This a follow up to my previous article here<\/a>, where you can find additional, very different images, the theory behind it, and relevance to machine learning techniques. What is surprising is that all these images were produced with a formula with a single parameter \u03bb<\/em>, and they look very different depending on the value of \u03bb<\/em>. More precisely, they are generated using the following recursion:<\/p>\n x<\/em>n<\/em>+1<\/span>\u00a0<\/span>=\u00a0<\/span>xn<\/span><\/em>\u00a0<\/span>+\u00a0\u03bb<\/em>\u00a0<\/span>sin(yn<\/span><\/em>),<\/p>\n y<\/em>n<\/em>+1<\/span>\u00a0<\/span>=\u00a0<\/span>xn<\/span><\/em>\u00a0<\/span>+\u00a0\u03bb<\/em>\u00a0<\/span>sin(xn<\/span><\/em>),<\/p>\n with initial conditions x<\/em>0<\/span>, y<\/em>0<\/span>.\u00a0<\/p>\n Seven different groups of three images are displayed. In each group, the leftmost image, a scatterplot (in blue) corresponds to the orbit of (xn<\/span><\/em>, yn<\/span><\/em>) in two dimensions, given the initial conditions. The central images features xn<\/span><\/em> and yn<\/span><\/em> as two time series, with xn<\/span><\/em> in blue and yn<\/span><\/em> in red. In both cases, 20,000 iterations are used. The rightmost image is the same as the leftmost one, except that only the first 25 iterations are displayed, and a green curve connects the 25 dots, to show how the orbit looks like at the beginning. The initial vector (x<\/em>0<\/span>, y<\/em>0<\/span>) is not included in that image.<\/p>\n<\/p>\n Figure 1<\/strong>: x0<\/span> = 1, y0<\/span> = 4,\u00a0\u03bb = 0.04<\/em><\/p>\n \n Figure 2<\/strong>: x0<\/span> = 1, y0<\/span> = 4,\u00a0\u03bb = 0.06<\/em><\/p>\n \n Figure 3<\/strong>: x0<\/span> = 3, y0<\/span> = 4,\u00a0\u03bb = 1.5<\/em><\/p>\n<\/p>\n Figure 4<\/strong>: x0<\/span> = 56, y0<\/span> = 4,\u00a0\u03bb = 0.04<\/em><\/p>\n \n Figure 5<\/strong>: x0<\/span> = 2, y0<\/span> = 4,\u00a0\u03bb = 10<\/em><\/p>\n<\/p>\n Figure 6<\/strong>: x0<\/span> = 1, y0<\/span> = 4,\u00a0\u03bb = 2.5<\/em><\/p>\n<\/p>\n Figure 7<\/strong>: x0<\/span> = 3, y0<\/span> = 4,\u00a0\u03bb = 2<\/em><\/p>\n<\/p>\n About the author<\/strong>:\u00a0 Vincent Granville is a data science pioneer, mathematician, book author (Wiley), patent owner, former post-doc at Cambridge University, former VC-funded executive, with 20+ years of corporate experience including CNET, NBC, Visa, Wells Fargo, Microsoft, eBay. Vincent also founded and co-founded a few start-ups, including one with a successful exit (Data Science Central acquired by Tech Target).<\/span>\u00a0He is also the founder and investor in\u00a0<\/span>Paris Restaurant<\/a>\u00a0<\/span>in Anacortes, WA. You can access Vincent’s articles and books,\u00a0<\/span>here<\/a>.<\/em><\/p>\n<\/div>\n Go to Source<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<\/a><\/p>\n
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