What Is Stability in the Context of the Hill Equation for a Mass-Spring System?

[Drokz]

When trying to solve a problem I arrive at the following equation of motion / Hill equation:

\frac{d^{2}y}{dx^2} + \frac{4 k_0}{m w^2} cos(2x)y = 0

There exists a value x_0 such that for all x>x_0 the motion is stable.

I actually don't know what is meant by this 'stability'. Can someone help, please?

Thanks, Drokz

 

[marcusl]

I've usually seen Hill's equation with omega^2, not omega^(-2). Is this a typo?

 

[Drokz]

I don't think it is a typo. Omega is just a constant here, I think.

 

[marcusl]

Ok. Solutions are products of exponential terms and periodic functions, so your stability condition is needed to keep the exponential parts of the solutions bounded. Do you have access to a reference on Mathieu functions like Whittaker and Watson, or Erdelyi's Higher Transcendental Functions? The latter one has a clear discussion of the stable and unstable solution regions (p. 101 vol. 3).