text: Rigorously, the oscillation is periodic only at . Indeed, in the realization of the Toda oscillator as a self-pulsing laser, these parameters may have values of order of ; during several pulses, the amplitude of pulsation does not change much. In this case, we can speak about the period of pulsation, since the function  is almost periodic.
  In the case , the energy of the oscillator  does not depend on , and can be treated as a constant of motion. Then, during one period of pulsation, the relation between  and  can be expressed analytically:
  where
   and  are minimal and maximal values of ; this solution is written for the case when .
  however, other solutions may be obtained using the principle of translational invariance.
  The ratio  is a convenient parameter to characterize the amplitude of pulsation.  Using this, we can express the median value
  as
  and the energy
  is also an elementary function of .
  In application, the quantity  need not be the physical energy of the system; in these cases, this dimensionless quantity may be called quasienergy.