Nonlinear system response

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Joined Jun 13, 2014

Suppose we a system
sin(w*t) = m*y'' + C*y' + k*y
For small values of C we get a response like this:

Now consider this system:
sin(w*t) = m*y'' + C*y' + k(y)*y
Are there known functions for k(y) that reduce the gain across parts or all of the frequency range? Is there anything that can be said about the location of the resonant frequency e.g. that it becomes dependent of the forcing amplitude or things like that?