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The nonlinear dynamics of a system can be analyzed using a square matrix. Ifoff resonance, the lead vector of a Jordan chain in a left eigenspace of thesquare matrix is an accurate action- angle variable for sufficiently high powerorder. The deviation from constancy of the action-angle variable provides ameasure of the stability of a trajectory. However, near resonance or thestability boundary, the fluctuation increases rapidly and the lead vector nolonger represents an accurate action-angle variable. In this paper we show thatnear resonance or stability boundary, it is possible to find a set of linearcombinations of the vectors in the degenerate Jordan chains as the action-anglevariables by an iteration procedure so that the fluctuation is minimized. Usingthe Henon-Heiles problem as an example on resonance, we show that when comparedwith conventional canonical perturbation theory, the iteration leads to resultin much more close agreement with the forward integration, and the iteration isconvergent even very close to the stability boundary. It is further shown thatthe action-angle variables found in the iteration can be used to find anotheraction-angle variables even much more closer to rigid rotations (KAMinvariants). The fast convergent result is not in the form of polynomials, itis an exponential function with a rational function in the exponent moresimilar to a Laurent series than a Taylor series. Hence the method provides anew way to probe the boundary of the region where KAM tori exist.

A new method to probe the boundary where KAM tori persist by square matrix