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# In some cases it may be much easier to study the

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Unformatted text preview: value problem for an autonomous linear system , , can arise from linearization about an equilibrium point. As we shall see in chapter 5, when hyperbolic this system gives a good approximation to the behavior of nearby trajectories. sketch The solution of [1] is given by the Fundamental Solution Theorem . The initial value problem for the non-autonomous linear system [1] is 13 Chapter 2 part B , , [2] can arise from linearization about a periodic orbit. In this case , where is the periodic orbit and is another trajectory close to the orbit. sketch Higher order terms are dropped to arrive at [2], but the solution of [2] may give a good approximation to the behavior of the nearby trajectory. See chapter 4 . Floquet theory discusses the solution of [2] when is periodic. Let the period be . The fundamental matrix solution corresponding to [2] is the solution of the initial value problem . [3] is the solution at time of the initial value problem that begins at time solves [3] then solves [2]. Note th...
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