DESIGN FOR A BRAIN 20/4 



20/4. A state of equilibrium can be denned in several ways. In 

 the field it is a terminating point of a line of behaviour. In the 

 equations of S. 19/7 the state of equilibrium X v . . . , X n is 

 given by the equations 



X t = Lim F^x ; t) {i = 1, . . . , n) . (1) 



t— >00 



if the n limits exist. In the canonical equations the values satisfy 

 f(X v . . . , X n ) = (i = 1, . . . , n) . (2) 



A state of equilibrium is an invariant of the group, for a change 

 of t does not alter its value. 



3fi 



If the Jacobian of the/'s, i.e. the determinant 



dxj 



which will 



be symbolised by J, is not identically zero, then there will be 

 isolated states of equilibrium. If, J =0, but not all its first minors 

 are zero, then the equations define a curve, every point of which 

 is a state of equilibrium. If J = and all first minors but not all 

 second minors are zero, then a two-way surface exists composed 

 of states of equilibrium; and so on. 



20/5. Theorem: If the f's are continuous and differ entiable, a 

 state- determined system tends to the linear form (S. 19/10) in the 

 neighbourhood of a state of equilibrium. 

 Let the system, specified by 



dxjdt = f i (x 1 , . . . , x n ) (i == 1, . . . , n) 



have a state of equilibrium X lf . . . , X n , so that 



f i (X li . . . , X n ) = (i = 1, . . . , n>. 



Put x t = X t ; -f £,. (i = t, . . . , n) so that x t is measured as a 

 deviation £ t from its equilibrial value. Then 



^(A-< + I,.) =f i (X 1 + ft, . . . , X n + £,) (i = 1, . . . , n) 



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

 dXJdt = and tYi&tf^X) = 0, we find, if the £'s are infinitesimal, 

 that 



W = ^ + • • • + W J n (t - 1, . . . , »). 



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

 numerical constants. So the system is linear. 



254 



