I am working with a 3:1 sub-harmonic forced duffing equation

$$\ddot{x}+x+\epsilon x^{3}=cos\Omega t$$

where $0<\epsilon \ll 1$ and and $\Omega =3$, $\Omega ^{2}=9(1+\epsilon \delta )$ and choosing slow time $T=\epsilon t$,

and writing $\ddot{x} + x= \ddot{x} +(\Omega /9 -\epsilon \delta)$. I have shown that the zeroth order equation is

$$\partial^{2}_{t}x_{0}+\frac{\Omega}{9}x_{0}=cos\Omega t $$

and I have shown that the first order equation can be written as

$$\partial^{2}_{t}x_{1}+\frac{\Omega}{9}x_{1}=-2\partial_{t}\partial_{T}x_{0} + \delta x_{0} - x_{0}^{3}$$

I have then gone on to find the solution to the zeroth order equation, which is

$$x_{0}(t, T)=C(T)e^{i(\Omega t/3)}+C(T)^{*}e^{-i(\Omega t/3)} + \gamma e^{i(\Omega t)}$$

where $\gamma = -9/16 \Omega^2$ and $C*$ is the complex conjugate of $C$.

I then need to find the solution of the first order equation. I do this by subbing in $x_{0}$ into the first order equation such that

$$\partial^{2}_{t}x_{1}+\frac{\Omega}{9} x_{1}=-(2i/3) \Omega C_{T}e^{2i(\Omega t/3)}+ (2i/3) \Omega C^{*}_{T}e^{-2i(\Omega t/3)}+i\gamma \Omega e^{i(\Omega t)}-\delta C(T)e^{2i(\Omega t/3)}+ \delta C(T)^{*}e^{-2i(\Omega t/3)} + \delta \gamma e^{i(\Omega t)} -(C(T)e^{2i(\Omega t/3)}+C(T)^{*}e^{-2i(\Omega t/3)} + \gamma e^{i(\Omega t)})^{3}$$.

I know that I need to reduce the equation in order to 'avoid resonance', and that I need terms to vanish, by I am not sure which ones or 'why'? Which terms need to disappear, what is the resonance/ non resonance term in this situation? I think it is to do with secular variation?

Many Thanks, any help is appreciated.

P.S I believe (although not entirely sure) the answer can be reduced to:

$$(2i \Omega /3) \partial_{T}C=C(\delta -6\gamma^{2}-3|C|^{2})+ 3\gamma C^{*2}$$

Basically, why is the answer the answer above?


I believe you are missing the square off some omega terms e.g. in the zeroth order equation.

  • $\begingroup$ Thanks, that was an error in my typing. But this does not answer my question of why I make certain terms 'vanish'. Why $e^{2i\Omega /3}$ but not some others? $\endgroup$ – Uzai Mar 27 '17 at 16:06
  • $\begingroup$ Another thing, I believe you need a $$\gamma e^{-i(\Omega t)}$$ term in your solution to $x_{0}$ in order to follow the exponential definition of $cos{\Omega t}$ correctly. $\endgroup$ – 777777 Mar 28 '17 at 0:46

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