21/13 PARAMETERS 



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

 dx/dt = x + 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 : Stable systems may form an unstable whole when 

 wined. Join the three systems 



dx/dt = — x — 2q — 2r 

 dy/dt = - 2p — y + r 

 dz/dt = p + q — 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 : Unstable systems may form a stable whole when 

 joined. Join the 2-variable system 



dx/dt = 3x — Sy — 3p > 

 dy/dt = Sx — 9y - 



which is unstable, to dz/dt = 2lq + Sr + 3z, which is also un- 

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



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The state-determined system 



21/13. It is now clear that there are, in general, two ways of 

 getting to know a complex dynamic system (i.e. one made of 

 many parts). 



One way is to know the parts (ultimately the individual vari- 

 ables) in isolation, and how they are joined. ' Knowing ' each 

 part, or variable, means being able to write down the correspond- 

 ing lines of the canonical representation (if not in mathematical 

 symbolism then in any other way that gives an unambiguous 

 statement of the same facts). Knowing how they are joined 

 means that certain parameters to the parts can be eliminated (for 

 they are functions of the variables). In this way the canonical 

 representation of the whole is obtained. Integration will then 

 give the lines of behaviour of the whole. Thus we can work 

 from an empirical knowledge of the parts and their joining to a 

 deduced knowledge of the whole. 



The other way is to observe the whole and its lines of behaviour. 



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