 2018 Nonlinear Dyn. 93 1681 Google Scholar
[6]	Wang C H, Liu X M, Xia H 2017 Chaos 27 033114 Google Scholar
[7]	Peng G Y, Min F H 2017 Nonlinear Dyn. 90 1607 Google Scholar
[8]	Li C, Min F H, Li C B 2018 Nonlinear Dyn. 94 2785 Google Scholar
[9]	Peng G, Min F 2018 Computer Electr. Eng. Article ID 86492 9
[10]	Feng W, He Y G, Li C L 2018 Complexity 2018 1
[11]	Abuelma'Atti M T, Khalifa Z J 2016 Int.J. Electr. Eng. 53 280 Google Scholar
[12]	Bao B C, Li Q D, Wang N 2016 Chaos 26 043111 Google Scholar
[13]	Leonov G A, Kuznetsov N V, Vagaitsev V I 2011 Phys. Lett. A 375 2230 Google Scholar
[14]	Kengne J, Negou A N, Tchiotsop D 2017 Nonlinear Dyn. 27 1
[15]	Bao B C, Xu L, Wang N, Bao H, Xu Q, Chen M 2018 Int. J. Electr. Com. 94 26 Google Scholar
[16]	王伟, 曾以成, 孙睿婷 2017 物理学报 66 040502 Google Scholar Wang W, Zeng Y C, Sun R T 2017 Acta Phys. Sin. 66 040502 Google Scholar
[17]	Min F H, Li C, Zhang L, Li C B 2019 Chin. J. Phys. 58 117 Google Scholar
[18]	Feudel U, Kraut S 2008 Int. J. Bifurcat. Chaos 18 1607 Google Scholar
[19]	Ngonghala C N, Feudel U 2011 Phys. Rev. E: Stat. Nonlinear Soft. Matter Phys. 83 056206 Google Scholar
[20]	Bao H, Wang N, Bao B C, Chen M, Jin P P, Wang G Y 2018 Commun. Nonlinear Sci. Numer. Simulat. 57 264 Google Scholar
[21]	Wang G Y, Yuan F, Chen G R, Zhang Y 2018 Chaos 28 013125 Google Scholar
[22]	Da C, Rafael A, Eisencraf M 2019 Commun. Nonlinear Sci. Numer. Simulat. 72 441 Google Scholar
[23]	Yavuz O, Erdem E 2019 Opt. Laser Technol. 114 224 Google Scholar
[24]	Njitacke Z T, Kengne J 2017 Chaos, Sol. Frac. 105 77 Google Scholar
[25]	Li C B, Akgul A, Sprott J C, Lu H H C, Thio W J C 2018 Int. J. Circ. Theor. Appl. 46 2434 Google Scholar
[26]	Li C B, Sprott J C, Liu Y J, Gu Z Y, Zhang J W 2018 Int. J. Bifurcat. Chaos 28 1850163 Google Scholar
[27]	Bao B C, Xu J P, Zhou G H, Ma Z H, Zou L 2011 Chin. Phys. B 20 109

施引文献

图 1 电路模型　(a)磁控忆阻电路; (b)磁控忆阻等效电路

Fig. 1. Circuit schematic: (a) Flux-controlled memristor circuit; (b) equivalent Circuit of flux-controlled memristor.

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图 2 $y-z$平面上典型混沌吸引子的相图与Poincaré截面图　(a)相图; (b) Poincaré截面图

Fig. 2. Phase portrait and Poincaré map of typical chaotic attractor in $y-z$ plane: (a) Phase portrait; (b) Poincaré map

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图 3 随参数$\gamma $变化的分岔图与Lyapunov指数谱　(a)分岔图; (b)Lyapunov指数谱

Fig. 3. Bifurcation and Lyapunov exponent spectrum with parameter $\gamma $: (a) Bifurcation diagram; (b) Lyapunov exponent spectrum

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图 4 随参数c变化的分岔图与Lyapunov指数谱　(a)分岔图; (b) Lyapunov指数谱

Fig. 4. Bifurcation and Lyapunov exponent spectrum with parameter $c$: (a) Bifurcation diagram; (b) Lyapunov exponent spectrum.

