Vol., No.3, 04-09 (009) do:0.436/ns.009.307 Natural Scence Wavelet chaotc neural networks and ther applcaton to contnuous functon optmzaton Ja-Ha Zhang, Yao-Qun Xu College of Electrcal and Automatc Engneerng, Sanjng Unversty, Nanjng, Chna; zhangjh6688@sohu.com Insttute of Computer and Informaton Engneerng Harbn Commercal Unversty, Harbn, Chna; xuyaoq@hrbcu.edu.cn Receved 7 September 009; revsed 0 October 009; accepted October 009. ABSTRACT Neural networks have been shown to be powerful tools for solvng optmzaton problems. In ths paper, we frst retrospect Chen s chaotc neural network and then propose several novel chaotc neural networks. Second, we plot the fgures of the state bfurcaton and the tme evoluton of most postve Thrd, we apply all of them to search global mnma of contnuous functons, and respectvely plot ther tme evoluton fgures of most postve Lyapunov exponent and energy At last, we make an analyss of the performance of these chaotc neural networks. Keywords: Wavelet Chaotc Neural Networks; Wavelet; Optmzaton. INTRODUCTION Hopfeld and Tank frst appled the contnuous-tme, contnuous-output Hopfeld neural network (HNN) to solve TSP [], thereby ntatng a new approach to optmzaton problems [,3]. The Hopfeld neural network, one of the well-known models of ths type, converges to a stable equlbrum pont due to ts gradent decent dynamcs; however, t causes sever local-mnmum problems whenever t s appled to optmzaton problems. M-SCNN has been proved to be more power than Chen s chaotc neural network n solvng optmzaton problems, especally n searchng global mnma of contnuous functon and travelng salesman problems [4]. In ths paper, we frst revew the Chen s chaotc neural network. Second, we propose several novel chaotc neural networks. Thrd, we plot the fgures of the state bfurcaton and the tme evoluton of most postve Fourth, we apply all of them to search global mnma of contnuous functons, and respectvely plot ther tme evoluton fgures of most postve Lyapunov exponent and energy At last, smulaton results are summarzed n a Table n order to make an analyss of ther performance.. CHAOTIC NEURAL NETWORK MODELS In ths secton, several chaotc neural networks are gven. And the frst s proposed by Chen, the rest proposed by ourselves... Chen s Chaotc Neural Network Chen and Ahara s transently chaotc neural network [5] s descrbed as follows: x () t f ( y ()) t y ()/ e () t y( t ) ky( t) Wjxj I z( t)( x( t) I0) j () z ( t ) ( ) z ( t) (3) where x () t s output of neuron ; y () t denotes nternal state of neuron ;W descrbes connecton weght from j neuron j to neuron, Wj W j ; I s nput bas of neuron, a a postve scalng parameter for neural nputs, k dampng factor of nerve membrane, 0 k, z () t self-feedback connecton weght (refractory strength) 0, dampng factor of z () t, 0< <, I 0 a postve parameter, steepness parameter of the output functon, >0... Morlet Wavelet Chaotc Neural Network (MWCNN) Morlet wavelet chaotc neural network s descrbed as follows: ( uy ( t)) / x () t f( y ()) t e cos(5 uy ()) t (4) ScRes Copyrght 009 Openly accessble at http://www.scrp.org/journal/ns/
J. H. Zhang et al. / Natural Scence (009) 04-09 05 y( t ) ky( t) Wjxj I z( t)( x( t) I0) j (5) z ( t ) ( ) z ( t) (6) where x () t, y () t, Wj,, k, I, z () t, I 0 are the same wth the above. And the Eq.4 s the Morlet wavelet u s a steepness parameter of the output functon whch s vared wth dfferent optmzaton problems..3. Mexcan Hat Wavelet Chaotc Neural Network (MHWCNN) Mexcan hat wavelet chaotc neural network s descrbed as follows: ( ( )) / ( ) ( ( )) ( ( ( )) ) uy t x t f y t uy t e (7) 3 y( t ) ky( t) Wjxj I z( t)( x( t) I0) j (8) z ( t ) ( ) z ( t) (9) where x () t, y () t, Wj,,k, I, z () t, I 0, u are the same wth the above. And the Eq.7 s the Shannon wavelet y(0,0) as follows: =0.00, z(0,0) =[0.8, 0.8], y(0,0) =[0.83, 0.83]. Meanwhle, we set the teraton as large as 5000 so as to get stable state of a global mnmum. 3.. Chen s Chaotc Neural Network ) Smulaton on the Frst Contnuous Functon k =, =0.5, =/0, I 0 =0.85. The tme evoluton fgures of the bggest postve Lyapunov exponent and energy functon of Chen s n solvng the frst contnuous functon are shown as Fgure, Fgure. smulaton are respectvely -0.99989 and (0.0073653, 0.0073653). ) Smulaton on the Second Contnuous Functon k =, =0.0, =/0, I 0 =0.85. The tme evoluton fgures of most postve Lyapunov exponent and energy functon of Chen s n 3. RESEARCH ON CONTINUOUS FUNCTION PROBLEMS In ths secton, we apply all the above chaotc neural networks to search global mnma of the followng three contnuous functons. The three contnuous functons are descrbed as follows [6]: sn x x 0.5 f (, ) 0.5 x x [ 0.00( x x)] x 00 (0) 4 6 f( x, x) 4x. x x /3 xx 4x 4x 4 x 5 () f 5. 5 4( x, x ) ( x 6) 0( )cos 0 4 x x 8 x 5 x 0,0 x 5 () The mnmum value of Eq.0,, respectvely are -, -.03685, 0, 0.398 and ts respondng pont are (0, 0), (0.08983, -0.76) or (-0.08983, 0.76), (-3.4,.75) or (3.4,.75) or (9.45,.45). In order to make comparson convenently, we set some parameters such as the annealng speed, the self-feedback z(0,0) and the ntal value of nternal state Fgure. Tme evoluton fgure of Lyapunov exponent. Fgure. Tme evoluton fgure of energy ScRes Copyrght 009 Openly accessble at http://www.scrp.org/journal/ns/
