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    Floquet Multipliers

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    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.

    © BrainMass Inc. brainmass.com December 24, 2021, 8:26 pm ad1c9bdddf
    https://brainmass.com/math/matrices/periodic-coefficients-floquet-multipliers-276655

    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.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 8:26 pm ad1c9bdddf>
    https://brainmass.com/math/matrices/periodic-coefficients-floquet-multipliers-276655

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