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# Special Topics in Phys PHY 150

UCD

GPA 3.82

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This 23 page Class Notes was uploaded by Horace McClure on Tuesday September 8, 2015. The Class Notes belongs to PHY 150 at University of California - Davis taught by Staff in Fall. Since its upload, it has received 4 views. For similar materials see /class/191839/phy-150-university-of-california-davis in Physics 2 at University of California - Davis.

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Date Created: 09/08/15

Example Chaotic Maps that you can analyze Reading for this lecture NDAC Sections 05 07 Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld Example ID Maps Shift Map n 6 01 n1 2 mod 1 Fixed Point 0 Unstable f 2 gt 1 Period2 Orbit 13 23 Unstable f2 4 gt 1 All periodic orbits unstable Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld ZEn l l Example ID Maps Shift Map n 6 01 1 n1 2n mod 1 Solvable mm 2 2mm mod 1 0 Chaotic mechanism 0 71 1 shift up least signi cant digits 10 2 0110101111 11 2 010101 1110 Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld Example ID Maps Lyapunov Characteristic Exponent for ID Maps CURH 1 5NfN U50 fN93 same 1 Ansatz 5N 50 39N Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld Example ID Maps Lyapunov Characteristic Exponent for ID Maps De nition LCE 1 5N Azjggnoo logZ E SO gt0 Compare iterates of 510 and 0 60 fN5130 50 fN550 1 A lim log2 N gtooN CEOF60 ZEO 60 gt 0 1 N A ngnoo N10g2 iltf coi Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld Example ID Maps Lyapunov Characteristic Exponent for ID Maps Chain rule fN5130 fN 1fN1IE0 f0f1 fIEN 1 LCE for ID Maps 1 N 1 39 A ngnoo N 21 If rem Interpretation A lt 0 stable Stable xed point or periodic orbit A gt 0 unstable Chaotic attractor Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld Example ID Maps Back to Shift Map Its LCE 1 N 1 A Alina N Z 10g2 lf 710 1 Independent of state f a 2 l 2A9 Ampli cation per step or bits of resolution lost I A 1 0 0 ALI 1 Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutchfield Example ID Maps Tent Map n 6 01 f L DIl i can 0 g n 3 71 1 a1 71 lt n g Slopea E 0 2 1 ZEn l l O O 37m Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld Example ID Maps Tent Map Bifurcation Diagram 1 Inn O 05 10 a 15 20 Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld Example ID Maps Tent Map Stable xed point 0 0 g a lt 1 a Unstable xed oints 0 1 lt lt 2 P 7 1 a 7 a A periodic orbits unstable period gt 1 No periodic windows 1 f mn l Z if ft f K 0 O 13 1 Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld Example ID Maps Tent Map LCE N 1 A lim E 10 iazlo a NgtOON g2i i g2 710 1 0539 Chaotic O O539 Periodic 1 076 O8 II 12 1I4 1 6 18 Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld Example ID Maps Logistic map 33n1 7le ajn State space mm E 0 1 Parameter height 7 E 0 4 1 inn kl 0 0 33m 1 Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld Example ID Maps Logistic map bifurcation diagram 10 xn 00 39 30 7 40 Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld Example ID Maps Logistic map LC E Local stability depends on state f r1 2 1 A Iggnoo N logg if m N l A jggnoo 210 ir1 Zmn Period00 r 1 Azlong Period I f AzlogZiZ ri Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld Example ID Maps Logistic map LCE Superstable orbit fm 0 A gt oo Example r 2 Bifurcations neutral stability A 0 Examples r 1 and r 3 Onset of chaos A 0 Chaos A gt 0 Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld Example ID Maps LCE for ID Maps Rather than time average over 0 1 2 1 N l A Jng N 210 If SEMI Average over attractor s distribution Pr E A Invariant distribution Pr f o Pr Statespace averaged LCE A A da Pres logg If vI Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld Example ID Maps Logistic map LCE r 4 Invariant distribution 1 39quot PI E 7r 1 10g2 Pr1 1 A d10g2 4 8m 0 7n1 m A 1 bit per step Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld Example ID Maps LCE view of perioddoubling route to chaos 1 I I I I I 7 113 T2B gt13 05 TC 5 3569945 A 0 742 4 A 0 o 1 l O5 1 I I I I I I I I 3 31 32 33 34 35 36 37 38 39 4 7a 383 Period3 Window Superstable Period2 A gt oo Superstable Period4 Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld Example ID Maps 1laogistic map bifurcation diagram selfsimilarity mm 01 31 r ii593 Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009im Crutchfield Example ID Maps Bifurcation Theory of ID Maps Scaling analysis of perioddoubling cascade 10 T233 T433 T833 621m A 24669 Universal constants a Tn1 oz hm n 25029 n gtoo dn1 Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009im Crutch eld Example ID Maps Bifurcation Theory of ID Maps Renormalization group analysis of perioddoubling TO gt T1 f27rl f7T0 lt AL T I ozme f 2 33 7T0O f Earl Universal Map 2 1 2 4 2 n 2 517 r zoz r 1 11m 04 7 f a 1 f 12 2 90 THOO f an n for N Emax mm m anflt2 gtlt m an Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld 21 Example ID Maps Bifurcation Theory of ID Maps Renormalization group analysis of perioddoubling 1 Too froo ozf2aroo for N max Limiting functional equation choose max 0 gm eta2 90 o amp g ltogt 0 Ga Solve byTaylor expansion g a bar2 m4 o Find 04 25029 Parameter rescaling more work Find 6 4669 Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld 22 Example ID Maps Reading for next lecture Lecture Notes Lecture 7 Nonlinear Physics Physics ISO250 Spring 2009Jim Crutch eld 23

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