Behavior Modes, Pathways and Overall Trajectories: Eigenvector and Eigenvalue Analysis of Dynamic Systems
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 sy...
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| Format: | Working Paper |
| Language: | en_US |
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Camrbidge, MA; Alfred P. Sloan School of Management, Massachusetts Institute of Technology
2011
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| Online Access: | http://hdl.handle.net/1721.1/65429 |
| _version_ | 1811089329506746368 |
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| author | Goncalves, Paulo |
| author_facet | Goncalves, Paulo |
| author_sort | Goncalves, Paulo |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T14:17:29Z |
| format | Working Paper |
| id | mit-1721.1/65429 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T14:17:29Z |
| publishDate | 2011 |
| publisher | Camrbidge, MA; Alfred P. Sloan School of Management, Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/654292019-04-10T19:05:37Z Behavior Modes, Pathways and Overall Trajectories: Eigenvector and Eigenvalue Analysis of Dynamic Systems Goncalves, Paulo Formal model analysis overall trajectories behavior modes eigenvectors eigenvalues eigenvalue elasticit loop dominance behavior contribution 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. 2011-08-29T15:17:55Z 2011-08-29T15:17:55Z 2008-05-09 Working Paper http://hdl.handle.net/1721.1/65429 en_US MIT Sloan School of Management Working Paper;4703-08 application/pdf Camrbidge, MA; Alfred P. Sloan School of Management, Massachusetts Institute of Technology |
| spellingShingle | Formal model analysis overall trajectories behavior modes eigenvectors eigenvalues eigenvalue elasticit loop dominance behavior contribution Goncalves, Paulo Behavior Modes, Pathways and Overall Trajectories: Eigenvector and Eigenvalue Analysis of Dynamic Systems |
| title | Behavior Modes, Pathways and Overall Trajectories: Eigenvector and Eigenvalue Analysis of Dynamic Systems |
| title_full | Behavior Modes, Pathways and Overall Trajectories: Eigenvector and Eigenvalue Analysis of Dynamic Systems |
| title_fullStr | Behavior Modes, Pathways and Overall Trajectories: Eigenvector and Eigenvalue Analysis of Dynamic Systems |
| title_full_unstemmed | Behavior Modes, Pathways and Overall Trajectories: Eigenvector and Eigenvalue Analysis of Dynamic Systems |
| title_short | Behavior Modes, Pathways and Overall Trajectories: Eigenvector and Eigenvalue Analysis of Dynamic Systems |
| title_sort | behavior modes pathways and overall trajectories eigenvector and eigenvalue analysis of dynamic systems |
| topic | Formal model analysis overall trajectories behavior modes eigenvectors eigenvalues eigenvalue elasticit loop dominance behavior contribution |
| url | http://hdl.handle.net/1721.1/65429 |
| work_keys_str_mv | AT goncalvespaulo behaviormodespathwaysandoveralltrajectorieseigenvectorandeigenvalueanalysisofdynamicsystems |