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).

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.