20/6 STABILITY 



20/6. In general the only test for stability is to observe or 

 compute the given line of behaviour and to see what happens 

 as t — ► oo. For the linear system, however, there are tests that 

 do not involve the line of behaviour explicitly. Since, by the 

 previous section, many systems approximate to the linear within 

 the region in which we are interested, the methods to be described 

 are often applicable. 



Let the linear system be 

 dxjdt = a ix x x + a i2 x 2 + . . . + a in x n (i = 1, . . . , n) (1) 



or, in the concise matrix notation (S. 19/21) 



x = Ax . . . . (2) 



Constant terms on the right-hand side make no difference to 

 the stability and can be ignored. If the determinant of A is not 

 zero, there is a single state of equilibrium. The determinant 

 a n — A a 12 • • • dm 



#21 #22 A • • • a 2n 



a nl a n2 ... a nn —X 



when expanded, gives a polynomial in X of degree n which, when 

 equated to 0, and, if necessary, multiplied by —1, gives the 

 characteristic equation of the matrix A : 



X n + mj?- 1 + m 2 X n ~ 2 + . . . + ™> n = 0. 

 Each coefficient ra t is the sum of all i-rowed principal (co-axial) 

 minors of A, multiplied by (— 1)*. Thus, 



m i = — (%1 + «22 + • • • + a nn)\ ™n = (~ !)" \ A l 



Example: The linear system 



dxjdt = — 5x 1 + 4cT 2 — 6^ 3 "j 

 dxjdt = 7x ± — 6x 2 + 8# 3 V 

 dxjdt = — 2x-l + 4# 2 — 4# 3 J 



has the characteristic equation 



P + 15A 2 + 2A + 8 = 0. 

 Of this equation, the roots A l9 . . . , X n are the latent roots 

 of A. The integral of the canonical representation gives each 

 x { as a linear function of the exponentials e\*, . . . , e x n. For 

 the sum to be convergent, every real part of X x , . . . , A n must be 

 negative, and this criterion provides a test for the stability of 

 the system. 



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