STABILITY 



20/5 



Let the system, specified by 



dxi/dt =fi( x v • • • » x n) (t = 1, . . . , n) 

 have a resting state Xj, . • . , X n , so that 



fi(X v . . . , X n ) = (t = 1 n) 



Put Xi = Xi -f- & (i = 1, . . . , m) so that xi is measured as a 



deviation ft from its resting value. Then 



d 



dt 



(Xi + ft) =f i (X 1 + & x n + ft.) 



(i = 1, . . . , n) 



Expanding the right-hand side by Taylor's theorem, noting that 

 dXi/dt = and that/i(Z) = 0, we find, if the £'s are infinitesimal, 

 that 



d£i _ dfi dft 



si T" • • • ~r 37?n 



(* = L 



., n) 



* aii * ' ' " ' ' din- 



The partial derivatives, taken at the point X v . . . , X n , are 

 numerical constants. So the system is linear. 



20/5. 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 de- 

 scribed are widely applicable. 



Let the linear system be 

 dxi 

 dt 



&%-iX-, 



0>i2p2 T" • • • i Min^n 



(i = i, 



n) (1) 



or, in the concise matrix notation (S. 19/27) 



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 resting state. The determinant 



'ii 



■X a 



12 



21 



22 ■ 



-;. . 



a n 



when expanded gives a polynomial 



'\n 

 ' 2 n 



(Inn A. 



in A of degree n which, when 



equated to 0, gives the characteristic equation of the matrix A 



217 



0. 



