(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: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 |
| 1,0 | -0.26813663648754 | 0.98546084298505 | 0.08818611671542 | 0.99030722865609 |
| 0,1 | -0.91067559396390 | 0.50446045609351 | -0.56502889980448 | 0.33630697012268 |
| 1,1 | 0.31172026382793 | 0.94707472523078 | 0.16299548727086 | 0.29804921230971 |
| 1,-1 | -0.04003977835470 | 0.23350105508507 | -0.80398881978155 | 0.15506467277737 |
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 |
| 3-frequency QP | Filling | 0,0 |
| 2-frequency QP | Curve | 0,- |
| Periodic | Points | -,- |
| Chaotic | Filling? Fractal? | +,? |
(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.
| a | Applet |
| 0.375 | |
| 0.75 | |
| 1.125 |