06 J. H. Zhang et al. / Natural Scence (009) 04-09 k =, =0., =, I 0 =0.5. The tme evoluton fgures of most postve Lyapunov exponent and energy functon of Chen s n solvng the frst contnuous functon are shown as Fgure 5, Fgure 6. smulaton are respectvely 0.39789 and (9.446,.4747). Fgure 3. Tme evoluton fgure of Fgure 4. Tme evoluton fgure of energy solvng the frst contnuous functon are shown as Fgure 3, Fgure 4. smulaton are respectvely - and (0, 0.707). 3) Smulaton on the Thrd Contnuous Functon 3.. Morlet Wavelet Chaotc Neural Network (Mwcnn) ) Smulaton on the Frst Contnuous Functon k =, =0.5, u =0.5, I 0 =0.65. The tme evoluton fgures of most postve Lyapunov exponent and energy functon of MWCNN n solvng the frst contnuous functon are shown as Fgure 7, Fgure 8. smulaton are respectvely -0.99997 and (0.0038638, 0.0038638). ) Smulaton on the Second Contnuous Functon k =, =0.05, u =0.7, I 0 =0.. The tme evoluton fgures of most postve Lyapunov exponent and energy functon of MWCNN n solvng the frst contnuous functon are shown as Fgure 9, Fgure 0. Fgure 5. Tme evoluton fgure of Fgure 7. Tme evoluton fgure of Lyapunov exponent. Fgure 6. Tme evoluton fgure of energy Fgure 8.Tme evoluton fgure of energy ScRes Copyrght 009 Openly accessble at http://www.scrp.org/journal/ns/
J. H. Zhang et al. / Natural Scence (009) 04-09 07 Fgure 9. Tme evoluton fgure of k =, =0.0, u =0.09, I 0 =0.. The tme evoluton fgures of most postve Lyapunov exponent and energy functon of MWCNN n solvng the frst contnuous functon are shown as Fgure, Fgure. smulaton are respectvely 0.39789 and (3.43,.733). 3.3. Mexcan Hat Wavelet Chaotc Neural Network (MHWCNN) ) Smulaton on the Frst Contnuous Functon k =, =0.5, u =0., I 0 =0.8. Fgure 0. Tme evoluton fgure of energy Fgure 3. Tme evoluton fgure of Fgure. Tme evoluton fgure of Fgure 4. Tme evoluton fgure of energy Fgure. Tme evoluton fgure of energy smulaton are respectvely -.00 and (-0.074007, 0.76863). 3) Smulaton on the Thrd Contnuous Functon Fgure 5. Tme evoluton fgure of ScRes Copyrght 009 Openly accessble at http://www.scrp.org/journal/ns/
08 J. H. Zhang et al. / Natural Scence (009) 04-09 The tme evoluton fgures of most postve Lyapunov exponent and energy functon of MHWCNN n solvng the frst contnuous functon are shown as Fgure 3, Fgure 4. smulaton are respectvely -0.99996 and (0.004359, 0.004359). ) Smulaton on the Second Contnuous Functon k =, =0.05, u =.8, I 0 =0.05. The tme evoluton fgures of most postve Lyapunov exponent and energy functon of MHWCNN n solvng the frst contnuous functon are shown as Fgure 5, Fgure 6. smulaton are respectvely -.036 and (-0.08985, Table. the smulaton results of the chaotc neural networks. Fu n f f f 4 Model Chen s MWCNN MHWCNN GM/ER TGM - - - PGM -0.99989-0.99997-0.99996 ER 0.000 0.00003 0.00004 TGM -.03685 -.03685 -.03685 PGM - -.000 -.036 ER 0.03685 0.0348 0.000085 5 TGM 0.398 0.398 0.398 PGM 0.3789 0.3789 0.3789 ER 0.09 0.09 0.09 AVE AVER 0.07096 0.0637 0.00479 0.763). 3) Smulaton on the Thrd Contnuous Functon. k =, =0.05, u =0.3, I 0 =0.. Fgure 6. Tme evoluton fgure of energy The tme evoluton fgures of most postve Lyapunov exponent and energy functon of MHWCNN n solvng the frst contnuous functon are shown as Fgure 7, Fgure 8. smulaton are respectvely 0.39789 and (3.45,.743). 4. ANALYSIS OF THE SIMULATION RESULTS Fgure 7. Tme evoluton fgure of Smulaton results are summarzed n Table. The columns GM/ER, TGM, PGM and AVER represent, respectvely, global mnmum/error rate; theoretcal global mnmum; practcal global mnmum; average error. Seen from the Table, we can conclude that the wavelet chaotc neural networks are superor to Chen s n AVER 5. CONCLUSION Fgure 8. Tme evoluton fgure of energy We have ntroduced Chen s and wavelet chaotc neural networks. We make an analyss of them n solvng contnuous functon optmzaton problems, and fnd out that wavelet chaotc neural networks are superor to Chen s n general. ScRes Copyrght 009 Openly accessble at http://www.scrp.org/journal/ns/
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