Generalized expression of potential function in catastrophe model and its application in eutrophication evaluation

Usually,elementary catastrophe models can only result in potential function forms and the corresponding normalization formulae with control variable dimensions n≤4.While for the complex system of n≥5,the system has to be first decomposed into several multi-layer subsystems of n≤4,and then the recursive approach is used to resolve the subsystems.The evaluation results of the complex system are generally given in the context of relative ordering.A generalized expression of potential function in catastrophe model and the corresponding normalization formula with unlimited dimensions for control variables are arrived through observation,comparison,analysis,and induction of the results of elementary catastrophe models.The complex systems with multi-indexes(control variables) can be evaluated and classified using the combined approach of the corresponding normalization formula resulted from the generalized expression and the hierarchical dimension reduction technique.The combined approach is applied to evaluate eutrophication in 21 Beijing lakes.The results are compared with the several reports on the same subject and for the same lakes,showing that 20 out of 21 eutrophication evaluations are consistent with other studies and our generalized expression of potential function and the corresponding normalization formula are reasonable and simple,practical to apply in the eutrophication evaluation.