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.

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.