The Lorenz System and Chaos in Nonlinear DEs

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Danuta Halina Socha
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1 The Lorenz System and Chaos in Nonlinear DEs April 30, 2019 Math 333 p. 71 in Chaos: Making a New Science by James Gleick

2 Adding a dimension adds new possible layers of complexity in the phase space of a DE. Today we ll explore what can happen in 3+ dimensions!

3 Chaos Chaos is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions. Ex. The Butterfly effect Butterfly#/media/ File:Necyria_bellona_manco_No varaexpzoologischetheillepido pteraatlastaf36.jpg

4 Randomness Randomness is an indeterministic unpredictable process which may lead to a probability distribution of outcome possibilities. ex. Flipping a coin. wiki/coin_flipping#/ media/ File:Coin_Toss_( ).jpg

5 Philosophical Analogy: Randomness is to luck as chaos is to fate.

6 Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the atmospheric sciences, 20(2),

7 <latexit sha1_base64="f5aqtwu+evaroeqdxmt89hqrcfi=">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</latexit> <latexit sha1_base64="zmdekty7bdscv0znw+23yiajfyc=">aaacdhicbvdlsgmxfm34rpvvdekmwaqxumdaww4krtcuk9ghdiasstntab5dkhhk0a9w46+4cagiwz/anx9j2s5cww8jnjx7ljf3hdgj2rjut7oyura+sznbym/v7o7tfw4ow1omcpmmlkyqtog0yvsqpqggku6scoihi+1wddottx+i0lskezooscdrqnciymss1csufu0hhnu89xz69oteofr1ojj/qagslavw5zbcgeay8tjsbbkavckx35c44uqyzjdwxc+ntzaizshmzjl3e01ihedoqlqwcssjdtlzmhn4apu+jksyvxg4u393pihrpeahdxjkhnqxnhx/q3ute1wdlio4mutg+aaoydbioe0g9qki2lcxjqgrav8k8raphi3nl29d8bzxxiatcsmrlmp3l8x6drzhdhyde3agphaf6uawneatypainserehoenbfn3fmyw1ecroci/ihz+qohvjgb</latexit> The Lorenz System dx = dt (y x) dy = x dt y xz dz dt = z + xy Famous parameter set: = 10, =8/3, = 28 Very few nonlinear terms Seems very simple

8 <latexit sha1_base64="qb0tmyjcktrth+om+dt0vl8o9ye=">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</latexit> Three equilibrium points: <latexit sha1_base64="1sfhitan/scdexajeua0gmbgi94=">aaacehicbvc7tsmwfhv4lvikmljyvihwqqokimpywcjyjpqqmqhyxke1aifbdpcqqj/awq+wmiaqkymbf4pbzqatx7j0fm69ur7hjxmvyrj+jjxvtfwnzdxwfntnd2/fpdhsyigrmdrwxclr9pekjiakoahipb0lgrjpsmsf3kz81imrkkbhvrrfxoooh9kayqs01dxpilzzn1izft2yx7ryqajugzfnhk611dulvswaai4toymfkkhenb/dxoqttkkfgzkyy1ux8likfmwmjpnuikmm8bd1sufteheivxs60bieaquhg0jogyo4vf92pihloek+ruridesinxh/8zqjcq68lizxokiiz4ocheevwuk6secfwyqnnefyup1xiadiikx0hnkdgr248jjpohx7volcxrrq11kcoxamtkar2kakauaw1eedypaexsabedeejvfjw/icla4ywc8rmipx9qtp85pk</latexit> (0, 0, 0), (±6 p 2, ±6 p 2, 27) J(x, y, z) = z 1 x y x At the origin:

9 At C + =(6 p 2, 6 p 2, 27) <latexit sha1_base64="v6nymonoeydddafov/de4me7ztu=">aaacd3icbzdlsgmxfiyz9vbrbdslm2brkkqzqwldcnvuxfawf2hryarpg5q5mjwry9a3cooruhghifu37nwb03bas/0h8pgfczg5vxmirscypo3ezozc/ejymbw0vlk6zq5vvjqfssrk1be+rdlemce9vgyogtucyyjrcfz1+svrvxrlpok+dwwdgdvd0vv4h1mc2mqzuw1gdxcdar4wr/dpm8cndsmhyg0pvimx32uzastrjywnwy4hjwkvwuzho+3t0gueueguqttwam2isobusggqesowenonxvbx6bgxqwy0vmeid7ttxh1f6ucbhrs/jylikjvwhd3peuipv7wr+v+thklnpblxlwibexsyqbmkdd4ehypbxdikyqcbumn1xzhteuko6ahtogt778ntumll7cns7vioxtip40iilbsnmshgevraf6ieyoiie/sintgl8wa8ga/g26q1ycqzm+ixjpcv186bqa==</latexit>

10 At C =( 6 p 2, 6 p 2, 27) <latexit sha1_base64="m1uof7s5cvuagplfmwicd/iioos=">aaacexicbzc7sgnbfiznvcz4i1radayhggm7uyynee1jgcfcilug2ckkgtj7ceasgja8go2vymohik2dnw/j5flexb8gpv5zdmfo74ackzdnh2nhcwl5ztwxllzf2nzatu3svluqscoqnbcbrltemcf9vgeogtvdyyjnclzze6vhvfbapokbfwv9kdke6fi8zskbbtvtgrvyi8sxgaelu+xfjntmq3sjcx5wpix5wlezltzz5kh4hqwjpnfe5wbq224fnpkyd1qqprqwgyitewmccjzi2pfiiae90menjt7xmhli0uudfkidfm4huj8f8midnoijp1tfc3wnr6crzmtd879ai4l2urnzp4ya+xs8qb0jdaeexonbxdikoq+bumn1xzhtekko6bctogrr9ur5qozz1kkuf3oall5n4kigfxsamshcbvre16imkoiij/sc3tc78wy8gh/g57h1wzjm7ke/mr5+ackqm7a=</latexit>

11 p. 536 in Blanchard, Devaney and Hall

13 Strange Attractor

$experiments suggest the strange attractor has fractal$

14 From Lorenz s original paper: Strogatz: Numerical experiments suggest the strange attractor has fractal dimension 2.05.

15 Sensitive Dependence on Initial Conditions Two ICs: (0,1,0) and (0,1.01,0)

16 Sensitive Dependence on ICs Topological Mixing (Ergocity) from Sarah Iams, Harvard University

17 If this is interesting and you want to learn more:

18 References Alligood, Kathleen T, Tim Sauer, and James A. Yorke. Chaos: An Introduction to Dynamical Systems. New York: Springer, Blanchard, Paul, Robert L. Devaney, and Glen R. Hall. Differential Equations. Boston, MA: Brooks/ Cole, Cengage Learning, Gleick, James. Chaos: Making a New Science. New York, N.Y., U.S.A: Penguin, Iams, Sarah. Lorenz Evolution.nb, (2019). Lorenz, Edward N. "Deterministic nonperiodic flow." Journal of the atmospheric sciences 20.2 (1963): Strogatz, Steven. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Boulder, Colo: Westview Press, The Internet for some pictures :D Weisstein, Eric W. "Lorenz Attractor." From MathWorld--A Wolfram Web Resource. mathworld.wolfram.com/lorenzattractor.html Wikipedia: Wikipedia:

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