Question

Consider the system:

[ dx/dt = αx - βy - x(x² + y²) ]
[ dy/dt = βx + αy - y(x² + y²) ]
with β > 0. Find the critical points of this system.

Answer 1

The critical point for the given system is (0,0). To find additional critical points, we need to solve two equations derived from setting the derivatives to zero, which requires further analysis or more specific values for α and β.

Step-by-step explanation:

To find the critical points of the given system:

• dx/dt = αx - βy - x(x² + y²)
• dy/dt = βx + αy - y(x² + y²)

We need to set the derivatives dx/dt and dy/dt to zero, as critical points occur where both derivatives are zero.

Solving for critical points:

1. Let dx/dt = 0, which gives us αx - βy - x(x² + y²) = 0.
2. Let dy/dt = 0, which yields βx + αy - y(x² + y²) = 0.
3. From the first equation, we can factor out an x to get x(α - βy/x - (x² + y²)) = 0. This implies either x = 0 or α - βy/x - (x² + y²) = 0.
4. From the second equation, factoring out a y gives us y(βx/y + α - (x² + y²)) = 0, which implies either y = 0 or βx/y + α - (x² + y²) = 0.
5. Setting x and y to zero in both equations gives us one critical point (0,0).
6. Other potential critical points come from solving the equations α - βy/x - (x² + y²) = 0 and βx/y + α - (x² + y²) = 0 jointly, but without specific values for α and β, we cannot determine the exact critical points.

Therefore, the critical point we can confirm is (0,0), and more analysis or information would be needed to find additional critical points, if they exist.