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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.
- Spring '09