Aggregation Methods in Analysis of Complex Multiple Scale Systems

Abstract

Background and Significance of the topic: Complexity of many advanced models practically precludes its robust analysis. Fortunately, in many cases the models involve multiple time or size scales and thus yield themselves to asymptotic analysis that allows for significant simplifications of them without losing essential features of their dynamics. Methodology: We apply various methods of asymptotic analysis, such as the Tikhonov-Vasilieva theory, geometric singular perturbation theory, asymptotic expansions, or the degenerate convergence theory. Application/Relevance to systems analysis: The presented theory allows for significant simplifications of multiscale complex systems in a way that preserves the main features of their dynamics. Policy and/or practice implications: In many cases a smart aggregation of the equations of a model leads to a complete solution of the problem at a lower computational cost. Discussion and conclusion: We presented two models and two ways of their asymptotic analysis. It is important to note that not always does the naive approach work–applicability of the asymptotic theories often requires a subtle mathematical analysis. Sometimes only the approximation of macroscopic variables is available. Whenever possible, the analysis of complex multiscale systems should be preceded by assessing the possibility of their simplification through the aggregation of equations and variables. This often requires application of advanced analytic methods but, if successful, leads to significantly simpler systems that can be solved at a lower computational cost.