朱建良等一个新的四维混沌系统及其电路实现第2 期 x 1 = a x2 x1 + x2 x3 x4 x 2 = b x1 + x2 x1 x3 x4 2 x 3 = c x1 x3 + x1 x2 x4 x 4 = ex3 dx4 + x1 x2 x3 系统具有自然的对称性即作如下变换后系统

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1 Vol 1 No 2 JOURNAL OF HARBIN UNIVERSITY OF SCIENCE AND TECHNOLOGY Apr O41 TM1 A A New Four-dimensional Chaotic System and Its Circuitry Implementation ZHU Jian-liang ZHAO Hong-chao School of Electrical and Electronic Engineering Harbin University of Science and Technology Harbin 18 China Abstract In order to generate complex chaotic attractors a new four-dimensional chaotic system is built in this paper which gives birth to a true four-winged chaotic attractor graph in time domain and power spectrum were drawn electronic circuit was designed to realize the chaotic system Experimental chaotic behaviors of the system were found to be identical to the simulations and it is easy to realize in circuit The circuit is important to secure communication and practical applications in communication encryption Key words four-dimensional chaotic system numerical simulation circuitry implementation x 1 = a x 2 - x 1 + x 2 x 3 x 4 x 2 = b x 1 + x 2 - x 1 x 3 x 4 1 x 3 = - cx 3 + ex 2 + x 1 x 2 x x 4 = - dx 4 + x 1 x 2 x zhu_jianliang@ 163 com 1984

朱建良等一个新的四维混沌系统及其电路实现第2 期 x 1 = a x2 x1 + x2 x3 x4 x 2 = b x1 + x2 x1 x3 x4 2 x 3 = c x1 x3 + x1 x2 x4 x 4

如图 1 所示分别为 LE1 = 24 983 LE2 = 12 996 6 LE3 = 2 48 7 LE4 = 9 936 4 可知该系统产生混沌 x1 x2 x3 x4 x1 x2 x3 x4 3 变换可表示为

1%%%%2%%%%3%%%%4%%%%%%%6%%%%7%%%%8%%%%9%%%1 Time/ms 混沌系统的 Lyapunov 指数 Lyapunov 指数是衡量系统动力学特性的一个图1 系统的

混沌系统的 MATLAB 仿真令 a = b = 1 c = 12 4 d = 2 e = 3 此时 2 2 1 1 重要定量指标对于系统是否存在动力学混沌可以从最大 Lyapunov

1-2%%%%%%%-1%%%%%%%%%%%%%%%%%%%%1%%%%%%%%%%2 a x1-x2-1 -1-2 -2-2%%%%%%%-1%%%%%%%%%%%%%%%%%%%%1%%%%%%%%%%2 d

2 朱建良等一个新的四维混沌系统及其电路实现第2 期 x 1 = a x2 x1 + x2 x3 x4 x 2 = b x1 + x2 x1 x3 x4 2 x 3 = c x1 x3 + x1 x2 x4 x 4 = ex3 dx4 + x1 x2 x3 系统具有自然的对称性即作如下变换后系统保持不变会随着时间的演化而成指数率的增加以致达到无法预测这就是混沌现象通过数值计算知该系统具有四个 Lyapunov 指数如图 1 所示分别为 LE1 = LE2 = LE3 = LE4 = 可知该系统产生混沌 x1 x2 x3 x4 x1 x2 x3 x4 3 变换可表示为 Lyapunov%Exponents P R4 R4 x PxP = 1 它满足方程 4 4 f Px = Pf x %%%%2%%%%3%%%%4%%%%%%%6%%%%7%%%%8%%%%9%%%1 Time/ms 混沌系统的 Lyapunov 指数 Lyapunov 指数是衡量系统动力学特性的一个图1 系统的 Lyapunov 指数 1 3 断出来一个正的 Lyapunov 指数意味着在系统相空间中无论初始两条轨线的间距多么小其差别都该系统出现混沌现象用 MATLAB 对其进行仿真混沌吸引子如图 2 所示混沌系统的 MATLAB 仿真令 a = b = 1 c = 12 4 d = 2 e = 3 此时重要定量指标对于系统是否存在动力学混沌可以从最大 Lyapunov 指数是否大于零非常直观的判 %%%%%%%-1%%%%%%%%%%%%%%%%%%%%1%%%%%%%%%%2 b x1-x %%%%%%%-1%%%%%%%%%%%%%%%%%%%%1%%%%%%%%%%2 a x1-x %%%%%%%-1%%%%%%%%%%%%%%%%%%%%1%%%%%%%%%%2 d x2-x %%%%%%%-1%%%%%%%%%%%%%%%%%%%%1%%%%%%%%%%2 c x1-x %%%%%%%-1%%%%%%%%%%%%%%%%%%%%1%%%%%%%%%%2 e x2-x4 图2-2%%%%%%%-1%%%%%%%%%%%%%%%%%%%%1%%%%%%%%%%2 f x3-x4 各平面混沌吸引子投影

