hw5_6243_2003

# hw5_6243_2003 - of the equilibrium at the origin of the...

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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Problem Set 5 1 Problem 5.1 y ( t ) a is an equilibrium solution of the diﬀerential equation y (3) ( t ) + ¨ y ( t ) + ˙ y ( t ) + 2 sin( y ( t )) = 2 sin( a ) , where a R and y (3) denotes the third derivative of y . For which values of a R is this equilibrium locally exponentially stable? Problem 5.2 In order to solve a quadratic matrix equation X 2 + AX + B = 0, where A, B are given n -by- n matrices and X is an n -by- n matrix to be found, it is proposed to use an iterative scheme X k +1 = X k 2 + AX k + X k + B. Assume that matrix X satisﬁes X 2 + AX + B = 0. What should be required of the eigenvalues of X and A + X in order to guarantee that X k ± X exponentially as k ± ² when X 0 X is small enough? You are allowed to use the fact that matrix equation ay + yb = 0 , where a, b, y are n -by- n matrices, has a non-zero solution y if and only if det( sI a ) = det( sI + b ) for some s C . 1 Posted October 22, 2003. Due date October 29, 2003

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± 2 Problem 5.3 Use the Center manifold theory to prove local asymptotic stability
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Unformatted text preview: of the equilibrium at the origin of the Lorentz system ² x ˙ = − x + yz, y ˙ = − ±y + ±z, z ˙ = − yx + ²y − z, where , ± are positive parameters and ² = 1. Estimate the rate of convergence of x ( t ) , y ( t ) , z ( t ) to zero. Problem 5.4 Check local asymptotic stability of the periodic trajectory y ( t ) = sin( t ) of system y ¨( t ) + ˙ y ( t ) + y 3 = − sin( t ) + cos( t ) + sin 3 ( t ) . Problem 5.5 Find all values of parameter a R such that every solution x : [0 , ± ) ∞² R 2 of the ODE ⎧ ⎨ cos(2 t ) a x ˙( t ) = β cos 4 ( t ) sin 4 ( t ) x ( t ) converges to zero as t ² ± when β > is a suﬃciently small constant....
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## This note was uploaded on 11/07/2011 for the course AERO 16.36 taught by Professor Alexandremegretski during the Spring '09 term at MIT.

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hw5_6243_2003 - of the equilibrium at the origin of the...

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