Gross Thilo, Feudel Ulrike
AG Nichtlineare Dynamik, Universität Potsdam, am neuen Palais 10, 14469 Potsdam, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2006 Jan;73(1 Pt 2):016205. doi: 10.1103/PhysRevE.73.016205. Epub 2006 Jan 6.
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.
我们提出了一种研究广义模型动力学的通用方法。在这些模型中，所考虑的过程不限于特定的函数形式。因此，单个广义模型可以描述一类具有相似结构的系统。尽管具有这种通用性，但所提出的方法使我们能够在局部分岔理论的框架内有效地研究广义模型的动力学性质。该方法基于一种归一化程序，用于识别系统的自然参数。然后，稳态下的雅可比矩阵作为这些参数的函数导出。使用计算机代数进行局部分岔的解析计算揭示了稳态局部渐近稳定性的条件，并为系统的全局动力学提供了一定的见解。所提出的方法在建模与非线性动力学之间建立了紧密的联系。我们通过考虑来自三个不同科学学科的例子来说明广义模型的研究：中国朝代循环的社会经济模型、耦合激光系统模型和一般生态食物网模型。
