Change ODE system to Polar to apply Poincare-Bendixson

In summary, to prove that the given system has at least one periodic orbit, Poincare-Bendixson Theorem can be applied. To create a trapping region, the system can be switched to polar coordinates with the substitutions given. By analyzing the function V(x,y) = \frac12(x^2 + y^2), it can be shown that there exist values of r which form the trapping region for the system. It is not necessary to calculate these values, but it is enough to prove their existence.
  • #1
wrolsr
2
0
Question:
Show that the system

x'= x-y-x[x^2 + (3/2)y^2]
y'= x+y -y[x^2 + (1/2)y^2]

has at least one periodic orbit.


I know that I need to apply Poincare-Bendixson Theorem. I can prove the first three points of it easily, but to create a trapping region, I believe that I need to switch this system to polar. I know that I need to make the substitutions y=r*sin(θ), dy= sin(θ)*dr + r*cos(θ) θ', x= r*cos(θ), dx= cos(θ)*dr - r*sin(θ)θ'. But when I do make the substitution, it makes the equations worse than before. Is there some other way to come up with a r min and r max for the trapping region? I just assumed it was a polar type question because its phase plane has a limit cycle on it.
 
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  • #2
If [itex]V(x,y) = \frac12(x^2 + y^2) = \frac12 r^2[/itex] then
[tex]\dot V = x\dot x + y \dot y
= x^2 + y^2 - x^4 - \frac{5}{2} x^2y^2 - \frac12 y^4[/tex]
which after some rearrangement yields
[tex]\dot V = x^2 + y^2 - \left(x^2 + \frac54y^2\right)^2 + \frac{17}{16}y^4[/tex]
Looking at that, you can see that there exist [itex]0 < m < M[/itex] such that if [itex]0 < r < m[/itex] then [itex]\dot V > 0[/itex] (because the origin is unstable, so trajectories are locally away from it) and if [itex]r > M[/itex] then [itex]\dot V < 0[/itex] (because if [itex]|y|[/itex] is fixed then [itex]\dot V < 0[/itex] for [itex]|x|[/itex] sufficiently large), so that the trapping region is [itex]m \leq r \leq M[/itex].

I don't think it's necessary to calculate [itex]m[/itex] and [itex]M[/itex]; it is enough to show that they exist.
 
Last edited:

1. What is the purpose of changing an ODE system to polar coordinates?

Changing an ODE system to polar coordinates can make it easier to analyze and solve the system, especially in cases where the system has a circular or spiral behavior. It can also help identify and understand the dynamics of the system, such as attracting or repelling fixed points.

2. How do you convert an ODE system to polar coordinates?

To convert an ODE system to polar coordinates, you can use the following substitutions: x = rcosθ and y = rsinθ. Then, use the chain rule to express the derivatives of x and y in terms of the derivatives of r and θ. Finally, substitute these expressions into the original ODE system.

3. What is the significance of applying the Poincare-Bendixson theorem?

The Poincare-Bendixson theorem is a powerful tool for analyzing the behavior of nonlinear systems. It states that under certain conditions, a system will exhibit a periodic or chaotic behavior, rather than converging to a fixed point or spiraling towards infinity. By applying this theorem to a polar ODE system, we can determine the existence and stability of periodic solutions.

4. What are the conditions for applying the Poincare-Bendixson theorem?

The Poincare-Bendixson theorem can be applied to a polar ODE system if the system is autonomous, has a bounded region of attraction, and satisfies the Poincare-Bendixson criterion. This criterion states that there must be no fixed points or limit cycles within the bounded region of attraction, and the system must have a continuous vector field within this region.

5. Are there any limitations to using polar coordinates and the Poincare-Bendixson theorem?

While converting an ODE system to polar coordinates and applying the Poincare-Bendixson theorem can be useful, it is not always a feasible approach. Some systems may not have a simple polar form, and the theorem only applies to certain types of nonlinear systems. In addition, the analysis can become quite complex for higher-dimensional systems. Therefore, it is important to consider the limitations and applicability of this approach when using it to analyze a system.

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