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 philosophical position has fostered the development of a number of formal methods aimed at understanding the causes of model behavior. To most in the field of system dynamics, behavior is commonly understood 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 in the overall trajectory of a state variable -a broader definition of behavior than that of a specific behavior mode. When we consider overall behavior trajectories, contributions from eigenvectors are just as central as those from eigenvalues. Our approach to understanding model behavior derives an equation describing overall behavior trajectories in terms of both eigenvalues and eigenvectors. We then use the derivatives of both eigenvalues and eigenvectors with respect to link (or loop) gains to measure how they affect overall behavior trajectories over time. The direct consequence of focusing on behavior trajectories is that system dynamics researchers' reliance on eigenvalue elasticities can be seen as too-narrow a focus on model behavior -a focus that has excluded the short term impact of a change in loop (or link) gain in its analysis.

Knowledge Management Patterns Model for a Flight Test Environment

“…SD response is usually represented by particular behavior and pattern like exponential growth and oscillations and by its direct relation with feedback loops characterized as reinforced, balanced, delayed balanced and their combination (Ford 1999;Gonçalves 2009). …”

“…These loops provide the possibility of studying the causal relations in the KM system that are traduced into equations and into curves of behavior over time. Once the main stock-and-flow structures (or levels and rates of change) and feedback processes characterizing the system are captured, it is possible to translate them into a mathematical simulation model (Gonçalves 2009). …”

“…It deals with complexity, non-linearity and feedback structures, inherent to social and physical systems. It is a methodology that provides better understanding of complex systems evolution (Forrester 1994;Sterman 2002;Meadows and Wright 2008).SD response is usually represented by particular behavior and pattern like exponential growth and oscillations and by its direct relation with feedback loops characterized as reinforced, balanced, delayed balanced and their combination (Ford 1999;Gonçalves 2009). …”

Power system modal analysis considering doubly-fed induction generators

“…The pathway of eigenvalues is subtly different for the two cases presented. This difference is related to parameter sensitivities [18] due to the increase of the power generated by the DFIG. Eigenvalues associated with ω r , ω and P m do not move when load and wind speed are varied.…”

Dynamic modeling of wind power generation

“…For the two presented cases, the pathway of eigenvalues are notably different. This difference may be related to parameter sensibilities on system stability [23] due to the increase of generated power of the DF IG. Note that the eigenvalues associated to ω r , ω and P m do not move when load and wind speed are varied.…”