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This a follow up to my previous article here, 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 λ, and they look very different depending on the value of λ. More precisely, they are generated using the following recursion:

xn+1 = xn λ sin(yn),

yn+1 = xn λ sin(xn),

with initial conditions x0, y0

Seven different groups of three images are displayed. In each group, the leftmost image, a scatterplot (in blue) corresponds to the orbit of (xn, yn) in two dimensions, given the initial conditions. The central images features xn and yn as two time series, with xn in blue and yn 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 (x0, y0) is not included in that image.

Figure 1: x0 = 1, y0 = 4, λ = 0.04

Figure 2: x0 = 1, y0 = 4, λ = 0.06

Figure 3: x0 = 3, y0 = 4, λ = 1.5

Figure 4: x0 = 56, y0 = 4, λ = 0.04

Figure 5: x0 = 2, y0 = 4, λ = 10

Figure 6: x0 = 1, y0 = 4, λ = 2.5

Figure 7: x0 = 3, y0 = 4, λ = 2

As a bonus, here is another picture produced with a different type of chaotic dynamical system. It is discussed here

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About the author:  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 is also self-publisher at, and founded and co-founded a few start-ups, including one with a successful exit (Data Science Central acquired by Tech Target). You can access Vincent's articles and books, here.

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