How Can You Analyze and Solve These Complex Differential Equations?

[DrWahoo]

1) Reduce the quadratic system to its normal form;
$x'=y+x^2 +y^2$
$y'=xy$

2) Consider the following vector field on $R^2$
$x'=\mu x -y-x(x^2 +y^2)$
$y'=x+\mu y -y(x^2 +y^2) $
where $\mu\in\Bbb{R}$. Define the cross section $\Sigma$ to the vector field by
$\Sigma=\{(r, \theta) \in \Bbb{R}$x$ [0, 2\pi]\}$ given $r >0, \theta =0$

3)
Consider the planar system
$x'=x-y-x^5$
$y'=x+y-y^5$
a) Show that $(0,0)$ is the only equilibrium point and study its stability.
b) Use the Poincare-Bendixson Thm to show there exists a periodic orbit to the above system in certain annular region, determine the inner and outer radius of this annular region as accurate as possible.

 

[DrWahoo]

These are problems my 600 level modeling students were given;

I will give some hints and a complete proof and overview of problem two. Hopefully someone else can chime in and voice their ideas as I love seeing analytical thinking.

On the 2nd problem we are asked;
2) Consider the following vector field on $R^2$
$x'=\mu x -y-x(x^2 +y^2)$
$y'=x+\mu y -y(x^2 +y^2) $
where $\mu\in\Bbb{R}$. Define the cross section $\Sigma$ to the vector field by
$\Sigma=\{(r, \theta) \in \Bbb{R}$x$ [0, 2\pi]\}$ given $r >0, \theta =0$We simply put the system into polar coordinates. The usual definitions work in our case, no need for spherical:
So we use
$x=r\cos\left({\theta}\right)$, $y=r\sin\left({\theta}\right)$, $\theta =\arctan\left({\frac{y}{x}}\right)$ ie; $r^2 = x^2 +y^2$
I will leave it to the reader to verify that in our system:
$r'=r(\mu -r^2)$
$\theta'=1$

This implies that the system has a periodic orbit given by $r=\sqrt{\mu}$.
Now we can construct the Poincare Map near this orbit. We then determine the flow of the given system.

By the flow of the system we have ${\varPhi}_{t} (r_{0}, {\theta}_{0})=([(\frac{1}{\mu} + (\frac{1}{(r^2)_{0}}-\frac{1}{\mu})e^{-2 \mu t}]^{\frac{-1}{2}}, t+ \theta_{0})$
Thus we can define the Poincare section denoted, "$\Sigma$" as;
$\sigma ={(r, \theta ) \in \Bbb{R} \times S' | r>0, \theta = \theta_{0}}$
$\implies$ we have a fixed point at the origin $(0,0)$ where $r=0$. Is this orbit periodic? Yes, $t=2*\pi$ returns the orbit, usually denoted with $\omega$ or $\alpha$ to denote which type of limit set. (Check for yourself)

We note that all periodic orbits correspond to a fixed point by def. Thus using $\sqrt{\mu}$, we check the periodic orbits corresponding to this fixed point;

That is $P(\sqrt{\mu})=(\frac{1}{\mu}+(\frac{1}{\mu}-\frac{1}{\mu})e^{-4 \pi \mu})^{\frac{-1}{2}}$.

Then finding the corresponding eigenvalue around the fixed point;
$r=\sqrt{\mu}$ which is $\lambda = e^{-4 \pi \mu} < 1 $

So what does this tell us about the system and in particular about our Poincare map, especially at $(0,0)$.

I will include the graph with some periodic orbits around the origin to show;

if $\mu < 0 \implies (0,0) =$ Spiral sink
if $\mu > 0 \implies (0,0) =$ Spiral Source

Can you guess which phase portrait corresponded to the hypothesis above?
View attachment 7633
View attachment 7632

2) Consider the following vector field on $R^2$
$x'=\mu x -y-x(x^2 +y^2)$
$y'=x+\mu y -y(x^2 +y^2) $
where $\mu\in\Bbb{R}$. Define the cross section $\Sigma$ to the vector field by
$\Sigma=\{(r, \theta) \in \Bbb{R}$x$ [0, 2\pi]\}$ given $r >0, \theta =0$

 

[DrWahoo]

Hint for #3.

Convert to polar coordinates. Set the RHS of each DE to 0 and solve. What possible equilibrium points do we have?
Can you follow the steps from #2 to get the answer?

Another method is to think of solutions to a similar DE, instead of to the 5th power, what if the largest power was 3. Would our phase portraits look any different? How would this affect a general solution to the model if there is one?