Ordinary differential equations with double resonance

Question

I want to know what is the definition of "resonance, double resonance" in

ordinary differential equations with double resonance

for exemple this : what it means the probleme is resonant in infity ?

Please,

Thank you.

Answer 1

Apologies but I don't have time to explain in more detail. The following link has a gentle explanation: https://www.google.co.uk/url?sa=t&source=web&cd=4&ved=0CEQQFjAD&url=http%3A%2F%2Fwww.math.pku.edu.cn%3A8000%2Fvar%2Fpreprint%2F562.pdf&ei=nR6rUfr2FMeiO8eNgJgD&usg=AFQjCNF8AqsnYZp0r3yj98-2Q2CKpuJnww&sig2=BalUlpIsDbxM4BUJn6gNZQ

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The following is my original answer.

Write $L$ for a linear differential operator. For example, the equation $\ddot{y}+\omega^2y = f(t)$ can be written as $$Ly = f(t) \qquad \text{where}\qquad L\equiv \frac{\mathrm d%2}{\mathrm d t^2} + \omega^2$$

Eigenfunctions $y_n$ such that $Ly_n=\lambda_n y_n$ are interesting because if someone gives you the equation $Ly = \sum_n c_n y_n(t)$ you can solve it by $$y(t) = \sum_n \frac{c_n}{\lambda_n}y_n(t)\quad\text{because}\quad L\sum_n \frac{c_n}{\lambda_n}y_n = \sum_n \frac{c_n}{\lambda_n} L y_n = \sum_n \frac{c_n}{\lambda_n} \lambda_n y_n = \sum_n c_n y_n$$

But this goes wrong when $\lambda_n=0$ for one of the forcing eigenfunctions, since you can't divide by zero. Therefore, forcing with zero-eigenvalue solutions is special, and called resonance. A particular example is a forced equation $Ly=f$ with $Lf=0$. For example, a $\sin (\omega t)$ forcing for the above example of $L$ is forced.

You can often fix this by multiplying by the independent variable $t$. This is because if $Lf=0$ then $$L(tf) = t L(f) + \text{something with }f,f',\ldots = \text{something}$$ in general. (To see this, just expand $L$ as a sum of derivative operators.) However, if the "something" is zero, because $tf$ is also a solution to $L(tf)=0$, then you can try $L(t^2f)$ and so on. It might be fun for you to figure out how this all works.

This is called 'resonance' because $tf$ grows faster than $f$; for example, $\sin(\omega t)$ forcing is a small, constant forcing but it leads to the max. of $|y|$ growing like $t$ over time.

An example is given by $Ly = y''$, which satisfies $L(1)=0$ and $L(t)=0$ but then $L(t^2)=2\propto 1$. That is, $L$ strips off the two $t$s we had to add to $1$. Then we can solve $Ly=1$ by $y=\frac{1}{2}t^2$.

Anyway, I don't know what 'double resonance' means; it might be this, or not!