final1

final1 - Applied Dynamical Systems - ME215A Fall 2010...

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Applied Dynamical Systems - ME215A Fall 2010 Take-Home Final - Part I - Due Wednesday December 1 1. (45 pts total) Consider the vector field ˙ x = ( x + 1)( x + 2 y )(1 - ( x + y - 1) / 5) , (1) ˙ y = - ( y + 1)(2 x + y ) . (2) (a) (5 pts) Verify that there are fixed points at ( x 0 , y 0 ) = (0 , 0), ( x 1 , y 1 ) = (2 , - 1), ( x 2 , y 2 ) = ( - 1 , - 1), and ( x 3 , y 3 ) = ( - 1 , 2). (b) (7 pts) Find the eigenvalues for the Jacobian evaluated at each fixed point; use this to determine the stability properties of each fixed point. (c) (8 pts) What are the stable and unstable eigenspaces of ( x 1 , y 1 ), ( x 2 , y 2 ), and ( x 3 , y 3 )? (d) (10 pts) Show that the lines lines { ( x, y ) | x = - 1 } , { ( x, y ) | y = - 1 } , and { ( x, y ) | x + y = 1 } are invariant under the flow. What does this allow us to conclude about the stable and unstable manifolds of ( x 1 , y 1 ), ( x 2 , y 2 ), and ( x 3 , y 3 )? (e) (10 pts) What does the trajectory starting at (
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final1 - Applied Dynamical Systems - ME215A Fall 2010...

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