Abstract: The nonlinear dynamics of the Gaussian beam propagation in resonator was studied by constructing the iterative map of q-parameter. From the Greene's residue theorem, there are specific configurations sensitive to nonlinear effect existing in the geometrically stable region. By applying to Kerr-lens mode-locking (KLM) resonators, we found that multiple solution, period doubling, period tripling and period quadrupling can occur at the configurations with product of cavity G-parameters equal to 0, 1/2, 1/4 (or 3/4) and (2 $POM $YLD@2)/4, respectively. Moreover, the systems will result in classical chaos if further increasing the nonlinear effect. We will also report appearance of unexpected transverse beam profiles and shrinkage of beam waist causes to lower laser threshold in an end-pumped Nd:YVO$-4$/ laser and chaotic behavior in the KLM laser around these critical resonator configurations.!5
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