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We have applied harmonic expansion to derive an analytical solution for theLorenz-Haken equations. This method is used to describe the regular andperiodic self-pulsing regime of the single mode homogeneously broadened laser.These periodic solutions emerge when the ratio of the population decay rates issmaller than 0.11. We have also demonstrated the tendency of the Lorenz-Hakendissipative system to behave periodic for a characteristic pumping rate "2CP"[4], close to the second laser threshold "2C2th" (threshold of instability).When the pumping parameter "2C" increases, the laser undergoes aperiod-doubling sequence. This cascade of period doubling leads towards chaos.We study this type of solutions and indicate the zone of the control parametersfor which the system undergoes irregular pulsing solutions. We had previouslyapplied this analytical procedure to derive the amplitude of the first, thirdand the fifth order harmonics for the laser-field expansion [4, 14]. In thiswork, we extend this method in the aim of obtaining the higher harmonics. Weshow that this iterative method is indeed limited to the fifth order, and thatabove, the obtained analytical solution diverges from the numerical directresolution of the equations.

The Lorenz model for single-mode homogeneously broadened laser: analytical determination of the unpredictible zone