Consider the Mathieu's Equation: $$\frac{d^{2}u}{dt^2}+[\omega^2 + 2\epsilon \cos(2t)]u=0$$ with $u(0)=1$ and $u'(0)=0$
What I have done is, assume $u(t)=u_{0}(t)+\epsilon u_{1}(t)+\cdots$, substitute into the DE, and we get $$u_{0}(t)=\cos(\omega t)$$ and $$u_{1}(t)=\frac{(1-\omega)\cos[(2+\omega)t]-2\cos(\omega t)+(1+\omega)\cos[(2-\omega)t]}{4-4\omega^2}$$.
The question is, what modes are resonant at order $\epsilon$? and what frequencies are resonant at the next order (i think it means order $\epsilon^2$)?
What are the definition of modes and resonance in this problem? I have no idea how to proceed because I do not know the definition. Can anyone tell me what does it mean by "modes" and "resonant"? Moreover, What is "frequency" here? The problem is just an ODE, where does the word "frequency" come from?
You can rewrite the last formula as $$ u_1(t)=\frac{\sin(t)}2\left(\frac{\sin((1-ω)t)}{1-ω}-\frac{\sin((1+ω)t)}{1+ω}\right) $$ which under $ω\to1$ has the limit $$ u_1(t)=\frac{\sin(t)}2\left(t-\frac{\sin(2t)}2\right) $$ which is growing without bound.
The recursion for the perturbation series is $$ \frac{d^2u_{k+1}}{dt^2}+ω^2u_{k+1}=-2\cos(2t)u_k, $$ using the trigonometric identity $2\cos(2t)\cos(at)=\cos((2+a)t)+\cos((2-a)t)$ for the single terms of $u_k$. This gives resonance in the integration if $2\pm a=\pm ω$.
As $u_1$ has frequencies $2+ω$, $ω$ and $|2-ω|$, the multiplication with the frequency $2$ term will lead in $u_2$ to terms with frequencies $4+ω$, $2+ω$, $ω$, $|2-ω|$ and $|4-ω|$. Resonance during integration will happen for the $ω$ term and additionally if $|2-ω|=ω\implies ω=1$, or $|4-ω|=ω\implies ω=2$ for their corresponding terms.