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图 5 双参数吸引盆　(a)参数$\gamma -b$; (b)参数$c-b$; (c)参数$\gamma -\xi $; (d)参数$c-\xi $

Fig. 5. Parameter mappings: (a) Parameter $\gamma $ and $b$; (b) parameter $c$ and $b$; (c) parameter $\gamma $ and $\xi $; (d) parameter $c$ and $\xi $

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图 6 不同变量组合下的共存吸引盆　(a) $\gamma -x\left(0\right)$平面, 初始条件为$\left(x\left(0\right),0,0,0\right)$; (b)$c-x\left(0\right)$平面, 初始条件为$\left(x\left(0\right),0,0,0\right)$

Fig. 6. Attraction basins of coexistence in different planes: (a) $\gamma -x\left(0\right)$ plane, with initial value of $\left(x\left( 0 \right),0,0,0\right)$; (b) $c-x\left(0\right)$ plane, with initial value of $\left(x\left(0\right),0,0,0\right)$

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图 7 不同变量组合下系统状态分布图　(a) $\gamma -w(0)$平面, 初始条件为$( - {10^{ - 9}},0,0,w(0))$; (b)$c-w(0)$平面, 初始条件为$( - {10^{ - 9}},0,0,w(0))$; (c)$\gamma -w(0)$平面, 初始条件为$({10^{ - 9}},0,0,w(0))$; (d)$c-w(0)$平面, 初始条件为$({10^{ - 9}},0,0,w(0))$

Fig. 7. Attraction basins of coexistence in different planes: (a) $\gamma -w(0)$ plane, with initial value of $( - {10^{ - 9}},0,0,w(0))$; (b) $c-w(0)$ plane, with initial value of $( - {10^{ - 9}},0,0,w(0))$; (c) $\gamma -w(0)$ plane, with initial value of $({10^{ - 9}},0,0,w(0))$; (d) $c-w(0)$ plane, with initial value of $({10^{ - 9}},0,0,w(0))$

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图 8 参数$\gamma {\rm{ = 0}}{\rm{.74}}$, $x-z$平面上不同初值下的多种共存吸引子　(a)左右共存周期3; (b)左右共存混沌与左右共存周期1

Fig. 8. For different initial value, phase diagram of coexisting attractors in $x-z$ planes when $\gamma {\rm{ = 0}}{\rm{.74}}$: (a) Coexisting attractors of period-3; (b) coexisting attractors of chaos and period-1

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图 9 参数$c = 1.274$, $x-z$平面上不同初值下的多种共存吸引子　(a)左右共存周期3; (b)左右共存混沌与左右共存周期1

Fig. 9. For different initial value, phase diagram of coexisting attractors in $x-z$ planes when $c = 1.274$: (a) Coexisting attractors of period-3; (b) coexisting attractors of chaos and period-1.

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图 10 参数$\gamma {\rm{ = 0}}{\rm{.74}}$, $x-z$平面上不同初值下的多种共存吸引子　(a)左右共存周期3、左右共存周期1与稳定不动点; (b)左右共存混沌与左右共存周期2

Fig. 10. For different initial value, phase diagram of coexisting attractors in $x-z$ planes when $\gamma {\rm{ = 0}}{\rm{.74}}$: (a) Coexisting attractors of period-3, period-1 and fixed point; (b) coexisting attractors of chaos and period-2

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图 11 参数$c = 1.274$, $x-z$平面上不同初值下的多种共存吸引子　(a)左右共存周期3与稳定不动点; (b)左右共存混沌与左右共存周期1

Fig. 11. For different initial value, phase diagram of coexisting attractors in $x-z$ planes when $c = 1.274$: (a) Coexisting attractors of period-3 and fixed point; (b) coexisting attractors of chaos and period-1

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图 12 顶层模块控制流程图

Fig. 12. The flow chart of calling order

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图 13 FPGA实物连接图与实现结果　(a)实物连接图; (b)y-z平面相图; (c)y, z两项时序图

Fig. 13. The hardware connection diagram and the result of implementation: (a) The hardware connection diagram; (b) phase diagram in y-z plane; (c) timing diagram of the term y and z

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图 14 固定参数$c = 1.274$时x-z平面内不同初值条件下的共存, Ch1 = 500 MV, Ch2 = 500 MV　(a)初值为$\left( {{\rm{1}}{{\rm{0}}^{ - {\rm{9}}}}{\rm{,0,0,0}}} \right)$, 左侧周期1; (b)初值为$\left( {{\rm{1}}{{\rm{0}}^{ - {\rm{9}}}}{\rm{,0,0,0}}{\rm{.45}}} \right)$, 左侧周期3; (c)初值为$\left( {{\rm{1}}{{\rm{0}}^{ - {\rm{9}}}}{\rm{,0,0,0}}{\rm{.9}}} \right)$, 左侧混沌; (d)初值为$\left( { - {\rm{1}}{{\rm{0}}^{ - {\rm{9}}}}{\rm{,0,0,0}}} \right)$, 右侧周期1; (e)初值为$\left( { - {\rm{1}}{{\rm{0}}^{ - {\rm{9}}}}{\rm{,0,0,}} - {\rm{0}}{\rm{.45}}} \right)$右侧周期3; (f)初值为$\left( { - {\rm{1}}{{\rm{0}}^{ - {\rm{9}}}}{\rm{,0,0,}} - {\rm{0}}{\rm{.9}}} \right)$, 右侧混沌

