Finding Equilibrium Points of Nonlinear Systems

[verd]

Hi,

So I keep making mistakes trying to find all of the equilibrium points of different simple nonlinear systems. These problems aren't difficult, it's just that I keep taking different approaches to finding the equilibrium points.

Is there a methodological way to know that I have found all of the equilibrium points for a system?

I have a few examples (below)

Example 1:
$$\dot{x}=x(3-x-2y)$$
$$\dot{y}=y(2-x-y)$$

Example 2:
$$\dot{x}=x^2-y$$
$$\dot{y}=x-y$$

Example 3:
$$\dot{x}=x(2-x-y)$$
$$\dot{y}=x-y$$

Example 4:
$$\dot{x}=x-y$$
$$\dot{y}=1-e^{x}$$

PS, this isn't homework. This is a component of an exam I need to pass, and I'm just looking for a structured way to approach this type of problem.

Thanks in advance
-H

 

[elibj123]

Equilibrium points are points where the derivative of both x and y equals zero.

So f.e. in this system:

$$\dot{x}=x-y$$
$$\dot{y}=1-e^{x}$$

The equilibrium points satisfy the system of (algebric) equations:

$$x-y=0$$
$$1-e^{x}=0$$

Which means you have only (0,0) as an equilibrium point.

 

[HallsofIvy]

Hi,

So I keep making mistakes trying to find all of the equilibrium points of different simple nonlinear systems. These problems aren't difficult, it's just that I keep taking different approaches to finding the equilibrium points.

Is there a methodological way to know that I have found all of the equilibrium points for a system?

I have a few examples (below)

Example 1:
$$\dot{x}=x(3-x-2y)$$
$$\dot{y}=y(2-x-y)$$

An "equilibrium solution" is simply a constant solution and so its derivative is 0. At an equilibrium point we must have
$$\dot{x}=x(3-x-2y)= 0$$
$$\dot{y}=y(2-x-y)= 0$$

x(3- x- 2y)= 0 when x= 0 or when 3- x- 2y= 0.

If x= 0, then y(2-x-y)= 0 becomes y(2- y)= 0 so either y= 0 or y= 2. So far, two equilibrium points are (0, 0) and (0, 2).

y(2- x- y)= 0 when y= 0 or 2- x- y= 0.

If y= 0, then x(3- x- 2y)= 0 becomes x(3- x)= 0 so either x= 0 or x= 3. We already had (0, 0) but now we have (3, 0) as a third equilibrium point.

If neither x nor y is 0 then we have 3- x- 2y= 0 and 2- x-y= 0. Subtracting the second equation from the first 1- y= 0 or y= 1. With y= 1, both equations become 1- x= 0 so x= 1. The fourth and last equilibrium point is (1, 1).

Do the same for the rest.

Example 2:
$$\dot{x}=x^2-y$$
$$\dot{y}=x-y$$

Example 3:
$$\dot{x}=x(2-x-y)$$
$$\dot{y}=x-y$$

Example 4:
$$\dot{x}=x-y$$
$$\dot{y}=1-e^{x}$$

PS, this isn't homework. This is a component of an exam I need to pass, and I'm just looking for a structured way to approach this type of problem.

Thanks in advance
-H