# hw3 - B x y = e-2 x What restrictions does Dulac’s...

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Applied Dynamical Systems - ME215A Fall 2010 Homework #3 - Due Wednesday October 20, in class 1. (15 pts total) Consider the map x n +1 = μx n (1 - x n ) g ( x n ) . (1) (a) (5 pts) Find the ﬁxed points of this map. Your expressions may depend on μ . (b) (5 pts) Over what ranges of μ are these ﬁxed points asymptotically stable and asymp- totically unstable? (c) (5 pts) Find the periodic orbits with period 2 for this map. 2. (10 pts total) Suppose ˙ x = y (2) ˙ y = - x - y + x 2 + y 2 (3) (a) (5 pts) What restrictions does Bendixson’s criterion place on the existence of periodic orbits for this system? (b) (5 pts) Let
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Unformatted text preview: B ( x, y ) = e-2 x . What restrictions does Dulac’s criterion (Thm 4.1.2 of the text by Wiggins) place on the existence of periodic orbits for this system? 3. (20 pts total) Suppose ˙ x = x (4) ˙ y =-y + xy + x 2 (5) (a) (5 pts) What is the stable eigenspace E s and the unstable eigenspace E u for the ﬁxed point at the origin? (b) (10 pts) Find the approximation y = h ( x ) for the local unstable manifold W u loc of the origin accurate to O ( x 3 ). (c) (5 pts) What is the stable manifold W s of the origin? 1...
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## This note was uploaded on 12/29/2011 for the course ME 215a taught by Professor Moehlis,j during the Fall '08 term at UCSB.

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