Behavior modes, pathways and overall trajectories: eigenvector and eigenvalue analysis of dynamic systems

Abstract One of the most fundamental principles in system dynamics is the premise that the structure of the system will generate its behavior. Such a philosophical position has fostered the development of a number of formal methods aimed at understanding the causes of model behavior. Behavior, to most in the field of system dynamics, is commonly interpreted as modes of behavior (e.g., exponential growth, exponential decay, and oscillation) because of their direct association with the feedback loops (e.g., reinforcing, balancing, and balancing with delays, respectively) that generate them. Hence, traditional research on formal model analysis has emphasized which loops cause a particular “mode” of behavior, with eigenvalues representing the most important link between structure and behavior. The main contribution of this work arises from a choice to focus our analysis on the overall trajectory of a state variable, instead of only a specific behavior mode. Since the overall behavior trajectory of state variable x i ( t ) is determined by a linear combination of the product of eigenvector components ( r ji ) and behavior modes ( $e^{l_jt}$ ) generated by eigenvalues ( λ j ), contributions from both eigenvalues and eigenvectors are important. By studying how the overall trajectory changes due to changes in link (or loop) gains, we observe that the derivatives of eigenvectors are more closely associated with the short‐term transient impact of those changes, whereas derivatives of eigenvalues are associated with the long‐term impact. Since we care deeply about both the short‐ and the long‐term impact of those changes, there is value in looking at the contributions from both eigenvalues and eigenvectors . Copyright © 2009 John Wiley & Sons, Ltd.