  The nonlinear dynamics of a system can be analyzed using a square matrix. If
off resonance, the lead vector of a Jordan chain in a left eigenspace of the
square matrix is an accurate action- angle variable for sufficiently high power
order. The deviation from constancy of the action-angle variable provides a
measure of the stability of a trajectory. However, near resonance or the
stability boundary, the fluctuation increases rapidly and the lead vector no
longer represents an accurate action-angle variable. In this paper we show that
near resonance or stability boundary, it is possible to find a set of linear
combinations of the vectors in the degenerate Jordan chains as the action-angle
variables by an iteration procedure so that the fluctuation is minimized. Using
the Henon-Heiles problem as an example on resonance, we show that when compared
with conventional canonical perturbation theory, the iteration leads to result
in much more close agreement with the forward integration, and the iteration is
convergent even very close to the stability boundary. It is further shown that
the action-angle variables found in the iteration can be used to find another
action-angle variables even much more closer to rigid rotations (KAM
invariants). The fast convergent result is not in the form of polynomials, it
is an exponential function with a rational function in the exponent more
similar to a Laurent series than a Taylor series. Hence the method provides a
new way to probe the boundary of the region where KAM tori exist.