Fig. 14. The phase diagram of coexistence attractors with different initial conditions at $c = 1.274$ in x-z plane, Ch1 = 500 MV, Ch2 = 500 MV: (a) The initial value as $\left( {{\rm{1}}{{\rm{0}}^{ - {\rm{9}}}}{\rm{,0,0,0}}} \right)$, left period-1; (b) the initial value as $\left( {{\rm{1}}{{\rm{0}}^{ - {\rm{9}}}}{\rm{,0,0,0}}{\rm{.45}}} \right)$, left period-3; (c) the initial value as $\left( {{\rm{1}}{{\rm{0}}^{ - {\rm{9}}}}{\rm{,0,0,0}}{\rm{.9}}} \right)$, left period-3; (d) the initial value as $\left( { - {\rm{1}}{{\rm{0}}^{ - {\rm{9}}}}{\rm{,0,0,0}}} \right)$, right chaos; (e) the initial value as $\left( { - {\rm{1}}{{\rm{0}}^{ - {\rm{9}}}}{\rm{,0,0,}} - {\rm{0}}{\rm{.45}}} \right)$, right period-3; (f) the initial value as $\left( { - {\rm{1}}{{\rm{0}}^{ - {\rm{9}}}}{\rm{,0,0,}} - {\rm{0}}{\rm{.9}}} \right)$, right chaos

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表 1 系统参数

Table 1. The valueof system parameters

参数	数值	参数	数值
$a$	1	$\xi $	0.12
$b$	3.5	$\alpha $	0.3
$c$	1	$\beta $	0.8
$\gamma $	0.86

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表 2 参数$\gamma $, c变化时系统运动状态与对应的Lyapunov指数

Table 2. The dynamic behavior and Lyapunov exponent with parameter $\gamma $ and $c$

参数$\gamma $	运动状态	Lyapunov指数
$(0.6{\rm{6}},0.{\rm{704}})$	稳定不动点	$( -, -, -, - )$
$(0.{\rm{704}},0.8{\rm{08}}) \cup (0.{\rm{829}},0.{\rm{845}})$	周期运动	$( + , -, -, - )$
${\rm{(0}}{\rm{.808,}}\,{\rm{0}}{\rm{.829)}} \cup {\rm{(0}}{\rm{.845,}}\,{\rm{0}}{\rm{.9)}}$	复杂运动(混沌, 多周期)	$( +,0, -, - )$
参数c	运动状态	Lyapunov指数
${\rm{(0}}{\rm{.9,}}\,{\rm{1}}{\rm{.02)}} \cup {\rm{(1}}{\rm{.07,1}}{\rm{.13)}}$	复杂运动(混沌, 多周期)	$( +,0, -, - )$
${\rm{(1}}{\rm{.02,}}\,{\rm{1}}{\rm{.07)}} \cup {\rm{(1}}{\rm{.13,}}\,{\rm{1}}{\rm{.41)}}$	周期运动	$( + , -, -, - )$
$({\rm{1}}{\rm{.41,1}}{\rm{.5}})$	稳定不动点	$( -, -, -, - )$

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表 3 不同颜色所对应的系统运动状态

Table 3. Colors and the corresponding system states

颜色	系统运动
紫色	稳定不动点
蓝色	周期1
绿色	周期2
黄色	周期3
红色	复杂运动(混沌, 多周期)

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表 4 运动状态与色标的对应表

Table 4. Different colors and the corresponding dynamical state

颜色	浅蓝	绿色	黄色	红色	紫色
共存类型	左侧周期1	左侧周期2	左侧周期3	左侧复杂运动(左侧多周期, 混沌)	稳定不动点
颜色	深蓝	青色	草绿	橙色
共存类型	右侧周期1	右侧周期2	右侧周期3	右侧复杂运动(右侧多周期, 混沌)

