Classification of Equlibrium Points

[Martyn Arthur]

Homework Statement

Find and classify equlibrium points of this system of non linear equations considering a Jacobian maitrix

Relevant Equations

dx/dt = -y+1
dy/dt = x^2- y^2

I hope this is more properly laid out?
We previously established that the stationery points were (1,1) and (-1,1)
For this first stage I now need to create the elements of a Jacobian maitrix using partial differentation.
I am confused by reference to the chain rule.
Am I correct that for dx/dt =-y +1 the elements are -y as a constant and 1 with y differentiated.
Then for dy/dt = x^2 - y^2 the elements are -2y with x ignored as a constant and 2x with -2y ignored as a constant.
Thanks
Martyn

 

[pasmith]

For the system $$\dot x = f(x,y), \quad \dot y = g(x,y)$$ the Jacobian matrix is $$\begin{pmatrix} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \\ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} \end{pmatrix}.$$

 

[Martyn Arthur]

Have I correctly calculated the elements that I need to use for the eigenvecor calculation as
dx/dt =-y +1 the elements are -y as a constant and 1 with y differentiated.
for dy/dt = x^2 - y^2 the elements are -2y with x ignored as a constant and 2x with -2y ignored as a constant.
Thanks
Martyn

 

[vela]

Have I correctly calculated the elements that I need to use for the eigenvecor calculation as
dx/dt =-y +1 the elements are -y as a constant and 1 with y differentiated.

No. If you're treating $y$ as a constant, its derivative with respect to $x$ is 0.

 

[Steve4Physics]

@Martyn Arthur, can I add this...

$f(x,y) = -y+1$ (If you don't like the absence of '$x$', think about the right hand side as $-y+1 +0x$)

What are $\frac {\partial f}{\partial x}$ and $\frac {\partial f}{\partial y}$?

$g(x,y) = x^2 - y^2$

What are $\frac {\partial g}{\partial x}$ and $\frac {\partial g}{\partial y}$?

Also, you might find this video useful as it walks you through a similar-ish problem.

 

[Martyn Arthur]

Thanks; hopefully this gets it and I can move to the Jacobian Maitrix?

 

[Martyn Arthur]

Herem from the video I am unclear about the process, are the functions f and g being integrated to their original form and the differentiated to find the partial derivatives?
Thanks
Martyn

 

[Steve4Physics]

I’m not sure if you are replying to my post #5 or to a different post. If to post #5, you didn't answer the questions I asked (i.e. what are the 4 required partial derivatives?). And you didn’t say if you watched the suggested video.

The handwriting in your attachment is hard to read but I note that it says $g(x,y)= x^2+y^2$ which is not the same as $g(x,y)= x^2-y^2$ in Post #1. So you need to check which one is correct.

If you post typed - not-hand-written - working, ideally using LatTex for equations, you will get more/better responses. See the forum guidelines: https://www.physicsforums.com/threads/homework-help-guidelines-for-students-and-helpers.686781/

And see the LaTex guide: https://www.physicsforums.com/help/latexhelp/ (which is the link just below the bottom-left of the edit window).

 

[Martyn Arthur]

Thanks; I will check it and improve my handwriting; I have looked at latex but I am finding the course difficult and struggling to find time to learn something else just now.
I did look at the video; he seems to be approaching the Jacobian maitrix differently from my tutor.
My overall question has been too vague alltogether; I will take a step back and rephrase it properly.
Thanks for your patience
Martyn

 

[Martyn Arthur]

Reviewing;
I am wasting your time sorry.
I have six weeks to get on top of this.
I need to get back to basics and work through piece by piece.
Maybe I can take a rain check with you at those stages please.
Thanks

 

[Martyn Arthur]

Would it be ok if I rework each of the sessions and check with you (with proper data) that I have an understanding?
I don't want to knock the OU system, itsmy University, but in large part it relies on peer to peer analysis - v - actual consulation and checking witha tutor.
Its ok but then getting difinitive solutions can be, and are abstract, not necessarlily correct or adequately constructive.
The interacting with a tutor takes ...a while.
Again not knocking the OU but tutor interactions are in large part substituted by peer - peer.
Here I find that your requirement for me to properly frame a question leads to a better understanding of the question; and of course thus the answer!
Anyway; thanks again
Martyn

 

[Martyn Arthur]

Its just so frustratting and unneccsary.
Just an ancilliary, mt lab partner (an English Teacher) and I did a 90 hour experiment following explicitly the instructions given.
We downloaded the data and inspected the files without finding the required data
My lab partner contacted his tutor, mine is reticent in replying and the data was downloadable on the TITR tab but we weren't told this.
I contacted my tutor but have not had a reply.
Knowing what we need to know my partner and I can redo the experiment if the System allows.
Upwards and Onwards
Sorry to be a Nag; thanks for patience as always.
Martyn