Standard Map
المؤلف:
Celletti, A. and Chierchia, L
المصدر:
"A Constructive Theory of Lagrangian Tori and Computer-Assisted Applications." Dynamics Rep. 4
الجزء والصفحة:
...
11-9-2021
2153
Standard Map
A two-dimensional map also called the Taylor-Greene-Chirikov map in some of the older literature and defined by
where 
and 
are computed mod 
and 
is a positive constant. Surfaces of section for various values of the constant 
are illustrated above.
An analytic estimate of the width of the chaotic zone (Chirikov 1979) finds
 |
(4)
|
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 |
exact |
approx. |
| Hermann |
 |
 |
0.029411764 |
| Celletti and Chierchia (1995) |
 |
 |
0.838 |
| Greene |
 |
- |
0.971635406 |
| MacKay and Percival (1985) |
 |
 |
0.984375000 |
| Mather |
 |
 |
1.333333333 |
Fixed points are found by requiring that
The first gives 
, so 
and
 |
(7)
|
The second requirement gives
 |
(8)
|
The fixed points are therefore 
and 
. In order to perform a linear stability analysis, take differentials of the variables
In matrix form,
![[deltaI_(n+1); deltatheta_(n+1)]=[1 Kcostheta_n; 1 1+Kcostheta_n][deltaI_n; deltatheta_n].](https://mathworld.wolfram.com/images/equations/StandardMap/NumberedEquation4.gif) |
(11)
|
The eigenvalues are found by solving the characteristic equation
 |
(12)
|
so
 |
(13)
|
![lambda_+/-=1/2[Kcostheta_n+2+/-sqrt((Kcostheta_n+2)^2-4)].](https://mathworld.wolfram.com/images/equations/StandardMap/NumberedEquation7.gif) |
(14)
|
For the fixed point 
,
The fixed point will be stable if 
Here, that means
 |
(17)
|
 |
(18)
|
 |
(19)
|
 |
(20)
|
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
 |
(23)
|
so
 |
(24)
|
 |
(25)
|
But 
, so the second part of the inequality cannot be true. Therefore, the map is unstable at the fixed point (0, 0).
REFERENCES:
Celletti, A. and Chierchia, L. "A Constructive Theory of Lagrangian Tori and Computer-Assisted Applications." Dynamics Rep. 4, 60-129, 1995.
Chirikov, B. V. "A Universal Instability of Many-Dimensional Oscillator Systems." Phys. Rep. 52, 264-379, 1979.
MacKay, R. S. and Percival, I. C. "Converse KAM: Theory and Practice." Comm. Math. Phys. 98, 469-512, 1985.
Rasband, S. N. "The Standard Map." §8.5 in Chaotic Dynamics of Nonlinear Systems. New York: Wiley, pp. 11 and 178-179, 1990.
Tabor, M. "The Hénon-Heiles Hamiltonian." §4.2.r in Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, pp. 134-135, 1989.
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