不连续保守系统和类不连续系统的特征动力学行为

Discontinuous models describe qualitative sudden changes after a long gradual change process in a system. They may display very different dynamical characteristics from what a continuous system displays. Based on the study on characteristics of discontinuous dissipative systems performed by our research group during the previous two National Science Foundation projects firstly the characteristics of discontinuous conservative systems have been investigated in the current NSF project. We have analytically deduced a discontinuous conservative model of an electronic relaxation oscillator and the simplified models of it, studied the so-called "quasi-dissipative" behaviors displayed by the models, defined some new concepts such as "quasi-attractor", "quasi-transience", "quasi-basin", and investigated the characteristics of period-doubling bifurcation and some new phenomena like "quasi-crisis", "quasi- intermittency" and the escaping from the strange "quasi-transient set". In another view of point, the sudden changes those are ideally described by the discontinuous models actually happen in finite time durations. One gets a continuous picture if using a large-enough number of parameters and variables and a fast-enough sampling speed to make a description with enough details. The only special thing is the relatively very different "slow change" and "quick change" behavior there. We have constructed two simplified models of so-called "quasi-discontinuous systems" based on Rose-Hindmarsh model describing the activities of nerve cells and model of Pikovsky circuit. Via studies on these simplified models and the corresponding practical models we defined some new concepts like "quasi-discontinuous system", "quasi-discontinuous region", "quasi- discontinuity" and investigated probably the most important characteristics in quasi-discontinuous systems, i.e. the extraordinary large Feigenbaum constants region in period-doubling bifurcation cascades and the intermittency between type I and type V. In summary, the project has been accomplished according to original research plan, some additional achievements, those were not mentioned in the original application, have been obtained as well.

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