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 λ j t ) 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.

Model behaviour and the concept of loop impact: A practical method

“…The loop impact can be contrasted with the understanding of loop strength in terms of the loop gain (Forrester, 1982;Gonçalves, 2009;Kampmann, 2012). This latter concept is independent of a target stock whereas the loop impact will be different on each target stock, as in the PPM method (Mojtahedzadeh et al, 2004) and Ford's (1999) behavioural method.…”

Psychobiological Model of Bipolar Disorder: Based on Imbalances of Glial-Neuronal Information Processing

“…On the behavioral level animals or humans must not only select action programs adapting to the environment situation, but they are also able to intend an action Open Journal of Medical Psychology in the environment. Basically, one of the most fundamental principles in system dynamics is the premise that the structure of the system will generate its behavior, commonly interpreted as modes of behavior [42]. and represents the period in which one of the six modes of behavior is activated.…”

Robust parameter design for system dynamics models: a formal approach based on goal‐seeking behavior

“…Kampmann (1996), Kampmann and Oliva (2006) and Saleh et al (2010) show that it is possible to analyze only a subset of the feedback loops in order to scale down the number of loops to be analyzed. Goncalves (2009) studies how eigenvectors, together with eigenvalues, can be used to explain overall behavior trajectories of the state variables.…”

“…As with previous studies (i.e. Forrester, 1982;Goncalves, 2009;Saleh et al, 2010), we focus the analysis on the endogenous dynamics of the system, thus assuming that the exogenous variables are zero or constant (Saleh et al, 2010). The following set of linear time-invariant state-space equations is then obtained:…”