aeroguy2008 said:Thanks...once I set these two equations I got y values of; y1,1=0, y1,2=-1, y1,3=2...how do i move on from here?
aeroguy2008 said:Well you can be a whole lot concerned. I am just trying to learn something.
The three critical points in a phase portrait are the stable point, unstable point, and saddle point. These points represent the behavior of a dynamical system at different equilibrium states.
The critical points in a phase portrait can be determined by setting the derivatives of the system's variables to zero and solving for the equilibrium points. Alternatively, they can be found by graphing the system's equations and identifying where the curves intersect.
The type of critical point in a phase portrait determines the behavior of the system near that point. A stable point indicates that nearby solutions will approach the equilibrium state, while an unstable point means that nearby solutions will move away from the equilibrium. A saddle point represents a mixture of stable and unstable behavior.
The type of local phase portrait can be determined by analyzing the eigenvalues of the Jacobian matrix at the critical point. If all eigenvalues are negative, the point is stable. If all eigenvalues are positive, the point is unstable. If there are both positive and negative eigenvalues, the point is a saddle.
A phase portrait provides a visual representation of the behavior of a dynamical system over time. It reveals the equilibrium points, their stability, and the direction of movement for solutions near those points. This information is crucial in understanding the long-term behavior of a system and predicting its future states.