# Bifurcation point of x' = r + x/2 - x/(x+1)

• Jamin2112
In summary, the conversation is about finding the ranges of r that affect the stationary solutions of an equation and how many solutions exist for different values of r. The equations and solutions are also discussed.
Jamin2112

As in title.

## Homework Equations

My book has a very shaky definition of what a bifurcation point. Basically, I need to play around with r and see how the system changes.

## The Attempt at a Solution

x' = 0 when x = 1/2 - r ± √((r-1/2)2 - 2r)

d/dx (x') = 1/2 - 1/(x+1)2, so
d/dx (x') > 0 when -1 - √2 < x < -1 + √2, and d/dx (x') < 0 when x < - 1 - √2 or x > -1 + √2.

I'm trying to combine these to find the ranges of r that I need to look at. Any ideas?

You have found that the equation has stationary solutions that depend on the parameter r. How many solutions are there at a given parameter value? Are there any special values?

Since we are talking about real values of x, you will have a problem where $(r- 1/2)^2- 2r< 0$. For what values of x is that negative? positive? 0?

How many solutions will you have in each case?

HallsofIvy said:
Since we are talking about real values of x, you will have a problem where $(r- 1/2)^2- 2r< 0$. For what values of x is that negative? positive? 0?

How many solutions will you have in each case?

0 (real) solutions when it's negative, 1 solution when it's 0, 2 solutions when it's positive.
Since (r - 1/2)2 - 2r = r2 - 3r + 1/4 = (r -3/2 + √2)(r - 3/2 - √2), we have that (r - 1/2)2 - 2r > 0 when r > 3/2 + √2 or r <3/2 - √2; r = 0 when x = 3/2 - √2 or x = 3/2 + √2; r < 0 when 3/2 - √2 < x < 3/2 + √2.

Ok. I've figured out this question. NEXT!

## 1. What is a bifurcation point?

A bifurcation point is a critical value in a system of differential equations, where the behavior of the system changes dramatically. It is the point at which the system transitions from one set of solutions to another.

## 2. How is the bifurcation point of x' = r + x/2 - x/(x+1) determined?

The bifurcation point of this equation is determined by finding the values of r where the denominator of x/(x+1) becomes zero, as this will cause the equation to be undefined and result in a change in the behavior of the system.

## 3. What does the r value represent in this equation?

The r value represents a parameter in the equation that controls the behavior of the system. It can be thought of as a measure of the system's stability, with different values of r resulting in different types of solutions.

## 4. How does the behavior of the system change at the bifurcation point?

At the bifurcation point, the system transitions from one set of solutions to another. This can manifest as a sudden change in stability, the appearance of new solutions, or the disappearance of existing solutions.

## 5. Can the bifurcation point be predicted for other systems of differential equations?

Yes, the bifurcation point can be predicted for other systems of differential equations by examining the equations and finding the values of parameters that result in changes in the behavior of the system. However, predicting bifurcation points can be complex and often requires advanced mathematical techniques.

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