Consider a linear 2 times 2 system x Ax a What type of crit

Consider a linear 2 times 2 system x\' = Ax a) What type of critical point is associated to a system that has solutions which neither converge nor diverge from the origin? b) What type of critical points are associated to systems that have at least 1 straight line solution? c) What is the stability of a critical point if both eigenvalues of A have negative real part? d) Is your answer in part c) still correct if only one of the eigenvalues has negative real part? e) What is the form of the matrix A if all of the solutions are straight lines? f) Suppose A has real eigenvalues. Is it true that the solutions in the direction of an eigenvector must be a straight line? Explain your reasoning.

Solution

a) The type of critical points is unstable if solution is neither converges nor diverges from the origin.

b) In this case type of critical point is stable or neutrally stable.

c)It is asymptotically stable if both eigen values of A have negative real part.

d)Yes, it is still correct and critical point is asymptotically stable.