steps: given the following parametric equations: \(x_{n+1} = a_0 + a_1x + a_2x^2 + a_3y + a_4y^2 + a_5xy\) \(y_{n+1} = a_6 + a_7x + a_8x^2 + a_9y + a_{10}y^2 + a_{11}xy\) 1. assign random values (from -1 to 1) to all values of \(a_i\) it will look something like this: \(x_{n+1} = -0.3 - 0.5x - 0.9x^2 + 0.9y + 0.8y^2 - 0.6xy\) \(y_{n+1} = -0.1 + 0.9x - 0.6x^2 - 0.3y - 0.7y^2 + 0.9xy\) we now have our attractor! ...but what do we actually do with it? 2. pick an initial point, lets call it \(P\) for simplicity we can start at \((0, 0)\), giving us \(P = (0, 0)\) using our \(x_{n+1}\) and \(y_{n+1}\) equations we can plug \(P\) into them as many times as we wish, generating a new point \(P_{n+1}\) using the equation \(P_{n+1} = (x_{n+1}(P_x, P_y), y_{n+1}(P_x, P_y))\) 3. calculate \(P_{n+1}\) and plot! 1 use the slider to generate points 2 through 10 this is what we get after repeating step 3 nine times!
the result of our work is a little cool--at least to me--but it's really nothing "strange". why then are they called "strange attractors" and what is an "attractor" anyways? put simply, an attractor is a system that attracts points to certain positions in 2D space. in the example above, our point \(P\) is being attracted to the point (-infinity, -infinity), which is called diverging. since we randomly generated our example, there's an infinite number of other attractors! some of which converge to a single point or certain paths. okay... but what makes them strange? well, 99% of the time an attractor will simply diverge to infinity, but the rest you will find act pretty strangely. that's really the only reason they're named that! idk abt you, but i really find these beautiful. at least some of them. you can use the "generate new attractor" button if you're displeased add dimension