Dynamical Systems: how to find equation for Poincare map?

In summary, the conversation discusses a question on Poincare maps in dynamical systems. The question involves finding the Poincare map for a system converted to polar coordinates. The solution is shown to tend towards a limit cycle and the Poincare map is defined as a plane where the trajectory intersects. The question asks to show that the solution for r_n is equal to P(r). The conversation ends with a suggestion to solve for r(2pi) instead of r_n and a question about the reasoning behind the equivalence.
  • #1
Master1022
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Homework Statement
Using the solution given in the previous part, show that for ## a > 0 ## one can define the Poincare map: ## P: \mathbb{R}^{+} \righarrow \mathbb{R}^{+} ## by: (expression below)
Relevant Equations
Differential equations
Hi,

I was attempting a question on the dynamical systems topic of Poincare maps, and was struggling to understand a certain part of it.

Knowledge from prior parts of the questions:
There was a system which we converted to polar coordinates to get: (## a ## is an arbitrary real constant)
[tex] \frac{dr}{dt} = r (a - r) [/tex]
[tex] \frac{d\theta}{dt} = -1 [/tex]

and the first equation was solved with initial condition: ## r(t = 0) = r_0 ## and ## \theta(t = 0 ) = 0 ## to give:
[tex] r(t) = \frac{a r_0}{r_0 + (a - r_0) e^{-at}} [/tex]
[tex] \theta(t) = -t [/tex]
We can see that all solutions tend to a limit cycle (## r = a ##) as ## t \rightarrow \infty## for ## a > 0 ##.

Question I am stuck on: Using the solution given in the previous part, show that for ## a > 0 ## one can define the Poincare map: ## P: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+} ## by ## r_n = P(r_{n - 1}) ## with:
[tex] P(r) = \frac{a r}{r + (a - r) e^{-2 a \pi}} [/tex]

Attempt:
This should be quite a simple thing to do, but I don't quite understand each step in the logic to get there. So I know the Poincare map can be thought of as a plane where we mark the intersections of the trajectory with the plane. This means that a fixed point on a Poincare map corresponds to a limit cycle for the full trajectory.

This plane will always be at the same angle ## \pm 2 n \pi ## (the x-axis). Are we interested in all the times ## t ## when ## = - 2 n \pi ##? These times ## t ## can be found by doing: ## -t = -2 n \pi \rightarrow t = 2 n \pi ##.

Substituting this into the expression for ## r(t) ## yields:
[tex] r(t) = \frac{a r_0}{r_0 + (a - r_0) e^{-2a n \pi}} \rightarrow r_n = \frac{a r_0}{r_0 + (a - r_0) e^{-2a n \pi}} [/tex]

However, I am not sure how to show that the expression is equal to ## P(r)##. What is the reasoning behind that?

Thanks in advance
 
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  • #2
Don't solve for [itex]r_n = r(2n\pi)[/itex] subject to [itex]r(0) = r_0[/itex]. Instead solve for [itex]r(2\pi)[/itex] subject to [itex]r(0) = r_{n-1}[/itex].
 

FAQ: Dynamical Systems: how to find equation for Poincare map?

How do I determine the Poincare map for a given dynamical system?

To find the Poincare map, you need to first identify the Poincare section, which is a two-dimensional surface that intersects the trajectory of the dynamical system. Then, you can use this section to define the Poincare map, which is a function that maps points on the Poincare section to their next intersection with the section. This map can be expressed as an equation using the coordinates of the points on the Poincare section.

What is the significance of the Poincare map in dynamical systems?

The Poincare map allows us to study the behavior of a dynamical system by reducing its dimensionality. By focusing on the intersections of the trajectory with the Poincare section, we can gain insight into the long-term behavior of the system without having to consider all of the points along the trajectory.

Can the Poincare map be used to predict the behavior of a dynamical system?

The Poincare map can provide information about the long-term behavior of a dynamical system, but it cannot predict the exact trajectory of the system. This is because it is based on a discrete mapping of points and does not take into account the continuous nature of the system.

How can I use the Poincare map to analyze the stability of a dynamical system?

The Poincare map can be used to determine the stability of a dynamical system by examining the fixed points of the map. If the fixed points are stable, the system is considered to be stable. If the fixed points are unstable, the system is considered to be unstable.

Are there any limitations to using the Poincare map in dynamical systems analysis?

While the Poincare map can provide valuable insights into the behavior of a dynamical system, it is not always applicable. This method is most useful for systems that exhibit periodic behavior, and may not be as effective for chaotic systems or systems with aperiodic behavior.

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