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# Standard Map

A two-dimensional map also called the Taylor-Greene-Chirikov map in some of the older literature and defined by

 (1) (2) (3)

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

 (5) (6)

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

 (9) (10)

In matrix form,

 (11)

The eigenvalues are found by solving the characteristic equation

 (12)

so

 (13)
 (14)

For the fixed point ,

 (15) (16)

The fixed point will be stable if Here, that means

 (17)
 (18)
 (19)
 (20)

so . For the fixed point (0, 0), the eigenvalues are

 (21) (22)

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).

Hénon-Heiles Equation

## 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.

Standard Map

## Cite this as:

Weisstein, Eric W. "Standard Map." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StandardMap.html