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Unformatted text preview: at, if . A trajectory that starts at an initial time and ends at time may be decomposed into two parts.
The first from time to time and the second from time to time . Mathematically, satisfies
.
Then and . Suppose that has period and consider the solution of [3] with
matrix is the solution of the initial value problem after one period . The monodromy .
Then the solution of [2] with
after one period is
. Consider the
trajectory during the second period. It is the solution of the initial value problem
.
Define a new time variable
, and use
to see that this is the same as
[2] with
and replaced by
. Therefore the solution of [2], with
, after two
periods is
. After periods
. The eigenvalues of are the Floquet multipliers. Suppose the initial condition
=0, is also an eigenvector of and is the corresponding eigenvalue. Then
, of [2], with 14
Chapter 2 part B where
is the corresponding Floquet exponent. Meiss discusses the fact that the monodromy
matrix is nonsingular; see Theorem 2.11. Therefore the Floquet multipliers are all nonzero and...
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 Fall '14
 MarcEvans

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