problem 1 let x0 ax and y 0 by be two 2 2

Question

Problem 1 Let ~x0 = A~x and y
0 = B~y be two 2  2 linear systems of ODE.
(1) Suppose that A and B have the same purely imaginary eigenvalues. Prove that these systems
are topologically conjugate.
(2) Suppose that A and B have dierent purely imaginary eigenvalues. Prove that the ODE
systems are not topologically conjugate.
(3) Suppose A has eigenvalues 0,  and B has eigenvalues 0, . Prove theta the ODE systems are
topologically conjugate if and only if  and  have the same sign.
(4) Prove that if A has purely imaginary eigenvalues, and B has real eigenvalues, then the ODE
systems are not topologically conjugate.
(5) Use the information above as well as the theorems from class to provide complete classication
of dynamics two-dimensional linear systems of ODE by conjugacy