Dynamic Systems: Question about Isoclines of Systems

[Master1022]

Homework Statement

Compute the isocline of the following system.

Relevant Equations

N/A

Hi,

I was doing some practice problems online for dynamical systems and came across the following question about isoclines. It left me with 2 questions that I hoping to get some insight to.

Question:
1. What are isoclines? (I have tried doing an internet search, but the results don't help me)
2. How can we calculate the isoclines?

Example:
Imagine we have the following dynamical system:
$$\dot x_1 =x_2 - x_2 ^3$$
$$\dot x_2 = -x_1 - x_2 ^2$$

Compute the isoclines of the system.

Attempt:
The solution simply does:
$$\frac{d x_2}{d x_1} = \frac{\dot x_2}{\dot x_1} = \frac{-x_1 - x_2 ^2}{x_2 - x_2 ^3}$$

and then evaluates different cases based on values. However, I don't quite understand why we are doing this method.

Any help would be greatly appreciated.

 

[Office_Shredder]

An isocline is defined to be the set of points on the $(x_1,x_2)$ graph where the solution curves have a fixed slope. For example, find all the points such that the solution going through that point has a slope of 2.

The slope in general is $d x_2 / dx_1$ which is why they computed that value.

 

[Master1022]

An isocline is defined to be the set of points on the $(x_1,x_2)$ graph where the solution curves have a fixed slope. For example, find all the points such that the solution going through that point has a slope of 2.

The slope in general is $d x_2 / dx_1$ which is why they computed that value.

Thanks @Office_Shredder ! That makes sense. Just to check my understanding, does that mean that if we wanted to, for example find all the points that have a slope of $k$ then we would solve the equation below?
$$k = \frac{-x_1 - x_2 ^2}{x_2 - x_2 ^3}$$

This would result in the curve/etc. of those points

 

[Office_Shredder]

That's right. The idea of these is if you draw some of them for different ks, then when you try to draw a solution curve, you know what the approximate slope is in each section of the plane.

 

[Master1022]

That's right. The idea of these is if you draw some of them for different ks, then when you try to draw a solution curve, you know what the approximate slope is in each section of the plane.

Many thanks! Really appreciate the help. Will have a think about it and have another go at the question