12.1.8 (Hnon's area-preserving quadratic map) The map Xn+1 = x, cosa-(y - x) sina Yn+1 =...

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12.1.8 (Hénon's area-preserving quadratic map) The map Xn+1 = x₁, cosa-(y₁ - x²) sina Yn+1 = x, sina +(y₁ - x²) cos a illustrates many of the remarkable properties of area-preserving maps (Hénon 1969, 1983). Here 0 ≤ a ≤ is a parameter. a) Verify that the map is area-preserving. 911 b) Find the inverse mapping. c) Explore the map on the computer for various a. For instance, try cos a = 0.24, and use initial conditions in the square-1 ≤x, y ≤ 1. You should be able to find a lovely chain of five islands surrounding the five points of a period-5 cycle. Then zoom in on the neighborhood of the point x = 0.57, y = 0.16. You'll see smaller islands, and maybe even smaller islands around them! The complexity extends all the way down to finer and finer scales. If you modify the parameter to cosa = 0.22, you'll still see a prominent chain of five islands, but it's now surrounded by a noticeable chaotic sea. 12.1.8 (Hénon's area-preserving quadratic map) The map Xn+1 = x₁, cosa-(y₁ - x²) sina Yn+1 = x, sina +(y₁ - x²) cos a illustrates many of the remarkable properties of area-preserving maps (Hénon 1969, 1983). Here 0 ≤ a ≤ is a parameter. a) Verify that the map is area-preserving. 911 b) Find the inverse mapping. c) Explore the map on the computer for various a. For instance, try cos a = 0.24, and use initial conditions in the square-1 ≤x, y ≤ 1. You should be able to find a lovely chain of five islands surrounding the five points of a period-5 cycle. Then zoom in on the neighborhood of the point x = 0.57, y = 0.16. You'll see smaller islands, and maybe even smaller islands around them! The complexity extends all the way down to finer and finer scales. If you modify the parameter to cosa = 0.22, you'll still see a prominent chain of five islands, but it's now surrounded by a noticeable chaotic sea.

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a To verify that the map is areapreserving you can check if the Jac... View the full answer