Time dependence of small perturbations

Equations (3.20) and (3.21) are typical examples of a general class of equations which can be written in the shorthand form 

Here the notation/(a, P) and gf(a, / ) is taken to represent that the derivatives are given by functions of the variables a and p. For the present model we have 

If we now imagine that the system is given a sudden small perturbation, so that the new concentrations of A and B become 

Linear ordinary differential equations are relatively easily manipulated and yield solutions which are simply the sum of exponential terms  

The coefficients Cj-c4 are merely constants which depend on the size and sign of the initial perturbation. More important here, however, are the exponents Al and A2. It is the sign (positive or negative) and character (real or complex) of these two quantities which determine whether and how the perturbations decay or grow. These exponents can be determined as the eigenvalues of the Jacobian matrix of eqns (3.38) and (3.39)  

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