SYSTEMS OF ODE, Problem 1 Let ~x0 = A~x and y 0 = B~y be two 2 2 linear systems of ODE., Mathematics

SYSTEMS OF ODE, Mathematics

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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 classication
of dynamics two-dimensional linear systems of ODE by conjugacy

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SYSTEMS OF ODE, Mathematics

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