Grebogi, Ott and Yorke considered the following map:

x_{n+1} |
= | x_{n} + w_{1} +a k P_{1}(x_{n},y_{n}) mod 1 |

y_{n+1} |
= | y_{n} + w_{2} +a k P_{2}(x_{n},y_{n}) mod 1 |

(think of *x* and *y* as two angle variables corresponding to motion around two sections of the 3-torus with "bare" frequencies *w _{1}, w_{2}*). The nonlinear functions

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 *w _{1}* and

a | Applet |

0.375 | |

0.75 | |

1.125 |

[Demos Introduction] [Introduction]

Last modified Friday, February 11, 2000

Michael Cross