Ruelle and Takens Theorem - Demonstration

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

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Last modified Friday, February 11, 2000
Michael Cross