2006/01/06 by Thilo Gross, Thilo Groß, Ulrike Feudel · 126 citations
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · Medicine · Physics and Astronomy · #Algorithm #Applied mathematics #Computation #Computer science #Dynamical systems theory #Evolution and Genetic Dynamics #Generality #Generalized function #Jacobian matrix and determinant #Mathematical analysis #Mathematical and Theoretical Epidemiology and Ecology Models #Mathematics #Nonlinear Dynamics and Pattern Formation #Nonlinear system #Normalization (sociology) #Stability (learning theory) #Symbolic computation #nlin.CD
paper · pdf · doi:10.1103/physreve.73.016205
published in Physical Review E 73(1), 016205 (American Physical Society) · 15 pages, 2 figures, (Fig. 2 in color)
openalex publication_date 2006/01/06 · arxiv created 2006/01/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We present a universal approach to the investigation of the dynamics in generalized models. In these models the processes that are taken into account are not restricted to specific functional forms. Therefore a single generalized models can describe a class of systems which share a similar structure. Despite this generality, the proposed approach allows us to study the dynamical properties of generalized models efficiently in the framework of local bifurcation theory. The approach is based on a normalization procedure that is used to identify natural parameters of the system. The Jacobian in a steady state is then derived as a function of these parameters. The analytical computation of local bifurcations using computer algebra reveals conditions for the local asymptotic stability of steady states and provides certain insights on the global dynamics of the system. The proposed approach yields a close connection between modelling and nonlinear dynamics. We illustrate the investigation of generalized models by considering examples from three different disciplines of science: a socioeconomic model of dynastic cycles in china, a model for a coupled laser system and a general ecological food web.