2d:null 2e:["$","title",null,{"children":"Assignment #6 | KarlieDoesNotExist | Observable"}] 33:T48b,md` ## Reading [Random Attractors Found using Lyapunov Exponents](http://paulbourke.net/fractals/lyapunov/) by Paul Bourke
[Fractals and Attractors](https://towardsdatascience.com/from-fractals-to-attractors-b0e4ce9dcc71) by Danielle Romanoff
In the mathematical field of dynamical systems, an attractor is **a set of states** toward which a system tends to evolve, for a wide variety of **starting conditions** of the system. We'll have a system of equations, and certain parameters to model the dynamic system. There are a number of ways the series may behave. - converge to a single point, called a fixed point - diverge to infinity - form a periodic orbit - exhibit chaos, filling in some region of the plane The first article above talks about more in-depth algorithms, including how to identify new systems and their coefficients with the Lyapunov exponent. The second shows us how attractors can be implemented with computer programming. Basically, we can get an abstract pattern by iterating the equation for thousands (millions) of times. The rendered image indicates the distribution of the points running in the system. `34:[{"id":"e0d586015ee49b36eaa8392a12eff90f0c895de48b594bb39c5631b1f74e3e03f48ccaaeba8267043b6b2ed1bb774fb361eecfd0c167ba14feb188f54e2f524b","href":"https://static.observableusercontent.com/files/e0d586015ee49b36eaa8392a12eff90f0c895de48b594bb39c5631b1f74e3e03f48ccaaeba8267043b6b2ed1bb774fb361eecfd0c167ba14feb188f54e2f524b","name":"12.png","type":"image/png","lastModified":1634648505958.656,"size":1618658},{"id":"f7847e27e9cf7f57b50317130f62d48cd0041821cdbf91edb5cdd85167f32538b30617824c3f408dd23bb706a75be0f04e1e3b1467b033087775dd8bf6bdb141","href":"https://static.observableusercontent.com/files/f7847e27e9cf7f57b50317130f62d48cd0041821cdbf91edb5cdd85167f32538b30617824c3f408dd23bb706a75be0f04e1e3b1467b033087775dd8bf6bdb141","name":"15.png","type":"image/png","lastModified":1634657108028.93,"size":1475364},{"id":"c254ccbf754b785c495ec3aeb2eabad64e941818bfa1dc83401fe65be7f496c5bbe69f2a4e1c5a4a7dbbff485267deaee5bc6590137a6515b5119c954ec2f84e","href":"https://static.observableusercontent.com/files/c254ccbf754b785c495ec3aeb2eabad64e941818bfa1dc83401fe65be7f496c5bbe69f2a4e1c5a4a7dbbff485267deaee5bc6590137a6515b5119c954ec2f84e","name":"14.png","type":"image/png","lastModified":1634648503260.869,"size":1242246},{"id":"b35af66ab724dc4f601675b05fe43e4dfbec5adde2cc11ddc7e9b9b413bb7887699e3e3038964d4e7ca4b564d0194b54aed31f75efa062e6b5199e0c28a4a6e9","href":"https://static.observableusercontent.com/files/b35af66ab724dc4f601675b05fe43e4dfbec5adde2cc11ddc7e9b9b413bb7887699e3e3038964d4e7ca4b564d0194b54aed31f75efa062e6b5199e0c28a4a6e9","name":"13.png","type":"image/png","lastModified":1634648505422.813,"size":1576939},{"id":"ba7f16a5812d85c54bbb7c90c851799d3e1f9f4812127fbb2f4b66482796c6f065f941b3524f976798671960f12c2640b3e9e5da54ee26537ffefbb2d214b2a3","href":"https://static.observableusercontent.com/files/ba7f16a5812d85c54bbb7c90c851799d3e1f9f4812127fbb2f4b66482796c6f065f941b3524f976798671960f12c2640b3e9e5da54ee26537ffefbb2d214b2a3","name":"5.png","type":"image/png","lastModified":1634648506339.402,"size":1711423}] 31:{"id":"21769fbb8ea6c854","slug":"assignment-6","title":"Assignment #6","thumbnail":"8089ef77d95a70c5e9bbf27cc11d9888f8987b78ea06b8433e108c93dd1991a6","access_level":"public","create_time":"$D2021-10-19T05:45:33.492Z","update_time":"$D2021-10-20T10:53:04.769Z","view_time":null,"first_public_time":"$D2021-10-19T13:25:07.965Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"b34c988451399e98","login":"wenrazhao2","name":"KarlieDoesNotExist","avatar_url":"https://avatars-next.observableusercontent.com/64edd542e55cff350a7503e5050267fb1768e8cc5052184b8a8cb49162d172f3","tier":"starter_2024"},"version":179,"publish_version":179,"role":null,"authored":false,"authors":[{"type":"individual","id":"302ba7635bb9b25d","login":"wenrazhao2","name":"KarlieDoesNotExist","avatar_url":"https://avatars-next.observableusercontent.com/64edd542e55cff350a7503e5050267fb1768e8cc5052184b8a8cb49162d172f3","tier":null}],"members":[{"type":"individual","id":"302ba7635bb9b25d","login":"wenrazhao2","name":"KarlieDoesNotExist","avatar_url":"https://avatars-next.observableusercontent.com/64edd542e55cff350a7503e5050267fb1768e8cc5052184b8a8cb49162d172f3","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":true,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":{"id":"a2245910e29a736f","slug":"generative-art","title":"Generative Art","owner":{"id":"b34c988451399e98","login":"wenrazhao2","name":"KarlieDoesNotExist"}},"type":"notebook","body":{"title":"Assignment #6","cells":[{"id":0,"value":"# Assignment #6","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":8,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":43,"value":"md`\n\n👉🏻 ** [more about attractors](http://paulbourke.net/fractals/peterdejong/peterdejong.pdf)**\n\n👉🏻 [Self-Organizing Two-Dimensional Cellular Automata](https://link.springer.com/content/pdf/10.1007/978-3-319-27270-2_1.pdf) by David Griffeath\n

\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":82,"value":"html`\n\n

Some experiments

\n\n

\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":104,"value":"md`\n~~~js\n// Variable-updating Function\n// x, y are variables, a,b,c,d are parameters defined by the attractors' positions\n// e, f are coefficients\n\nxn = sin(a * y) - cos(b * x);\nyn = sin(c * x) - cos(d * y);\nzn = sin(e * x) - cos(f * z);\n\n// zn (z-coordinate) is what we want to draw the curve\n// map z-Coordinate to color pallete\nr[0] = z * 0.9 + (1.0 - zn) * 0.6; \nr[1] = z * 0.2 + (1.0 - zn) * 0.4;\nr[2] = z * 0.5 + (1.0 - zn) * 0.9;\n\n~~~\n\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W34","stdlib":"1","titling":"auto"}} 32:"$31:body"