Nonlinear Dynamics&Chaos PHYS 4267
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This 0 page Class Notes was uploaded by Kylie Bartoletti DVM on Monday November 2, 2015. The Class Notes belongs to PHYS 4267 at Georgia Institute of Technology - Main Campus taught by Roman Grigoriev in Fall. Since its upload, it has received 20 views. For similar materials see /class/234280/phys-4267-georgia-institute-of-technology-main-campus in Physics 2 at Georgia Institute of Technology - Main Campus.
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Date Created: 11/02/15
P 4126 6t Lama 21 mom 253cm h 0ng no Wg Wacmcvvj g X XnCL XTA Xn H E M X 3km 7n r V L S i xn O i X M cm W Sm Como V kwg onz WWW SM Frcsa quaJchtckcxe dljnmag are IDENTICAL v b QMGKQ Or bCJv dw mmgl W W nguWCQ 0 bcguccd ons CW U SQquanQ MQCHQ vmfo ts CHE AJUL mimode maf 3M o Rowe CL wower 34an 07qu 4x232gtlt23 as 38gtlt2 C 6 9 gag uP f0 PML39Odf g 39 1000000 0875000 0750000 0375000 0250000 0125000 0000000 39 0700000 0760000 0820000 0880000 0940000 1000000 1000000 0875000 0750000 0625000 0500000J 0375000 0250000 0125000 0000000 1 I Y I I 2800000 3040000 3233000 3520000 3760000 4 000000 Figure 1062 Courtesy oi Andy Christian The beginning of this sequence is familiar periods 1 2 and 2X 2 are the first stages in the perioddoubling scenario The later perioddoublings give periods greater than 6 so they are omitted here Next periods 6 5 3 correspond to the large windows mentioned in the discussion of Figure 1062 Period 2 x3 is the was be 212 MwFQ i f Same Suwanee in BE read con Rwy 1701 462 I CorvHWrw aszQMM Q88M 39 7 MPMW F12ng hq quotfrangcen39og 016302 SgSaLamg W anHWLCUHVZ Carma Saga rn WW putm AMICdnd axbx roak WdaL W3 1531 ggzaq g q 900 CWBWWgt NH C W unwame COMM do VH gtok2503 Ygt0 Xmax W 11 coouaomn an maceUfa Cabana zr 40182 lt 6 Wm OW W MLW M 12mm at C paJed dodf Wuhan ww cgwwm X50452 JTWUW 553 00 jutnation UMP MAE a I Livy u tvxxuu uuuu111150 uo Lllv 1Cl 1L1511 llUlllUCl lb lllbl CGDCU 30km 347 352 362 365 l 1 l l 50 100 150 200 Ts Or Figure 39065 Libchaber et 0 1982 p 213 ries shows the temperature variations at one point in the uid For the temperature varies periodically This may be regarded as the basic Hugs a k N3 quot QMQ COOI HOLD Dma to Wg Fwdch PMS ct LonCMLOUA h ma SJWQ SodaZ Exampfgzj Gags2w Sgngem i3yg W X W A Lama OrbH g Q0 Qlb02 Z gt h Z Q I u 391 Loven2 ma gucc SSNevnaA Cmq 03 MB thmVOdCle l 39 E MW gtk lt w1 ro c W 0M WCMQ oil xa s vs C OfbH OQCCLSCCUM 1 gt UvS Mme M l k gt Wig2 Cbz okca7ym 39 39 u o 1 Lovingquotm3 L Whig L4 anal wcmoo a quot gt awed amp aggws 39 Consarlujmc g a VHVWP 5 W28 23901 n p M 0Uscpcutcxl 2 13 OH 41 DO 39 Dowo lc user 40C 0amp3ch gmx 39 l A MBJLWCQ mccum g Wk where a b and c are parameters This system contains only one nonlinear term at and is even simpler than the Lorenz system Chapter 9 which has two nonlin earities Figure 1066 shows twodimensional projections of the system s attractor for different values of c with a b 02 held fixed 44 39A A A A A A z A A A L 14 A A 4 A A A e A A quot C 25 quot A C 35 1 1b 0 1h 1 d V r p 1 0 1 gt h gt v gt1 gt gt y o h 1 1 b 1 1 1 o b F 0 1 L 1 1b v 4 3 i f 14 x 14 14 x 14 l4 A A A A A A A A A 14 A e 4 A A A c A A A A A c4 c5 1 1 1y 1 gt 1p 1 gt gt t b 4 0 gt o b 1 1 4L 0 i t t 4 t t 3 5 14 x 14 14 x 14 Figure 1066 Olsen and Degn 1985 p 185 At 6 25 the attractor is a simple limit cycle As 6 is increased to 35 the limit cycle goes around twice before closing and its period is approximately twice that of the original cycle This is what period doubling looks like in a continuous time system In fact somewhere between c 25 and 35 a perioddoubling bifurca tion of cycles must have occurred As Figure 1066 suggests such a bifurcation
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