There are two things wrong with this sentence.NiallBucks said:Homework Statement
X'=x-y^2 Y'=x^2 -xy -2xHomework Equations
Find the critical point of this system.
The Attempt at a Solution
I know one point is (0,0).
When I tried it I got sqrt(x)= 2 and sqrt(x)= 1 but the point (1,1) doesn't fit later parts of the question. I'm also aware that since a sqrt can have positive and negative values then (4, -2) can also equal zero.
Set both X' and Y' to 0 and solve the resulting equations simultaneously.NiallBucks said:Am I missing something obvious?
Critical points, also known as equilibrium points, are specific points in a phase plane that represent stable or unstable behavior of a system. At these points, the rate of change of the system is zero, meaning the system is at rest.
To find critical points, you need to set the differential equations of the system to zero and solve for the variables. This will give you the coordinates of the critical points in the phase plane.
Critical points are important in phase planes because they represent the stable and unstable behavior of a system. They can also give insights into the long-term behavior of a system.
No, a critical point can only be either stable or unstable. In some cases, a critical point can be both stable and unstable depending on the direction of the trajectories around it, but this is not common.
The stability of a critical point can be determined by analyzing the behavior of the trajectories around it. If the trajectories move towards the critical point, it is stable. If the trajectories move away from the critical point, it is unstable. The direction of the trajectories can be determined by plotting a few solutions to the differential equations in the phase plane.