MHB Finding Fixed Points for F, B, A

[Dustinsfl]

Is there a clean may to get the fixed points for
\begin{alignat*}{9}
F - 2B' - cB - \frac{3}{4}AB^2 - \frac{3}{4}A^3 & = & 0 & \quad & \Rightarrow & \quad & B' & = & \frac{1}{2}F - \frac{c}{2}B - \frac{3}{8}AB^2 - \frac{3}{8}A^3\\
2A' + cA - \frac{3}{4}A^2B - \frac{3}{4}B^3 & = & 0 & \quad & \Rightarrow & \quad & A' & = & \frac{3}{8}A^2B + \frac{3}{8}B^3 - \frac{c}{2}A
\end{alignat*}

 

[Sudharaka]

Is there a clean may to get the fixed points for
\begin{alignat*}{9}
F - 2B' - cB - \frac{3}{4}AB^2 - \frac{3}{4}A^3 & = & 0 & \quad & \Rightarrow & \quad & B' & = & \frac{1}{2}F - \frac{c}{2}B - \frac{3}{8}AB^2 - \frac{3}{8}A^3\\
2A' + cA - \frac{3}{4}A^2B - \frac{3}{4}B^3 & = & 0 & \quad & \Rightarrow & \quad & A' & = & \frac{3}{8}A^2B + \frac{3}{8}B^3 - \frac{c}{2}A
\end{alignat*}

Does \(F\) and \(c\) stand for constants?

 

[Dustinsfl]

Does \(F\) and \(c\) stand for constants?

Yes c must be positive because it is dampening. F is forcing. This comes from a weak nonlinear oscillator.

 

[Sudharaka]

Yes c must be positive because it is dampening. F is forcing. This comes from a weak nonlinear oscillator.

Since you have a system of polynomial equations you can try to solve it using a numerical method. Here are some articles describing about numerical methods to solve polynomial systems.

1) System of polynomial equations - Wikipedia, the free encyclopedia

2) http://math.berkeley.edu/~bernd/cbms.pdf

>>Here<< is the answer that Wolfram gives. :)

 

[Dustinsfl]

Since you have a system of polynomial equations you can try to solve it using a numerical method. Here are some articles describing about numerical methods to solve polynomial systems.

1) System of polynomial equations - Wikipedia, the free encyclopedia

2) http://math.berkeley.edu/~bernd/cbms.pdf

>>Here<< is the answer that Wolfram gives. :)

On page 176, why are doing what they are doing. Neglect the k_1 term since in my problem omega was 1.

http://audiophile.tam.cornell.edu/randdocs/nlvibe52.pdf

 

[Sudharaka]

On page 176, why are doing what they are doing. Neglect the k_1 term since in my problem omega was 1.

http://audiophile.tam.cornell.edu/randdocs/nlvibe52.pdf

There is only 153 pages in your attached pdf.

 

[Dustinsfl]

There is only 153 pages in your attached pdf.

I meant page 36

 

[Sudharaka]

On page 176, why are doing what they are doing. Neglect the k_1 term since in my problem omega was 1.

http://audiophile.tam.cornell.edu/randdocs/nlvibe52.pdf

I meant page 36

The author had obtained a function between \(\omega\) and \(R\). Is there any particular thing that you don't understand there?

 

[Dustinsfl]

The author had obtained a function between \(\omega\) and \(R\) Is there any particular thing that you don't understand there?

I am trying to relate what he did for when $\omega = 1$ but I can't figure it out. The equation I started with is
$$
x'' + x + \epsilon cx' + \epsilon x^3 = \epsilon F\cos t, \quad\quad\epsilon\ll 1,
$$

 

[Sudharaka]

I am trying to relate what he did for when $\omega = 1$ but I can't figure it out. The equation I started with is
$$
x'' + x + \epsilon cx' + \epsilon x^3 = \epsilon F\cos t, \quad\quad\epsilon\ll 1,
$$

Note that equation 181 gives a relation between \(\omega\) and \(R\). He had drawn the curves for each of the following situations.

1) \(c=0\mbox{ and }F=0\)

2) \(c=0\mbox{ and }F>0\)

i) \(A=R\)

ii)\(A=-R\)

3) \(c>0\mbox{ and }F>0\)

All of these three curves have a value when, \(\omega=1\). For the first situation \(R=0\) is the value at \(\omega=0\). For second and third situations you can obtain the value of \(R\) at \(\omega=0\) using equation 181.

 

[Dustinsfl]

Note that equation 181 gives a relation between \(\omega\) and \(R\). He had drawn the curves for each of the following situations.

1) \(c=0\mbox{ and }F=0\)

2) \(c=0\mbox{ and }F>0\)

i) \(A=R\)

ii)\(A=-R\)

3) \(c>0\mbox{ and }F>0\)

All of these three curves have a value when, \(\omega=1\). For the first situation \(R=0\) is the value at \(\omega=0\). For second and third situations you can obtain the value of \(R\) at \(\omega=0\) using equation 181.

He has a $k_1$ term that comes from expanding $\omega$. Since I don't need to expand omega, I don't have a $k_1$ term. I don't see how to move on from where I am at looking at his argument. I tried making $k_1 = 0$ but he solves for $k_1$ and uses it later on.

 

[Sudharaka]

He has a $k_1$ term that comes from expanding $\omega$. Since I don't need to expand omega, I don't have a $k_1$ term. I don't see how to move on from where I am at looking at his argument. I tried making $k_1 = 0$ but he solves for $k_1$ and uses it later on.

The idea in that section is to investigate the behavior of \(R\) for various values of \(\omega\), but not to solve the duffing equation for \(\omega=1\). What is your aim, to obtain a solution to the Duffing equation?

 

[Dustinsfl]

The idea in that section is to investigate the behavior of \(R\) for various values of \(\omega\), but not to solve the duffing equation for \(\omega=1\). What is your aim, to obtain a solution to the Duffing equation?

I am trying to determine the large-time solution dynamics