Two Dimensional Maps - Equations
Henon Map
| xn+1 |
= |
yn + 1 - a xn2 |
| yn+1 |
= |
b xn2 |
Circle Map
| xn+1 |
= |
xn + c + yn+1/2 |
| yn+1 |
= |
b yn - a sin(2xn) |
Duffing Map
| xn+1 |
= |
yn |
| yn+1 |
= |
- b xn + a yn - yn3 |
Baker's Map
| xn+1 = b xn |
yn+1 = yn/a |
if yn < a |
| xn+1 = (1 - c) + c xn |
yn+1 = (yn - a)/(1 - a) |
if yn > a |
Kaplan-Yorke Map
| xn+1 |
= |
a xn mod 1 |
| yn+1 |
= |
- b yn + cos(2xn) |
Standard Map
| xn+1 |
= |
xn + yn+1/2 |
| yn+1 |
= |
yn + a sin(2xn) |
Grebogi-Ott-Yorke Map
| xn+1 |
= |
xn + w1 +a P1(xn,yn) mod 1 |
| yn+1 |
= |
yn + w2 +a P2(xn,yn) mod 1 |
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 selected a set of A (i)rs,B (i)rs randomly in the range 0 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 parameters w1 and w2 are reset with random values between 0 and 1 each time the evolution is reset using the "Reset" button or by clicking outside the stopped plot.
Sinai Map
| xn+1 |
= |
xn + yn + a cos(2yn) mod 1 |
| yn+1 |
= |
xn + 2 yn mod 1 |
My Function
[Instructions]
Last modified Thursday, March 2, 2000
Michael Cross