HÂN TÍCH DAO ĐỘNG PHI TUYẾN TRONG HỆ CHỊU KÍCH ĐỘNG NGẪU NHIÊN VÀ TUẦN HOÀN

Header Page of 123 VIN HN LM KHOA HC V CễNG NGH VIT NAM VIN C HC -o0o - DNG NGC HO PHN TCH DAO NG PHI TUYN TRONG H CHU KCH NG NGU NHIấN V TUN HON LUN N TIN S K THUT H Ni - 2015 Footer Page of 123 Header Page of 123 ii VIN HN LM KHOA HC V CễNG NGH VIT NAM VIN C HC -o0o - DNG NGC HO PHN TCH DAO NG PHI TUYN TRONG H CHU KCH NG NGU NHIấN V TUN HON LUN N TIN S K THUT Chuyờn ngnh: C k thut Mó s: 62 52 01 01 Ngi hng dn khoa hc: GS TSKH Nguyn ụng Anh H Ni - 2015 Footer Page of 123 Header Page of 123 iii LI CM N Tỏc gi chõn thnh cỏm n thy hng dn GS.TSKH Nguyn ụng Anh ó tn tõm hng dn khoa hc, luụn ng viờn v giỳp tỏc gi c v vt cht ln tinh thn tỏc gi hon thnh lun ỏn ny Tỏc gi xin gi li cỏm n n Khoa o to sau i hc v cỏn b Vin C hc, bn bố v ng nghip ti trng i hc Cụng ngh thụng tin, HQG Tp HCM, ó ng viờn, giỳp v to iu kin thun li cho tỏc gi quỏ trỡnh lm lun ỏn Nhõn õy, tỏc gi cng gi li cỏm n n NCS Nguyn Nh Hiu, ngi ó lng nghe v chia s rt nhiu vi tỏc gi v chuyờn mụn, v c bit l PGS.TS Dng Anh c, ngi ó to iu kin tt nht tỏc gi an tõm thc hin nghiờn cu ca mỡnh Sau ht, tỏc gi chõn thnh cỏm n b m, v con, v gi li cỏm n n ngi thõn ó rt kiờn nhn ng viờn tỏc gi thi gian lm lun ỏn Tỏc gi lun ỏn, Dng Ngc Ho Footer Page of 123 Header Page of 123 iv LI CAM OAN Tụi cam oan õy l cụng trỡnh nghiờn cu ca riờng tụi, di s hng dn trc tip ca GS TSKH Nguyn ụng Anh Cỏc s liu, kt qu nờu lun ỏn l trung thc v cha tng c cụng b bt k cụng trỡnh no khỏc Tỏc gi lun ỏn, Dng Ngc Ho Footer Page of 123 Header Page of 123 v MC LC LI CM N iii LI CAM OAN iv MC LC v DANH MC HèNH V, TH viii DANH MC BNG x CC Kí HIU DNG TRONG LUN N xii M U .1 CHNG 1.TNG QUAN 1.1 Gii thiu 1.2 Cỏc phng phỏp nghiờn cu h dao ng ngu nhiờn phi tuyn 1.3 H dao ng chu kớch ng tun hon v ngu nhiờn 13 1.4 Mc tiờu ca lun ỏn 15 CHNG C S Lí THUYT 16 2.1 Cỏc khỏi nim c bn gii tớch ngu nhiờn 16 2.1.1 S lc v lý thuyt xỏc sut 16 2.1.1.1 Khụng gian xỏc sut 16 2.1.1.2 Bin ngu nhiờn 17 2.1.2 Quỏ trỡnh ngu nhiờn 21 2.1.2.1 nh ngha 21 2.1.2.2 Mt s quỏ trỡnh ngu nhiờn thng gp 22 2.1.3 Tớch phõn ngu nhiờn 26 2.1.3.1 M u 26 Footer Page of 123 Header Page of 123 vi 2.1.3.2 Tớch phõn Ito Tớch phõn Stratonovich 28 2.1.3.3 Tớnh cht ca tớch phõn Ito 29 2.1.4 Phng trỡnh vi phõn ngu nhiờn 31 2.2 C s lý thuyt nghiờn cu h dao ng ngu nhiờn 34 2.2.1 Phng phỏp trung bỡnh ngu nhiờn theo biờn v pha 34 2.2.2 Phng phỏp trung bỡnh ngu nhiờn h ta -cỏc 36 2.2.3 Phng phỏp hm b tr v li gii phng trỡnh Fokker-Planck (FP) 39 2.2.3.1 Phng phỏp hm b tr 39 2.2.3.2 Nghim ca phng trỡnh FP vi cỏc h s dch chuyn tuyn tớnh 40 2.2.3.3 Tuyn tớnh húa tng ng- gii xp x phng trỡnh FP 46 2.2.4 Phng phỏp mụ phng s 50 2.3 Kt lun chng 52 CHNG PHN TCH DAO NG TRONG H PHI TUYN CHU KCH NG NGU NHIấN V TUN HON 53 3.1 H dao ng Van der Pol 55 3.1.1 Tớnh toỏn lý thuyt 56 3.1.2 Kt qu v tho lun 58 3.1.3 So sỏnh vi phng phỏp phi tuyn tng ng 65 3.2 H dao ng Duffing 67 3.2.1 Tớnh toỏn lý thuyt 67 3.2.2 Kt qu v tho lun 69 3.3 Dao ng Van der Pol Duffing 74 3.3.1 Tớnh toỏn lý thuyt 74 3.3.2 Kt qu v tho lun 75 3.4 H dao ng Mathieu-Duffing 79 Footer Page of 123 Header Page of 123 vii 3.4.1 Tớnh toỏn lý thuyt 79 3.4.2 Kt qu v tho lun 82 3.5 Kt lun chng 87 CHNG PHN TCH BAN U P NG TH IU HềA TRONG H DAO NG PHI TUYN CHU KCH NG NGU NHIấN V TUN HON 89 4.1 Gii thiu 89 4.2 K thut phõn tớch 90 4.3 Kt qu v tho lun 97 4.4 Kt lun chng 100 KT LUN 102 DANH SCH CễNG TRèNH CA TC GI C CễNG B LIấN QUAN N LUN N 105 TI LIU THAM KHO 106 PH LC 112 Ph lc A 112 Ph lc B 116 Footer Page of 123 Header Page of 123 viii DANH MC HèNH V, TH Hỡnh 1.1 H mt bc t a) Kt cu to nh tng b) Mụ hỡnh tng ng Hỡnh 2.1 Mt qu o ca chuyn ng Brown (quỏ trỡnh Wiener) 23 Hỡnh 2.2 Qu o ca phng trỡnh vi phõn thng 27 Hỡnh 2.3 Qu o ca mt quỏ trỡnh ngu nhiờn 27 Hỡnh 3.1.1 th trung bỡnh theo thi gian ca trung bỡnh bỡnh phng ỏp ng theo tham s Q 61 Hỡnh 3.1.2 th trung bỡnh theo thi gian ca trung bỡnh bỡnh phng ỏp ng theo tham s Q so sỏnh vi kt qu mụ phng s 62 Hỡnh 3.1.3 th hm mt xỏc sut ng thi p ( x, x& ) ca h dao ng Van der Pol ti thi im t = 294s 63 Hỡnh 3.1.4 th ca hm mt xỏc sut ca dch chuyn x theo cỏc thi gian khỏc 64 Hỡnh 3.1.5 th ca hm mt xỏc sut ca dch chuyn x ti thi im t = 294 (s) 64 ( ) Hỡnh 3.1.6 th ng cong E x ca h Van der Pol theo n lõn cn w 65 Hỡnh 3.2.1 Kt qu tớnh toỏn E ộở x ( t ) ựỷ v E ộở x ( t ) ựỷ bng phng phỏp gii tớch v so vi kt qu mụ phng s 71 Hỡnh 3.2.2 th bỡnh phng biờn ca ỏp ng trung bỡnh theo tham s Q 71 Hỡnh 3.2.3 th bỡnh phng biờn ca ỏp ng trung bỡnh theo tham s s 72 Footer Page of 123 Header Page of 123 ix Hỡnh 3.2.4 th trung bỡnh theo thi gian ca trung bỡnh bỡnh phng ỏp ng E ( x ) theo tham s s 72 Hỡnh 3.2.5 th ng cong cng hng ca h Duffing 73 Hỡnh 3.3.1 th trung bỡnh theo thi gian ca E ộở x ( t ) ựỷ theo tham s phi tuyn g 78 Hỡnh 3.3.2 th trung bỡnh theo thi gian ca E ộở x ( t ) ựỷ theo biờn lc kớch ng tun hon Q 78 Hỡnh 3.4.1 Kt qu gii tớch E ộở x ( t ) ựỷ c so sỏnh vi cỏc kt qu s 84 Hỡnh 3.4.2 Kt qu gii tớch E ộở x ( t ) ựỷ c so sỏnh vi cỏc kt qu s 