Standard map: This is a demo implementation with 20 lines of Javascript. The browser actually computes the Lyapunov exponent of an orbit and the colors the orbit accordingly. For previous implementations in Java or C. Javascript is surprisingly fast when writing directly into the canvas matrix as pointer arithmetic does in C. One does not have to worry about memory allocations. The dynamical system ``` T(x,y) = (2x-y+G sin(x),x) ``` displayed here is not only one of the icons of chaos, it also remains an enigma. Even so measurements clearly indicate that the entropy should be bound below by log(G/2), we can not prove it. My pages on this from 1999, when I thought of having been able to find a proof. The entropy problem belongs to one of the most important open problems in ergodic theory. I burned myself several times on it: As an undergraduate, in 1987, I have defined the analytic map ```T(z,w,,u,v) = (z w e(z-u), w e(z-u), u v e(u-z), v e(u-z)) ``` on the complex 4 manifold C4 which has the 2-tori { (z,w,u,v) | |z|=|u|=G/2, |w|=|v|=1, z u=G2/4, w v=1 } invariant and displays there a conjugate T(x,y)=(x+y+ G sin(x),y+ G sin(x)) of the Standard map. I argued that since the Jacobean cocycle A(x,y) = z dT(x,y) is now analytic in z,w, that one can use the by Herman established fact that the Lyapunov exponent is subharmonic on complex parameters and therefore plurisubharmonic. I proudly wrote it down and showed it to my undergraduate advisor Jürgen Moser. He overnight found the mistake: the tori under considerations are not the boundaries of poly-discs. The estimate is not valid. I tried for many years more and crashed again, this time for good.
Oliver Knill, 7/15/2015