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ln23 - .M-W_— Sammieaflfi> ag £~>oo> and...

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Unformatted text preview: ,_.M.___.-W_.._— Sammieaflfi > ag £~>oo > and 0002+)"; fimsmfi We.” 01324?“ Om} W tmdgopog Sd'bfles 00ch Q, (Spicajg aomFQO-k) SQ)" 05¢ FCC/1‘58 (0 W th&13]9ace (ELM—f 3.9%) Dugoduq, Sigskewg Jacigxm/Uflgvfifle, oath) 6m QEEECGCQOC WWW Smb. diamag 6 swk aJQJrCae/Qor WH‘C, (5&3nt <—> gram 352, WW~ Smyi-NL £0 (:an COOO‘HyLQng COM‘Dtoszd ern mnflinacgfia OQW “0an aqufitons hum Mcoug Wth/QWPS {’49P W L meybw 0% W crLtde—OC Mk (ngéw 33mm ,5 L X“ ‘Y‘E :5- Qj:><+oxx/ . flak/:3)?“ QMQQ.1 a: b+2£x Q) -—————~———-~—~« TWOCJrQ/(fi’ W23” a » gfiw‘vmj (0:;wa owl’\ ~ Q010xvfié( ccossmé“ w/o CWQWOROD Jim W {,de ACMQW SCOO Z) - (mgmvkwn L<Qz¥wflxm3 £0 Uni» Xy ‘3QMQJB' ’ fi> AWWF- — 2A SWIG. g-OQDQQ’CQ CRQCAUVQ’ AtOQ‘LLW‘QS .gc/Uf 3W QCQUHLQL 0 W 1 (Jump/WOO "gun 12.3.1 Figure 12.3.2 Abraham and Show (i983), p. ‘23 (a) trajectory divergence along attractor compression mward attractor Figure 12.3.3 (I) (h) (C) Figure 12.3.4 Abrohum and Show (1983), pp V22 4'23 8 ““0de mndacgmd m Ram/mi QWLLCJ‘QXQ 490k we age ‘ ‘ 130an0: Mich. ””7“ guy/(ma: ; 935305;] ‘% was => 2-mmomscon’a52 ’POmQam/ wave 002 m (mam 15mm w 51mm“ to mm W <1}ch Ks QQcA) _______ ‘ W Vromgg of, sJWQXC/Mfi 010A QCQOQTE CS mg w:% waélMs‘rOOd ‘07 meé M W8. ogxh<% \ Q QQ$Q~I)§+ayn3) i~é \kné i E35 CED 619%: in skccpg 09: mama Udm‘Lchng a,“ Afircwmr) A = 3“? $3 » Caxéwc go} 08r QINL Sggmflnkg» ) $0,413) Q7 Figure 12.3.5 Abraham and Show (1983), p. 123 Making Pastry Figure 12.1.2 shows a process. used to make filo pastry or croissant. flatten and stretch re-in'ect J fold Figure 12.1.2 Figure 12.1.3 Figure 12.1.4 :— 2 3 K AC5 Moficmc: Sm “$339-$2ch gaksx s.— S m Shawl 2:”‘(35 Q.— 990% => A 9&3} mug wgop 04%“ => B °°C$~Cmmftami WWW“ .Qor 61/07 (my/(A (fligwee 9mm @MEXQJLO A i3 Opt “0‘5" 0*? ‘5 O x “0"” :> % “(SX—OfiHCQ’G/(‘NQ/ LA is MWMX JDoc ~wko Ute§r 10/37 *0 3WD) WE; Utmobiwnx 42%: <3) 2) => (>\i=L'\/L>O) ’,\?_=QA,\4, <0 Cam CUL$O DEW gcom é/klofbucfl’tahofY @nchhg and OXMSE ‘ ’\ emkfi'kg amp as Z“ a Q’nbnz = QM“ , Aways fifi‘ow as L223“: QMQM‘JJMQB QWHZXH _ L3Ww0¢ dmwgtpn; \D ‘ :(Q/nz "D msggeakm: x431: QmZ—ebnm\o (Oasis) :> Am» 0% >, Acm CELR\\ ) W2 2 S (1K 32— '; HVOLL/maq Qonfiad'oon .> S'H‘QWSQ, OJHXQDPOP =4: “\[oluwk‘ QWCA = No OHCOJJLOP.’ (Likachm/Lflonm 0" Q ' ng 53 > , gtdg’vQ/ms) Engg: (Mama M0,?» Vnfl = 3n 1' 4’ 07(0 b) T : "19:1,? R1 Bn-H =1 \DXn m Wk? Oil/n \DQ, {ngf QQ 02$ (2, Com?03ifivfl <53 ‘ekrge’ MW: OPWH$I T;VT”/0Tl/OT( )>T(.‘F> =T///<T//<TICF$) \‘3’ECQIWGN'. T/i XI=XS \7(: «+5 ~sz‘ 0 COLOLOAS': "T” 3 X0: 5%) "VI/‘3?! “ELM/OREGON; T!” 3 ”1:7 // 3 70/: X” [j LQL; :3; :3 x») mm; g__,_._..,.w wwhm . “w New _ Ram) 9&9;wa sacmohm min L choungx/Mge, fibraoi'wfl "‘ Swath ‘m JUMZ. Qggélw OWFU‘JWOO (Ebowm, mg goof/w, @50ng $3944“ Mo‘x Mari-,5 mag)» miflulshom 1.0% Lg m Wow-op “mom to 80% Gum/W5 mm ‘ Q0? dumfiewg? 