J j j j j several common types of matrices are normal

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Unformatted text preview: t→∞ for every initial vector c. In this case u = Au is said to be a stable system, and A is called a stable matrix. If Re (λi ) > 0 for some i, then components of u(t) can become unbounded as t → ∞, in which case the system u = Au as well as the underlying matrix A are said to be unstable. If Re (λi ) ≤ 0 for each i, then the components of u(t) remain finite for all t, but some can oscillate indefinitely. This is called a semistable situation. Example 7.4.2 Predator–Prey Application. Consider two species of which one is the predator and the other is the prey, and assume there are initially 100 in each population. Let u1 (t) and u2 (t) denote the respective population of the predator and Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.4 Systems of Differential Equations http://www.amazon.com/exec/obidos/ASIN/0898714540 545 prey species at time t, and suppose their growth rates are given by u1 = u1 + u2 , u2 = −u1 + u2 . Problem: Determine the size of each population at all future times, and decide if (and when) either population will eventually become extinct. It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu Solution: Write the system as u = Au, u(0) = c, where A= 1 −1 1 1 , u= u1 u2 , and 100 100 c= . D E The characteristic equation for A is p(λ) = λ2 − 2λ + 2 = 0, so the eigenvalues for A are λ1 = 1 + i and λ2 = 1 − i. We know from (7.4.7) that u(t) = eλ1 t v1 + eλ2 t v2 (where vi = Gi c ) (7.4.9) T H is the solution to u = Au, u(0) = c. The spectral theorem on p. 517 implies A − λ2 I = (λ1 − λ2 )G1 and I = G1 + G2 , so (A − λ2 I)c = (λ1 − λ2 )v1 and c = v1 + v2 , and consequently (A − λ2 I)c v1 = = 50 (λ1 − λ2 ) λ2 λ1 and λ1 λ2 v2 = c − v1 = 50 IG R . With the aid of (7.4.8) we obtain the solution components from (7.4.9) as u1 (t) = 50 λ2 eλ1 t + λ1 eλ2 t = 100et (cos t + sin t) and u2 (t) = 50 λ1 eλ1 t + λ2 eλ2 t = 100et...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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