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表 5 不同初值对应的共存多吸引子类型

Table 5. Coexisting multiple attractor with different initial condition

参数	吸引子类型	初始条件
$\gamma = 0.{\rm{74}}$	左右共存点吸引子	$\left( \pm {10^{ - 9}},0,0, \mp 0.{\rm{45}}\right)$
	左右共存周期1	$\left( \pm 0.1,0,0,0\right)$,$\left( \pm {10^{ - 9}},0,0, \pm {\rm{0}}{\rm{.45}}\right)$
	左右共存周期2, 左右共存周期3	$\left( \pm {\rm{1}}{{\rm{0}}^{ - {\rm{9}}}},0,0, \pm {\rm{0}}{\rm{.45}}\right)$, $ \left( \pm 0.{\rm{4}},0,0,0\right) $, $\left( \pm {10^{ - 9}},0,0 \pm 0.{\rm{5}}\right)$
	左右共存混沌	$\left( \pm 0.{\rm{8}},0,0,0\right)$, $\left( \pm {10^{ - 9}},0,0, \pm 0.9\right)$
$c = 1.274$	左右共存点吸引子	$\left( \pm {10^{ - 9}},0,0, \mp 0.{\rm{45}}\right)$
	左右共存周期1	$\left( \pm 0.1,0,0,0\right)$, $\left( \pm {10^{ - 9}},0,0,0\right)$
	左右共存周期3	$\left( \pm 0.45,0,0,0\right)$, $\left( \pm {10^{ - 9}},0,0 \pm 0.4{\rm{5}}\right)$
	左右共存混沌	$\left( \pm 0.{\rm{8}},0,0,0\right)$, $\left( \pm {10^{ - 9}},0,0, \pm 0.9\right)$

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[1]	Chua L O 1971 IEEE Trans. Circ. Theory 18 507 Google Scholar
[2]	林毅, 刘文波, 沈骞 2018 物理学报 67 230502 Google Scholar Lin Y, Liu W B, Shen Q 2018 Acta Phys. Sin. 67 230502 Google Scholar
[3]	Olumodeji O A, Gottardi M 2017 Integration 58 438 Google Scholar
[4]	Joglekar Y N, Wolf S J 2009 Eur. J. Phys. 30 661 Google Scholar
[5]	Guo M, Gao Z H, Xue Y B, Dou G, Li Y X 2018 Nonlinear Dyn. 93 1681 Google Scholar
[6]	Wang C H, Liu X M, Xia H 2017 Chaos 27 033114 Google Scholar
[7]	Peng G Y, Min F H 2017 Nonlinear Dyn. 90 1607 Google Scholar
[8]	Li C, Min F H, Li C B 2018 Nonlinear Dyn. 94 2785 Google Scholar
[9]	Peng G, Min F 2018 Computer Electr. Eng. Article ID 86492 9
[10]	Feng W, He Y G, Li C L 2018 Complexity 2018 1
[11]	Abuelma'Atti M T, Khalifa Z J 2016 Int.J. Electr. Eng. 53 280 Google Scholar
[12]	Bao B C, Li Q D, Wang N 2016 Chaos 26 043111 Google Scholar
[13]	Leonov G A, Kuznetsov N V, Vagaitsev V I 2011 Phys. Lett. A 375 2230 Google Scholar
[14]	Kengne J, Negou A N, Tchiotsop D 2017 Nonlinear Dyn. 27 1
[15]	Bao B C, Xu L, Wang N, Bao H, Xu Q, Chen M 2018 Int. J. Electr. Com. 94 26 Google Scholar
[16]	王伟, 曾以成, 孙睿婷 2017 物理学报 66 040502 Google Scholar Wang W, Zeng Y C, Sun R T 2017 Acta Phys. Sin. 66 040502 Google Scholar
[17]	Min F H, Li C, Zhang L, Li C B 2019 Chin. J. Phys. 58 117 Google Scholar
[18]	Feudel U, Kraut S 2008 Int. J. Bifurcat. Chaos 18 1607 Google Scholar
[19]	Ngonghala C N, Feudel U 2011 Phys. Rev. E: Stat. Nonlinear Soft. Matter Phys. 83 056206 Google Scholar
[20]	Bao H, Wang N, Bao B C, Chen M, Jin P P, Wang G Y 2018 Commun. Nonlinear Sci. Numer. Simulat. 57 264 Google Scholar
[21]	Wang G Y, Yuan F, Chen G R, Zhang Y 2018 Chaos 28 013125 Google Scholar
[22]	Da C, Rafael A, Eisencraf M 2019 Commun. Nonlinear Sci. Numer. Simulat. 72 441 Google Scholar
[23]	Yavuz O, Erdem E 2019 Opt. Laser Technol. 114 224 Google Scholar
[24]	Njitacke Z T, Kengne J 2017 Chaos, Sol. Frac. 105 77 Google Scholar
[25]	Li C B, Akgul A, Sprott J C, Lu H H C, Thio W J C 2018 Int. J. Circ. Theor. Appl. 46 2434 Google Scholar
[26]	Li C B, Sprott J C, Liu Y J, Gu Z Y, Zhang J W 2018 Int. J. Bifurcat. Chaos 28 1850163 Google Scholar
[27]	Bao B C, Xu J P, Zhou G H, Ma Z H, Zou L 2011 Chin. Phys. B 20 109

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基于现场可编程逻辑门阵列的磁控忆阻电路对称动力学行为分析