84 Hỡnh 3.4.3 th hm mt xỏc sut ng thi ca h Mathieu-Dufing ti thi im t = 294 s 85 Hỡnh 3.4.4 th hm mt xỏc sut ca x ti thi im t = 294 (s) 86 Hỡnh 3.4.5 th hm mt xỏc sut ca x ti vi thi im (s) 86 Hỡnh 4.1 th trung bỡnh theo thi gian ca trung bỡnh bỡnh phng ỏp ng th iu hũa theo tham s s 99 Hỡnh 4.2 nh hng ca s v Q0 lờn trung bỡnh bỡnh phng ỏp ng th iu hũa 99 Hỡnh 4.3 nh hng s v h lờn trung bỡnh bỡnh phng ỏp ng th iu hũa 100 Footer Page of 123 Header Page 10 of 123 x DANH MC BNG Bng 3.1.1 Sai s gia kt qu mụ phng v cỏc giỏ tr xp x ca trung bỡnh theo thi gian ca trung bỡnh bỡnh phng ỏp ng E ộở x ( t ) ựỷ theo tham s e 58 Bng 3.1.2 Sai s gia kt qu mụ phng v cỏc giỏ tr xp x ca trung bỡnh theo thi gian ca trung bỡnh bỡnh phng ỏp ng E ộở x ( t ) ựỷ theo tham s n 59 Bng 3.1.3 Sai s gia kt qu mụ phng v cỏc giỏ tr xp x ca trung bỡnh theo thi gian ca trung bỡnh bỡnh phng ỏp ng E ộở x ( t ) ựỷ theo tham s Q 60 Bng 3.1.4 Sai s gia kt qu mụ phng v cỏc giỏ tr xp x ca trung bỡnh theo thi gian ca trung bỡnh bỡnh phng ỏp ng E ộở x ( t ) ựỷ theo tham s s 61 Bng 3.1.5 Sai s gia kt qu mụ phng v cỏc giỏ tr xp x ca trung bỡnh theo thi gian ca trung bỡnh bỡnh phng ỏp ng E ộở x ( t ) ựỷ theo k thut ca lun ỏn v phng phỏp phi tuyn tng ng theo tham s 66 Bng 3.2.1 Sai s gia kt qu mụ phng v cỏc giỏ tr xp x ca trung bỡnh theo thi gian ca trung bỡnh bỡnh phng ỏp ng E ộở x ( t ) ựỷ theo tham s g 69 Bng 3.2.2 Sai s gia kt qu mụ phng v cỏc giỏ tr xp x ca trung bỡnh theo thi gian ca trung bỡnh bỡnh phng ỏp ng E ộở x ( t ) ựỷ theo tham s s 69 Bng 3.2.3 Sai s gia kt qu mụ phng v cỏc giỏ tr xp x ca trung bỡnh theo thi gian ca trung bỡnh bỡnh phng ỏp ng E ộở x ( t ) ựỷ theo tham s s vi cỏc giỏ tr e khỏc 70 Bng 3.3.1 Sai s gia kt qu mụ phng v cỏc giỏ tr xp x ca trung bỡnh theo thi gian ca trung bỡnh bỡnh phng ỏp ng E ộở x ( t ) ựỷ theo tham s e 75 Footer Page 10 of 123 Header Page 119 of 123 106 TI LIU THAM KHO Ting Vit: [1] Nguyn ụng Anh (1999), Mt s kt qu nghiờn cu lnh vc dao ng ngu nhiờn thc hin ti Vin C hc, Mt s thnh tu ca Vin C hc sau 20 nm thnh 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strongly non-linear oscillators under wide-band random excitation, Non-Linear Mechanics, 36, pp 1235-1250 [83] Zhu WQ, Wu YJ (2003), First passage time of Duffing oscillator under combined harmonic and white noise excitations, Nonlinear Dynamics, 32, pp 291-305 Trang web v phn mm: [84] John MC (2010), Probability Density Functions , www.mne.psu.edu/me345/ Lectures/Probability_density_functions.pdf [85] Laurence CE (2002), An introduction to stochastic differential equations (version 1.2), Department of Mathematics, UC Berkeley (math.berkeley.edu/ ~evans/SDE.course.pdf) [86] Jonathan MB, Matthew PS (2011), An Introduction to Modern Mathematical Computing With Maple, Springer [87] Jaan Kiusalaas (2010), Numerical methods in engineering with Matlab (Second Edition), Cambridge University Press Footer Page 124 of 123 Header Page 125 of 123 112 PH LC Ph lc A Xõy dng v gii h phng trỡnh phi tuyn cho cỏc h s tuyn tớnh hoỏ Chng trỡnh Maple tớnh cỏc h s tuyn tớnh hoỏ theo cỏc mụ men ca a1 v a2 ( trỏnh nhiu ch s, cỏc chng trỡnh di õy lun ỏn dựng ký hiu b v d thay cho a1 v a2 ) on chng trỡnh tớnh cỏc mụ men bc cao ca a1 v a2 theo trung bỡnh, phng sai v hip phng sai ca a1 v a2 Footer Page 125 of 123 Header Page 126 of 123 113 Tớnh cỏc h s ca hm mt dng Chng trỡnh Matlab tớnh cỏc h s tuyn tớnh hoỏ function tuyentinhhoa_vanderpol % chuong trinh tim cac he so tuyen tinh hoa phan Pol global a B sig2 P nu Delta omega epsilon; % clear all clc a = 1; % alpha B = 4; % beta epsilon = 0.2; omega = 1; nu = 1.01; Delta = (omega^2-nu^2)/epsilon; val=[1] % co the dua nhieu gia tri vao day de co day num=length(val); sig2=1; X2=zeros(num,1); for m=1:1:num P=val(m); x0=[-2,-0.5,2,1.5,-1,2]; L1b=x0(1); L1d=x0(2); L10=x0(3); % he so eta11, L2b=x0(4); L2d=x0(5); L20=x0(6); % he so eta21, alpha1 = (1/2)*a+L1b; Footer Page 126 of 123 tich he Van der tinh 12, 13 22, 23 Header Page 127 of 123 114 beta1 = Delta/(2*nu)+L1d; lambda1 = L10; alpha2 = -Delta/(2*nu)+L2b; beta2 = (1/2)*a+L2d; lambda2 = P/(2*nu)+L20; Ab2= -(2*(alpha1^2+alpha1*beta2+alpha2^2- alpha2*beta1))*nu^2*(alpha1+beta2)/(sig2*(alpha2^2- 2*alpha2*beta1+beta1^2+alpha1^2+2*alpha1*beta2+beta2^2)); % he so tau1 Ab1 = ((2*(2*lambda1*alpha1+2*lambda1*beta2+2*alpha2* lambda2-2*beta1*lambda2))*nu^2*(alpha1+beta2)/ (sig2*(alpha2^2- 2*alpha2*beta1+beta1^2+alpha1^2+ 2*alpha1* beta2+beta2^2))); % he so tau4 Abd = ((2*(2*alpha2*beta2+2*beta1*alpha1))*nu^2* (alpha1+beta2)/(sig2*(alpha2^2-2*alpha2*b eta1+beta1^2+alpha1^2+2*alpha1*beta2+beta2^2))); % he so tau3 Ad2 = (-(2*(alpha1*beta2+beta2^2-alpha2*beta1+beta1^2))* nu^2*(alpha1+beta2)/(sig2*(alpha2^2-2*alpha2* beta1+beta1^2+alpha1^2+2*alpha1*beta2+beta2^2))); % he so tau2 Ad1 = ((2*(-2*alpha2*lambda1+2*lambda2*alpha1+ 2*lambda2*beta2+2*beta1*lambda1))*nu^2* (alpha1+beta2)/(sig2*(alpha2^2-2*alpha2*beta1+ beta1^2+alpha1^2+2*alpha1*beta2+beta2^2))); % he so tau5 psb = 2*Ad2/(4*Ab2*Ad2-Abd^2); % phuong sai cua b if ((Ab2