Scaling relations and self-similarity conditions in strongly coupled dynamical systems

Dynamical equations arising in a number of physical areas typically involve many dependent variables as well as numerous parameters. In some cases these equations may contain a dominant dependent variable (e.g., the temperature in flame systems, etc.), and the consequences of such an identification are examined in this paper. Particular emphasis is placed on the behavior of dynamical Green's functions and system sensitivity coefficients with respect to the parameters residing in the particular model. In the case of a single dominant dependent variable for a system of ordinary differential equations, it is possible to reduce the Green's function matrix to knowledge of one of its columns and often to one independent element. Furthermore, when there are N dependent variables and M parameters, the N X M matrix of sensitivity coefficients reduces to knowledge of only two characteristic vectors of lengths N and M, respectively. These various reductions are referred to as scaling relations and self-similarity conditions. The consequences of a dominant dependent variable are illustrated with examples drawn from various areas of combustion and kinetics modeling. A brief discussion in the Appendix is also presented on similar organizing principles in multidominant dependent variable systems and cases described by partial differential equations.