The text we are using is "A Second Course in Elementary Differential equations" by Paul Waltman. This section begins on page 77. Here is the link to the book
Let p(t) be a continuous function of period omega. consider the system x'=Ax, where A=[0,1;p(t),0].
(a.) Let Phi(t)be a fundamental matrix such that Phi(0)=I, where I is the 2X2 identity matrix. Show that the determinant of the fundamental matrix is one for all t.
(b.) Show that the Floquet Multipliers are the roots of f^2-nf+1=0, where n is the trace of the fundamental matrix.
(c.) If n=2, show that the differential equation has a solution of period omega.© BrainMass Inc. brainmass.com December 24, 2021, 8:26 pm ad1c9bdddf
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Consider the equation , where and is a periodical function with period .
(a) From the condition, is a fundamental matrix with . Let be the determinant of , then we have the formula
where and is the trace of .
So we get
Therefore, the determinant of is for all .
(b) From your given resources, we know that , where is a non-singular matrix with period and is a constant matrix. The Floquet Multipliers are the eigenvalues of . We note
To find the eigenvalues of , we set
From (a), we know that . Let be the trace of .
Then the Floquet Multipliers are the roots of the equation
(c) If , then . So the Floquet Multipliers contains 1. From your given resources, we must conclude that the equation has a periodic solution with period .