{"title":"Properties of the Julia Set","authors":"R. Devaney","doi":"10.1201/9780429280801-21","DOIUrl":"https://doi.org/10.1201/9780429280801-21","url":null,"abstract":"","PeriodicalId":314009,"journal":{"name":"An Introduction to Chaotic Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130381598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dynamics of Linear Maps","authors":"R. Devaney","doi":"10.1201/9780429280801-27","DOIUrl":"https://doi.org/10.1201/9780429280801-27","url":null,"abstract":"Create a variable A initialized to be the above matrix (check the command matrix): sage: # edit here Create a variable v that contains the vector (1,1) (check the command vector): sage: # edit here To compute the image of a vector under a matrix you need to use the multiplication *. Compute the image A * v: sage: # edit here Write a function draw_orbit(A, v0, n) that draw the n-th first iterate of the orbit of v0 under A. That is the sequence of vectors v0, Av0, Av0, ..., Av0 (you can use the graphics primitives point2d or line2d): sage: # edit here On the same graphics, draw several of these orbits: sage: # edit here What is happening to them? The object AA in Sage stands for \"real algebraic numbers\". It is one way to consider exact numbers beyond integers and rationals.: sage: AA In the following cell, we create the algebraic number √ 5 and the golden ratio φ = 1+ √ 5 2 : sage: a = AA(5).sqrt() sage: phi = (1 + a) / 2 Check with Sage that the vectors u+ = (1,−φ) and u− = (1,φ −1) are eigenvectors of the matrix A: sage: up = vector([1, -phi]) sage: um = vector([1, phi-1]) sage: # edit here Create a graphics with several orbits together with the lines Ru+ and Ru−: sage: # edit here","PeriodicalId":314009,"journal":{"name":"An Introduction to Chaotic Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131897135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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