Standard Map
A two-dimensional map also called the Taylor-Greene-Chirikov map in some of the older literature and defined by
where I and theta are computed mod 2pi and K is a positive constant. Surfaces of section for various values of the constant K are illustrated above.
An analytic estimate of the width of the chaotic zone (Chirikov 1979) finds
| deltaI=Be^(-AK^(-1/2)). |
(4)
|
Numerical experiments give A approx 5.26 and B approx 240. The value of K at which global chaos occurs has been bounded by various authors. Greene's Method is the most accurate method so far devised.
Fixed points are found by requiring that
The first gives Ksintheta_n=0, so sintheta_n=0 and
| theta_n=0,pi. |
(7)
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The second requirement gives
| I_n+Ksintheta_n=I_n=0. |
(8)
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The fixed points are therefore (I,theta)=(0,0) and (0,pi). In order to perform a linear stability analysis, take differentials of the variables
In matrix form,
| [画像: [deltaI_(n+1); deltatheta_(n+1)]=[1 Kcostheta_n; 1 1+Kcostheta_n][deltaI_n; deltatheta_n]. ] |
(11)
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![画像: [deltaI_(n+1); deltatheta_(n+1)]=[1 Kcostheta_n; 1 1+Kcostheta_n][deltaI_n; deltatheta_n].](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/StandardMap/NumberedEquation4.svg){kind=link}
The eigenvalues are found by solving the characteristic equation
so
| lambda^2-lambda(Kcostheta_n+2)+1=0 |
(13)
|
| lambda_+/-=1/2[Kcostheta_n+2+/-sqrt((Kcostheta_n+2)^2-4)]. |
(14)
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For the fixed point (0,pi),
The fixed point will be stable if |R(lambda^((0,pi)))|<2. Here, that means
| 1/2|2-K|<1 |
(17)
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| |2-K|<2 |
(18)
|
| -2<2-K<2 |
(19)
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| -4<-K<0 |
(20)
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so K in [0,4). For the fixed point (0, 0), the eigenvalues are
If the map is unstable for the larger eigenvalue, it is unstable. Therefore, examine lambda_+^((0,0)). We have
| [画像: 1/2|2+K+sqrt(K^2+4K)|<1, ] |
(23)
|
so
| -2<2+K+sqrt(K^2+4K)<2 |
(24)
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| -4-K<sqrt(K^2+4K)<-K. |
(25)
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But K>0, so the second part of the inequality cannot be true. Therefore, the map is unstable at the fixed point (0, 0).
See also
Hénon-Heiles EquationExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Standard MapCite this as:
Weisstein, Eric W. "Standard Map." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StandardMap.html