Linear systems whose parameters are independent of time possess, as has been abundantly illustrated already, well-defined normal modes from which their motion can be synthesized by superposition; and the response to an applied force, varying with time, can be written in terms of the response to each separate Fourier component of the force. The same is not true of non-linear systems, since superposition is no longer a valid procedure for synthesizing the response. Every anharmonic system responds differently to a given form of time-dependent force, and even when the response has been found in any special case it will not scale up unchanged in response to an amplification of the force. Thus the response to a sinusoidal force is in general non-sinusoidal, the waveform changing with the amplitude of the force. There are very few general statements that can be made about the character of the response. One cannot even assert that the oscillations of the vibrator will have the same fundamental frequency as the applied force – it may respond at a subharmonic frequency, i.e. an integral submultiple of that of the force, or the response may be asynchronous to the point of randomness. Even when order prevails, with regular vibration at the fundamental or subharmonic frequency, changing the amplitude or frequency of the applied force to an infinitesimal degree may have the effect of throwing the response into an entirely new pattern.