[SOLVED]Fixed points

dwsmith

Well-known member
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
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
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.

MHB Math Helper

dwsmith

Well-known member
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

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Last edited:

dwsmith

Well-known member
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
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
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
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
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