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[SOLVED] Fixed points

dwsmith

Well-known member
Feb 1, 2012
1,673
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

Well-known member
MHB Math Helper
Feb 5, 2012
1,621
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?
 

dwsmith

Well-known member
Feb 1, 2012
1,673
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

Well-known member
MHB Math Helper
Feb 5, 2012
1,621

dwsmith

Well-known member
Feb 1, 2012
1,673
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

Well-known member
MHB Math Helper
Feb 5, 2012
1,621

dwsmith

Well-known member
Feb 1, 2012
1,673

Sudharaka

Well-known member
MHB Math Helper
Feb 5, 2012
1,621
Last edited:

dwsmith

Well-known member
Feb 1, 2012
1,673
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

Well-known member
MHB Math Helper
Feb 5, 2012
1,621
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.
 

dwsmith

Well-known member
Feb 1, 2012
1,673
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

Well-known member
MHB Math Helper
Feb 5, 2012
1,621
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?
 

dwsmith

Well-known member
Feb 1, 2012
1,673
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