Lecture 1

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Unformatted text preview: sitive real parts, asymptotically stable if all of the eigenvalues have negative real parts, and unstable otherwise. Unfortunately, not all matrices are diagonalizable. However, A can always be reduced to the Jordan canonical form 2 3 1 6 7 2 7 T 1AT = 6 (1.2.12a) 6 7 ... 4 5 ; l where each Jordan block has the form 2 6 6 i=6 4 i 1 i 3 7 7 ... 1 7 5 (1.2.12b) i The dimension of the Jordan block i corresponds to the multiplicity of the eigenvalue i . Thus, if i is simple, the block is a scalar. With this, it is relatively easy to show 2] that y = 0 is stable when either Re( i ) < 0 or Re( i ) = 0 and i is simple, i = 1 2 : : : m, asymptotically stable when Re( i ) < 0, i = 1 2 : : : m, and unstable otherwise. Example 1.2.1. Consider the predator-prey model of Section 1.1 y1 = y1(a ; y2) y2 = y2(;c + y1) 0 0 where a, , c, and are positive constants. This autonomous problem has two equilibrium points: y1 = 0 c= y 0 a= : 2 14 The point 0 0]T corresponds to extinction of predators and prey, whereas c= a= ]T implie...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.

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