The dissipative term in the Rayleigh oscillator contains a nonlinear (cubic) contribution. Usually, the coefficient, ε, of the dissipative term is assigned a small value. The solution evolves into a limit cycle with O(1) maximal amplitude, maximal velocity and period. In this paper, the case, in which this coefficient is extremely large (ε ≥ 1), is studied. As expected, for long times, the solution tends to a limit cycle. The characteristics of the solution are different from those encountered in energy conserving oscillatory systems in which the magnitude of the nonlinear term is increased indefinitely. Based on numerical solutions of the equation, it is found that the amplitude and period grow linearly with ε, whereas the maximal velocity remains O(1). The consequences of the linear growth with ε of amplitude and period are studied through the definition a scaled solution and a scaled time variable, an angle θ, in which the scaled form of the limit cycle is periodic.
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