It is the central problem of mathematical modelling to describe relationships
observed in nature in the language of mathematics. In order to model a given system the modeler has to formulate mathematical equations, which take experimental evidence and theoretical reasoning into account. Most models that are formulated in this way are specific models. That means, the equations describing the relationship between the model variables are restricted to specific functional forms. This approach is generally advantageous if the functional forms can be derived from first principles. Often, however, this is not possible. In this case the use of general models can be advantageous. In contrast to specific models, general models are based on functions which are not restricted to specific functional forms. General models have been studied and explained in literature for a long time. However, in contrast to earlier works, we have developed a new conceptual approach to the analysis of general models.
Formulation of general models
The main disadvantage of general models is that they cannot be studied by simulation. For this reason, general models have only rarely been used in the past. However, despite their generality general models can be studied very efficiently in the framework of local bifurcation theory. In this way, it is posssible to determine the stability of steady states in the general model. Since the functional forms in the model are not specified, the analysis of a single general model reveals the stability of all steady states in a whole class of similar systems.
Manuscript in preparation: A general model for interacting populations
Global dynamics in general models
Since our approach to general models is based on local bifurcation theory, it focuses primarily on local properties of the models. However, the analysis of local bifurcations enables us to draw certain conclusions on certain global properties of the system. For instance chaotic dynamics do generically occur close to a codimension-2 double Hopf bifurcation. This bifurcation can therefore serve as an indicator of chaotic dynamics. Likewise, a codimension-2 Takens-Bogdanov bifurcation indicates the presence of a homoclinic bifurcation. In this way, the computation of certain bifurcations of higher codimension enables us to draw very general conclusions about the global dynamics of a class of models.