PARAMETERS 21/7 



which all continuous systems approximate, S. 20/4) the meaning is 

 clear. Several examples will be given. 



Example 1 : Two systems may be stable if joined one way, and 

 unstable if joined another. Consider the 1 -variable systems 

 dx/dt = x + 2p x -f Vz an d dy/dt = — 2r — 3y. If they are 

 joined by putting r = x, p x = y, the system becomes 



dx/dt = x -f 2y + p, 



dy/dt = — 2x — 3y 



The latent roots of its matrix are — 1, — 1 ; so it is stable. But 



if they are joined by r - x, p 2 = y, the roots become + 0-414 



and — 2-414 ; and it is unstable. 



Example 2 : Several systems, all stable, may be unstable when 

 joined. Join the three systems 



dx/dt = — x — 2q — 2r 

 dy/dt = — 2p — y -f- r 

 dz/dt = p -f a — z 

 all of which are stable, by putting p = x, q = y, r = z. The 

 resulting system has latent roots +1, — 2, — 2. 



Example 3: Systems, each unstable, may be joined to form a 

 stable whole. Join the 2-variable system 



dx/dt = Sx — Sy — Sp 



dy/dt = 3x — 9y — 8p^ 

 which is unstable, to dz/dt = 21 g -j- 3r -J- 3^, which is also 

 unstable, by putting q — x, r = y, p = z. The whole is stable. 

 Example 4 : If a system 

 dxi/dt =fi{x v . . . , x n ; a l9 . . .) (i = 1, . . . , n) 



is joined to another system, of ?/'s, by equating various a's and i/'s, 

 then the resting states that were once given by certain com- 

 binations of x and a will still occur, so far as the ^-system is 

 concerned, when the ?/'s take the values the a's had before. The 

 zeros of the/'s are thus invariant for the operations of joining and 

 separating. 



231 



