A 2-D Map, also called the Taylor-Greene-Chirikov Map in some of the older literature.
where and are computed mod and is a Positive constant. An analytic estimate of the width of the
Chaotic zone (Chirikov 1979) finds
|
(3) |
Numerical experiments give and . The value of at
which global Chaos occurs has been bounded by various authors.
Greene's Method is the most accurate method so far devised.
Author |
Bound |
Fraction |
Decimal |
Hermann |
|
|
0.029411764 |
Italians |
|
- |
0.65 |
Greene |
|
- |
0.971635406 |
MacKay and Pearson |
|
|
0.984375000 |
Mather |
|
|
1.333333333 |
Fixed Points are found by requiring that
The first gives
, so
and
|
(6) |
The second requirement gives
|
(7) |
The Fixed Points are therefore
and . In order to perform a
Linear Stability analysis, take differentials of the variables
In Matrix form,
|
(10) |
The Eigenvalues are found by solving the Characteristic Equation
|
(11) |
so
|
(12) |
|
(13) |
For the Fixed Point ,
The Fixed Point will be stable if
Here, that means
|
(15) |
|
(16) |
|
(17) |
|
(18) |
so . For the Fixed Point (0, 0), the Eigenvalues are
If the map is unstable for the larger Eigenvalue, it is unstable. Therefore,
examine
. We have
|
(20) |
so
|
(21) |
|
(22) |
But , so the second part of the inequality cannot be true. Therefore, the map is unstable at the Fixed
Point (0, 0).
References
Chirikov, B. V. ``A Universal Instability of Many-Dimensional Oscillator Systems.'' Phys. Rep. 52,
264-379, 1979.
© 1996-9 Eric W. Weisstein
1999-05-26