# hwk3 - Phys. 7268 Assignment 3 Due: 2/17/11 Problem 1...

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Phys. 7268 Assignment 3 Due: 2/17/11 Problem 1 Stability Criteria. Let us derive the necessary and suﬃcient conditions for a 2 × 2 real matrix to have eigenvalues with negative real parts. (a) Consider a 2 × 2 real matrix A with matrix elements a ij . Show that the two conditions tr( A ) = a 11 + a 22 < 0 , det( A ) = a 11 a 22 - a 12 a 21 > 0 are necessary and suﬃcient conditions for the two eigenvalues σ i of A to both have negative real parts. (b) Explain why these conditions are equivalent to the statement that all solutions u ( t ) of the two-dimensional constant-coeﬃcient linear dynamical system d u /dt = A u decay to zero. Problem 2 Linearized Brusselator Equations. Consider the “Brusselator” equations t u 1 = a - ( b + 1) u 1 + u 2 1 u 2 + D 1 2 x u 1 , t u 2 = bu 1 - u 2 1 u 2 + D 2 2 x u 2 . (a) Derive the linearized equations for small perturbations about the stationary uniform solution. (b) Determine the numerical value of the coherence length

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## This note was uploaded on 08/25/2011 for the course PHYS 7268 taught by Professor Staff during the Spring '11 term at Georgia Institute of Technology.

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hwk3 - Phys. 7268 Assignment 3 Due: 2/17/11 Problem 1...

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