How Can You Analyze and Solve These Complex Differential Equations?

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In summary, we are given three problems involving systems of equations. The first problem asks us to reduce a quadratic system to its normal form. The second problem involves a vector field on $R^2$ and asks us to define a cross section to the vector field. The third problem asks us to consider a planar system and determine the stability of the equilibrium point, as well as use the Poincare-Bendixson Theorem to find a periodic orbit in a certain region. Hints and a complete proof for problem two are provided, and we are given hints for problem three such as converting to polar coordinates and considering solutions with a maximum power of 3.
  • #1
DrWahoo
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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.
 
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  • #2
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
DrWahoo said:
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 $r >0, \theta =0$
 

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  • #3
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?
 

FAQ: How Can You Analyze and Solve These Complex Differential Equations?

What is differential equations (DE)?

Differential equations are mathematical equations that describe how a system changes over time. They involve derivatives, which represent the rate of change of a function with respect to its variables.

How are DEs used in scientific research?

DEs are used in many different fields of science, including physics, engineering, and biology. They can be used to model and understand complex systems, make predictions, and solve real-world problems.

What are the different types of DEs?

There are several types of DEs, including ordinary differential equations (ODEs) and partial differential equations (PDEs). ODEs involve only one independent variable, while PDEs involve multiple independent variables.

How do you solve a DE?

The method used to solve a DE depends on its type and complexity. Some DEs can be solved analytically using mathematical techniques, while others may require numerical methods or computer simulations.

Are there any real-world applications of DEs?

Yes, there are many real-world applications of DEs. They are used to model and predict physical phenomena such as population growth, chemical reactions, and fluid flow. They are also used in engineering to design and optimize systems, such as in designing aircraft wings or predicting the behavior of electrical circuits.

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