hw3 - 1 2 V x 2 n of V (i.e., the matrix of second partial...

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ME/ECE859-Spring 2008 Homework 3, Due date: 2/06/08 Wed 1. (Gradient systems) A dynamical system ˙ x = F ( x ), where F ( x ) = grad V ( x ) = p ∂V ( x ) ∂x P T (minus gradient of a smooth function V ), is called a gradient system. a If x ( t ) is a trajectory of a gradient system ˙ x = grad V ( x ), show that d dt V ( x ( t )) = −b grad V ( x ( t )) b 2 . b If the Hessian D 2 V = 2 V ∂x 2 1 · · · 2 V ∂x 1 ∂x n . . . . . . . . . 2 V ∂x n ∂x
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Unformatted text preview: 1 2 V x 2 n of V (i.e., the matrix of second partial derivatives of V ) is nonsingular at the equilibria of x = grad V ( x ), show that the equilibria cannot be spirals or centers. 2. Exercise 4.3-part (2) of the textbook. 3. Exercise 4.4 of the textbook. 1...
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