Grebogi, Ott and Yorke considered the following map:
|xn+1||=||xn + w1 +a k P1(xn,yn) mod 1|
|yn+1||=||yn + w2 +a k P2(xn,yn) mod 1|
(think of x and y as two angle variables corresponding to motion around two sections of the 3-torus with "bare" frequencies w1, w2). The nonlinear functions Pi are sums of sinusoidal functions A (i)rs sin[2(rx+sy+B (i)rs )], with (r,s)=(0,1),(1,0),(1,1), or (1,-1). GOY used values of A (i)rs,B (i)rs selected randomly from the range -1 to 1:
|r,s||A (1)rs||B (1)rs||A (2)rs||B (2)rs|
The parameter k=0.107, and then a gives the strength of the nonlinearity with a=1 corresponding to the point at which the map (for these values of A,B) becomes noninvertible.
The question of interest is the relative probability of the following types of dynamics:
|Type of motion||Poincare section||Lyapunov Exponents|
(The Poincare section is not very useful in distinguishing the 3-frequency quasiperiodic motion and chaotic motion.) In the applet the values of w1 and w2 are rechosen at random between 0 and 1 each time the evolution is reset using the "Reset" button or by clicking outside the stopped plot. Do this several times for each value of a to get some idea of the probability of finding the different types of dynamics.