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It is well known that for the simple harmonic oscillator of the form

$$ \frac{d^{2}x}{dt^{2}} = -\omega_{s}^{2}x $$ where $\omega_{s}$ is the resonant frequency of the oscillation, has the general solution $$ x(t) = x_{0}\text{cos}(\omega_{s}t) + \frac{v_{0}}{\omega_{s}}\text{sin}(\omega_{s}t) $$

Suppose now I modulate my resonant frequency periodically about some frequency $\omega_{0}$ with a pump frequency $\omega_{p}$, such that $\omega_{s}=\omega_{s}(t)=\omega_{0} + \delta\omega\text{sin}(\omega_pt)$, where we assume that $\delta\omega\ll\omega_{0}$, how do I find the general solution of $x(t)$ for $\omega_p = 2\omega_{s}$?

Attempt: Here is my initial attempt at the problem. I first substitute the time-dependent $\omega_{s}$ back into the first equation, such that $$ \frac{d^{2}x}{dt^{2}} = -\omega_{s}^{2}(t)x \approx -\omega_{0}^{2}\Big(1 + 2\frac{\delta_{\omega}}{\omega_{0}}\text{sin}(\omega_{p}t)\Big)x $$ where I have neglected terms which are quadratic in $\delta\omega$ (since $\delta\omega^{2}\ll1$). I am unsure of how to proceed from here to solve the preceding equation. I tried guessing an ansatz solution of the form $$ x(t) = x_{0}e^{\lambda t}\text{cos}(\omega_{s}t) $$ for some constant $\lambda$ (probably a function of $\delta\omega$), but substituting the ansatz solution back into the modified differential equation shows that it is not a valid solution (or I've made a mistake that I couldn't find). I do know that the solution must have some exponential dependence that shows that the $x(t)$ grows over time, signifying amplification. In general, how can I solve this problem and what should be the correct approach here?

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    $\begingroup$ see en.wikipedia.org/wiki/Mathieu_function $\endgroup$
    – hyportnex
    Commented Nov 15, 2023 at 3:48
  • $\begingroup$ @hyportnex Thank you very much for your response. I took a look at it and noticed that what I have is indeed a Mathieu function, but what I should infer from the page and if there is anything on there that is helpful? $\endgroup$
    – kowalski
    Commented Nov 15, 2023 at 3:58
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    $\begingroup$ The Mathieu functions are the solutions of your equation, and if that is what you are interested in, then study them. I have only given you a general reference to start, but there are also many references on the bottom of the page. If you are worried about $\delta \omega$ not being sufficiently small then here is something else, en.wikipedia.org/wiki/Hill_differential_equation, this has been a very well studied subject for some 200 years now. $\endgroup$
    – hyportnex
    Commented Nov 15, 2023 at 10:24

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The usual approach for getting the exponential growth is to use the Floquet method. You calculate the eigenvalues of the linear flow in the 2D phase space $(x,y)=(x,\dot x)$ after a period. In your case, you cannot compute it exactly unless you introduce Matthieu functions. You'll need to calculate the eigenvalues perturbatively in the limit $\delta\omega/\omega_0\ll 1$.

Note that you can do this exactly in the case when $\omega_s^2$ is a square wave. It turns out that the qualitative behaviour is the same. You can find more details in V. Arnold's Ordinary Differential Equations.

Btw, if you're familiar with quantum mechanics, this is the same approach as for Bloch waves in a periodic potential. The time is replaced by space and you reinterpret the equation of motion as a Schrödinger equation, with $\omega_s^2$ the periodic potential. Your parametric resonances correspond to the regions where there is a gap in the spectrum as the wave function cannot have an exponential growth.

Hope this helps.

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