3/8 AN INTRODUCTION TO CYBERNETICS 



(3) Repeat the process for each variable in turn until the whole 

 transformation is written down. 



The set of equations so obtained — giving, for each variable in the 

 system, what it will be as a function of the present values of the 

 variables and of any other necessary factors — is the canonical 

 representation of the system. // is a standard form to which all 

 descriptions of a determinate dynamic system may be brought. 



If the functions in the canonical representation are all linear, the 

 system is said to be linear. 



Given an initial state, the trajectory or line of behaviour may now 

 be computed by finding the powers of the transformation, as in 

 S.3/9. 



*Ex. 1 : Convert the transformation (now in canonical form) 



dxidt = y 

 dyjdt = z 

 dzjdt = z + 2xy — x"^ 



to a differential equation of the third order in one variable, x. (Hint: 

 Eliminate y and z and their derivatives.) 



*Ex. 2: The equation of the simple harmonic oscillator is often written 



d2x 



dr2+-- = ^ 



Convert this to canonical form in two independent variables. (Hint: 

 Invert the process used in Ex. 1.) 

 *Ex. 3 : Convert the equation 



d2x dx 2 



to canonical form in two variables. 



3/8. After this discussion of differential equations, the reader who 

 is used to them may feel that he has now arrived at the "proper" 

 way of representing the effects of time, the arbitrary and discrete 

 tabular form of S.2/3 looking somewhat improper at first sight. He 

 should notice, however, that the algebraic way is a restricted way, 

 applicable only when the phenomena show the special property of 

 continuity (S.7/20). The tabular form, on the other hand, can be 

 used always; for the tabular form includes the algebraic. This is 

 of some importance to the biologist, who often has to deal with 

 phenomena that will not fit naturally into the algebraic form. When 

 this happens, he should remember that the tabular form can always 

 provide the generality, and the rigour, that he needs. The rest 

 of this book will illustrate in many ways how naturally and easily 

 the tabular form can be used to represent biological systems. 



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