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Generalized models as a universal approach to the analysis of nonlinear dynamical systems
1AG Nichtlineare Dynamik, Universität Potsdam, am neuen Palais 10, 14469 Potsdam, Germany. thilo.gross@physics.org
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 21, 2006
Summary
We developed a universal method to analyze generalized models, enabling efficient study of dynamical properties using local bifurcation theory. This approach connects system modeling with nonlinear dynamics across various scientific fields.
Area of Science:
- Applies to diverse scientific disciplines including socioeconomic, physics (laser systems), and ecology.
Background:
- Generalized models describe systems without pre-defined functional forms, allowing a single model to represent a class of similar structures.
- Investigating the dynamics of these flexible models is crucial for understanding complex systems.
Purpose of the Study:
- To present a universal and efficient approach for analyzing the dynamics of generalized models.
- To connect the fields of mathematical modeling and nonlinear dynamics.
Main Methods:
- A normalization procedure identifies natural system parameters.
- The Jacobian at a steady state is derived as a function of these parameters.
- Local bifurcation theory and computer algebra are employed for analytical computation.
Main Results:
- Efficient study of dynamical properties in generalized models.
- Conditions for local asymptotic stability of steady states are revealed.
- Insights into the global dynamics of the systems are provided.
Conclusions:
- The proposed approach offers a powerful tool for investigating generalized models across sciences.
- It establishes a strong link between the process of modeling and the study of nonlinear dynamics.
- Demonstrated applicability through examples from socioeconomic, laser physics, and ecological systems.
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