Equivalent non-linear potential systems: review of previous assumptions

In the field of stochastic dynamics there is a few exact solutions for the response of dynamical systems. Thus, the methods of the equivalent linearization and of the equivalent nonlinearization are often used. While the former yields a Gaussian response to a Gaussian excitation, the latter gives a non Gaussian response, which is nearer to the exact unknown response of a non linear system. Among the methods of equivalent non-linearization that based on the replacement of the actual dynamic system by means of a potential system stands up. In fact, this method leads to a general procedure differently from other non-linearization methods. The procedure is developed basing on the assumption that the ratio  is constant and equal to one, being  the ratio of the cross moment of the mechanical energy  powered to j and the square of the velocity and the moment E[j1] . This relationship lacks of an analytical demonstration. In this paper, numerical analysis are presented to ascertain its validity without resorting to Monte Carlo simulation. It is found that in most cases it holds true, but some others are doubtful. The effects of considering  constant when it is not are ascertained for a Duffing oscillator with linear plus cubic damping