# Floquet Multipliers

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

http://books.google.com/books?id=e1euiBF73yIC&dq=A+Second+Course+in+Differential+Equations+Paul+Waltman&printsec=frontcover&source=bl&ots=imup9X2PbZ&sig=g1__chkwNHxDDg-I7M48gjmjQL4&hl=en&ei=k5_dSu6KFM3l8QbEpJlh&sa=X&oi=book_result&ct=result&resnum=1&ved=0CA4Q6AEwAA#v=onepage&q=&f=false

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.

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

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 .

Done.

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