0.18 0.16 Fig". 72: a) The Hénon attractor for 104 iterations. Some successive iterates have been numbered to illustrate their erratic movement on the attractor. b), c) Enlargements of the squares in the preceding figure. d) The height of each bar is the relative probability to find a point in one of the six leaves in c). (After Farmer, 1982a, b.) GMQQ, mmtgom; 5;“ (x\—‘>P>: “#00 \‘XQNS W a “v: A00 VJ“ °'WMMOQ: Q‘WM N“ V“ Cad Vomfi A!“ 030M$ (LnOK Afifmi‘s “WW Caflm Jigyas 0,? QFe (0AA fl cjoqm “(MS @0‘ {£196 caffjwilg QLQ, on Um qflwoeflor_ ., wt: ’ g’wm ‘ £01 skangg, Mfcwkof$2 9 “62,4wa W§¢CO£ quaeéa QYQQAM W5240k¢4 “VOW“ ‘ Figure 22.1: Plot of the Hénon attractor (red) and the stable manifold (blue) of the fixed point that lies within the attractor. The stable component E; of the tangent space (i.e. the contracting direction) lies along this curve at the points x where it intersects the attractor. Since the expanding direction E f; lies along the attractor, we see that E; and E j: are parallel at points where the blue curve is tangent to the attractor. At these points they do not span the tangent space 7} indicating a breakdown of hyperbolicity. The calculation of the stable manifold is discussed in reference [5], and the figure was constructed using the program supplied there. 30 m0 Mb *MQE: flwuum‘wm n W LumskaMLX WCQoM Oanmtk CHILE/C gm} Uk$M 0C Sim CMQQML) WCQOKA 01% W in‘d 90mm \ m Momma) (J: awn mkumxk cm MSW (396L665) mat/040361 llamafififi $0000de or \Aadmuaxd \oQ, 39.0. M a 59%;: Cnflfwdim 0’0““ 5% fflgéfliifi meat-15$“ C(hLLfsQCHOOS => £qn3&\ a TKQ, my: mums Chaos: - NW“... Safim'ts \N x Cure/i _' gmw ~~ $010de me m3, Bum ~ ka {locomjr \libmfitmn MM ' gag o: mgni’ maflm putodu‘, (Cm IchQCdeta/ku macho, AMA/x98 Q0 5 EMU/m kmmflrw/‘Es SWOQSVS—DV‘C/Cxlz pefiiowc, Swg ‘bdmm maflmjrs (M009 3 HonQg 3 GWMWM Imam-Q5 ) y \ ‘ 21¢ ‘ Masha: DCNa/o dodeL‘we/M :Z/ 9%};ng ogctuasfiom °>.< +‘Q’SL - ><+x3 : ngbwt . W... m~———m—mm :w‘MWOL‘KQJW/{Oftm‘fit 4 W‘ MA wg‘ andef ‘> komww Maze, => W5 Mvmw—_—_4———-~W— --——-_._._.~~« M—"~- 7 ..M._____~ h.» -——.._- -__..‘____.__,_..__.— Kawam aficaofixor has? on ”cop as, w‘L \l _ mmm A} Mggmgwwgflq . «mm _,”W.M.W—v~.——~——~A '{NJ'O Q; Wz\ ' QQXCSH n3 éfibh Sew (1 g a => Ema/3 bar/Nun Mango; Mam FW 1.0 dosa J29 m bOwndA/Cts ~ gmmwagemmc => kmggad‘ W5. Lag. Lowfié “PM {DOW/Lon m%”3\ => kaal boundMJ K ,% ,chum: Q—rcw‘wpfi bow/ii) pm.“ Figure 12.5.4 (I) (b) (a) (b) Figure 23.5: Poincare section for Duffing Oscillator: (a) driving strength g = 0 and no damping y = 0; (b) stable and unstable manifolds for g = 0.10; (c) stable and unstable manifolds for g = 0.40; (d) attractor for g = 0.40. In (b)-(d) the damping is y = 0.25 and the drive freqeuncy is (Up = 1. (From Guckenheimer and Holmes) Eng"; .. ...
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