In this chapter we begin the second part of these notes, looking at some ideas based on the concept of monodromy. Very roughly, this is the study of how objects depending on a parameter, and which are locally constant in some sense, change as the parameter moves around a non-trivial path. This idea is particulary appropriate for the center-focus problem, as the essence of this problem is about trying to make global extensions of local information. For example, we might naïvely hope to be able to extend the local first integral at the origin to a global first integral. This is not possible in general, but even if we could do so, the first integral would certainly ramify as a global object. Our desire would then be to read off some important information about the system from this global ramification.
KeywordsPeriodic Solution Model Problem Algebraic Function Monodromy Group Dehn Twist
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