Question

Which one of these systems has a periodic solution? Support your claims
a) dx/dt=x^2-xy and dy/dt=y^2+xy+1
b) dx/dt= 3x^3+sin(y) and dy/dt= (x^4)y+y+tan-inverse(x)

Answer 1

@SithsAndGiggles ok I know you have started this question for me before and I have tried working it several times since but I feel utterly stuck. Is there any way we can do this together and you show me the steps I'm supposed to be taking? I can't seem to work the rest of the problems in this section if I cannot work one like this. Thanks!

Answer 2

*

Answer 3

I'll continue from where I left off. If you need a refresher on the process for getting to this point, refer to your previous post: http://openstudy.com/users/kdekle#/updates/54579a30e4b01024cba6e1e3

Also, it would probably be a good idea to brush up on any definitions you might not be familiar with. This link also gives a mostly worked out example:
http://www.mcs.csueastbay.edu/~malek/Class/nonlinear.pdf

Using the equilibrium point \((0,1)\):
\[\begin{pmatrix}x\\y\end{pmatrix}'=\begin{pmatrix}0&0\\-1&2\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}\]
Find the eigenvalues and -vectors:
\[\begin{vmatrix}-\lambda&0\\-1&2-\lambda\end{vmatrix}=\lambda(\lambda-2)=0~~\implies~~\lambda_1=0,~\lambda_2=2\]
For \(\lambda_1=0\),
\[\begin{pmatrix}0&0\\-1&2\end{pmatrix}\begin{pmatrix}\eta_{1,1}\\\eta_{1,2}\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}~~\implies~~\eta_{1,1}=2\eta_{1,2}\]
which gives an eigenvector \(\vec{\eta}_1=\begin{pmatrix}2\\1\end{pmatrix}\).

For \(\lambda_2=2\),
\[\begin{pmatrix}-2&0\\-1&0\end{pmatrix}\begin{pmatrix}\eta_{2,1}\\\eta_{2,2}\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}~~\implies~~\eta_{2,1}=0,~\eta_{2,2}\in\mathbb{R}\]
which gives an eigenvector \(\vec{\eta}_2=\begin{pmatrix}0\\1\end{pmatrix}\).

From here you can find a general solution, which will help you set up a phase portrait but only around the point \((0,1)\).

Answer 4

but since there are 4 equilibrium points I need to do this process for all of them?

Answer 5

Yes, find all the local solutions. Then, if I'm reading the pdf link correctly, you essentially compile the individual phase portraits.

Answer 6

ok I will try again tomorrow! Thanks