The Analysis of Weakly Coupled Dynamical Systems†

An approximate method of solution is developed for linear, stationary dynamical systems which is weakly coupled. A dynamical system which is built up by the interconnection of simpler sub-systems is termed weakly coupled when the degree of interaction between the constituent sub-systems is not large. For the purpose of the approximation weak coupling is defined quantitatively in terms of a set of conditions which involve the characteristics of the sub-systems and the strengths of the interconnections. The approximation allows the solution of the complete system to be expressed in terms of the solutions of modified sub-systems. Apart from the purely computational aspects, an application of the approximation helps towards an understanding of the way in which the sub-systems interact to produce the complete system behaviour. A number of examples are given of physical systems which exhibit weak coupling and one of these examples is developed in detail to demonstrate the application of the approximation.