THE APPLICATION OF CAUSE-AND-EFFECT ANALYSIS TO MATHEMATICAL-MODELS OF GEOPHYSICAL PHENOMENA .2. STABILITY ANALYSIS

In a recent paper (Andronova and Schlesinger, 1991) we formulated a "cause-and-effect analysis" (CEA) technique and applied it to determine the sensitivity of mathematical models of geophysical phenomena. Here we apply the CEA technique to investigate the stability of such mathematical models. If a process is discrete, the evolution of its system of interactions can often be described by the difference equation, x(k) = Bx(k - 1), where x is a vector of internal variables, B is a matrix, and k is an iteration index. If the process is continuous, then its evolution can often be described by the differential equation, dx/dt = Bx, where t is time. The stability characteristics of such systems are determined by the eigenvalues-lambda of the characteristic equation of matrix B, p(lambda) = 0. In this paper we show that the ith coefficient of p(lambda) is equal to -PL(i)), the ith order loop effect of the graph analog of matrix B - I, where I is the identity matrix. The classical conditions for stability for both discrete process (DP) and continuous process (CP) systems are then reformulated in terms of the PL(i) determined from the graph analog of the system. The use of the resulting graph analog stability conditions is then illustrated by application to two CP systems, the Chapman photochemical cycle and an energy balance climate model, and to a DP system, a finite-differenced differential equation. The use of CEA stability analysis permits one to visualize and thereby understand the interactions among a system's internal variables. As it is these interactions which determine the stability characteristics of a system, their visualization can enable one to determine the cause of an instability and facilitate modification of the system to make it stable.