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To show that the limit set for the Poincaré map Xn:=

ISBN: 9780321747730 43

Solution for problem 11E Chapter 5.8

Fundamentals of Differential Equations | 8th Edition

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Problem 11E

To show that the limit set for the Poincaré map Xn:= x(2?n), vn:= x’(2?n),where x(t) is a solution to equation (6), is an ellipse and that this ellipse is the same for any initial values x0,v0 , do the following:(a) Argue that since the initial values affect only the transient solution to (6), the limit set for the Poincaré map is independent of the initial values.(b) Now show that for n large, Where And (c) Use the result of part (b) to conclude that the Ellipse contains the limit set of the Poincaré map.

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