6 1 2 4 Runge-Kutta 9 s a = b = 1 c = 12 4 d = 2

3 Runge-Kutta 9 s a = b = 1 c = 12 4 d = 2 e 6 = %%%%%%%%%%%%2%%%%%%%%%%%4%%%%%%%%%%%6%%%%%%%%%%8%%%%%%%%% %%%%%%%%%%%2%%%%%%%%%%%4%%%%%%%%%%6%%%%%%%%%%8%%%%%%%%%1 (a)x (b)x %%%%%%%%%%%%2%%%%%%%%%%%4%%%%%%%%%%%6%%%%%%%%%%8%%%%%%%%%1 %%%%%%%%%%%2%%%%%%%%%4%%%%%%%%%%6%%%%%%%%%%8%%%%%%%%1 (c)x 3 (d)x %%%%%1%%%1%%2%%2%%3%%3%%4%%4%% (a)x 1 - %%%%%1%%1%%2%%2%%3%%3%%4%%4%% %%%%%1%%%1%%2%%2%%3%%3%%4%%4%% (b)x %%%%%1%%1%%2%2%%3%%3%%4%%4%% (c)x 3 4 (d)x 4

4 AD633 k 赘 k 赘 1k 赘 1nF 1k 赘 1k 赘 1k 赘 LF347 1k 赘 x1 4 1nF 1k 赘 1k 赘 1k 赘 1nF 1k 赘 x2 1k 赘 124k 赘 124k 赘 1k 赘 1nF 1k 赘 1k 赘 4 1k 赘 x3 1k 赘 6 2k 赘 2k 赘 1k 赘 1nF 1k 赘 1k 赘 634k 赘 1k 赘 x %%%%%%-3%%%%%-1%%%%%%1%%%%%%%%3%%%%%%% %%%%%%-3%%%%%-1%%%%%%1%%%%%%%%3%%%%%%% (a)x 1-x 2 (b)x 1-x %%%%%%-3%%%%%-1%%%%%%1%%%%%%%%3%%%%%%% -%%%%%%-3%%%%%-1%%%%%%1%%%%%%%%3%%%%%%% (c)x 1-x 4 (d)x 2-x %%%%%%-3%%%%%-1%%%%%%1%%%%%%%%3%%%%%%% -%%%%%%-3%%%%%-1%%%%%%1%%%%%%%%3%%%%%%% (e)x 2-x 4 (f)x 3-x 4 6

5 8 1 4 Physical J M 2 7 J PECORA L M CARROLLTl Synchronization in Chaotic Systems J Phys Rev lett OTTE GREBOGIC YORKEJA Driving Systems with Chaotic Signals J Phys Rev SPROOTTJC A New Class of Chaotic Circuit J Phys Rev Lett A SONG Yu-zhong ZHAO Guang-zhou QI Dong-lian Passive Control of Chaotic System with Multiple Strange Attractors J Chinese 8 Multisim9 M 27 9 J 29 S 檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪檪 3 规格定子外径 D 1/mm 67 定子内径 D i1/mm 37 铁芯长 L/mm 规格硅钢片重 G Fe%/kg 2 1 J 4%kW %kw 6%kW 63%kW 原方案现方案原方案现方案原方案现方案原方案现方案 %kW %kw 6%kW 63%kW 原方案现方案原方案现方案原方案现方案原方案现方案 1%433 1%882 1%966 1%28 1%866 1%323 2%16 1%47 定子绕组铜重 G Cu1/kg %%%246 %%%1628 %%%287 %%%1739 %%%216 %%%1844 %%%2274 %%%1989 转子铜条重 G Cu2/kg %%%%%41 %%%%%389 %%%%%481 %%%%%412 %%%%%3 %%%%%47 %%%%%72 %%%%%493 电机钢重量减少 /% 电机总铜重量减少 /% CHO Dong-Hyeok JUNG Hyun-Kyo CHUNG Tae-Kyung et al Design of a Short-Time Rating Interior Permanent Magnet Synchronous Motor Using a Niching Genetic Algorithm J IEEE Trans- actions on Magnetics M 27 3 CHOI Jae-Hak KIM Tae Heoung JANG K B Geometric and E- lectrical Optimization Design of SR Motor Based on Progressive Quadratic Response Surface Method Magnetics J IEEE Transactions M J J C / / J

u d = R s i d - ωl q i q u q = R s i q + ωl d i d + ωψ 1 u d u q d-q i d i q d q L d L q d q ψ f R s ω i 1 i 5th i th 5 θ 1 θ θ 3 5 5

u d = R s i d - ωl q i q u q = R s i q + ωl d i d + ωψ 1 u d u q d-q i d i q d q L d L q d q ψ f R s ω i 1 i 5th i th 5 θ 1 θ θ 3 5 5 3 5 018 10 Vol. 3 No. 5 JOURNAL OF HARBIN UNIVERSITY OF SCIENCE AND TECHNOLOGY Oct. 018 150080 Matlab /Simulink DOI 10. 15938 /j. jhust. 018. 05. 011 TM35 A 100-683 018 05-006- 06 Stator Harmonic Optimal

朱建良 等 一个新的四维混沌系统及其电路实现 第2 期 x 1 = a x2 x1 + x2 x3 x4 x 2 = b x1 + x2 x1 x3 x4 2 x 3 = c x1 x3 + x1 x2 x4 x 4 = ex3 dx4 + x1 x2 x3 系统具有自然的对称性 即作如下变换后系统

朱建良等一个新的四维混沌系统及其电路实现第2 期 x 1 = a x2 x1 + x2 x3 x4 x 2 = b x1 + x2 x1 x3 x4 2 x 3 = c x1 x3 + x1 x2 x4 x 4 = ex3 dx4 + x1 x2 x3 系统具有自然的对称性即作如下变换后系统