Generalized models as a universal approach to the analysis of nonlinear dynamical systems
Generalized models as a universal approach to the analysis of nonlinear dynamical systems
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DOI：
10.1103/physreve.73.016205
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发表时间：
2006-01-01
影响因子：
2.4
通讯作者：
Feudel, U
中科院分区：
文献类型：
--
作者：
Gross, T;Feudel, U
我们提出了一种通用的方法来研究广义模型中的动力学。在这些模型中，考虑的过程不限于特定的功能形式。因此，单个广义模型可以描述一类具有相似结构的系统。尽管具有这种普遍性，所提出的方法使我们能够在局部分岔理论的框架中有效地研究广义模型的动力学特性。该方法基于用于识别系统自然参数的标准化程序。然后根据这些参数导出稳定状态下的雅可比行列式。使用计算机代数对局部分岔的分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某些见解。所提出的方法在建模和非线性动力学之间建立了紧密的联系。我们通过考虑三个不同科学学科的例子来说明对广义模型的研究：中国王朝周期的社会经济模型、耦合激光系统模型和一般生态食物网模型。
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.