A new network data transmission strategy was proposed in Zhang & Chen [2005] (arXiv:1405.2404), where the resulting nonlinear system was analyzed and the effectiveness of the transmission strategy was demonstrated via simulations. In this paper, we further generalize the results of in the following ways: 1) Construct first-return maps of the nonlinear systems formulated in Zhang & Chen [2005] and derive several existence conditions of periodic orbits and study their properties. 2) Formulate the new system as a hybrid system, which will ease the succeeding analysis. 3) Prove that this type of hybrid systems is not structurally stable based on phase transition which can be applied to higher-dimensional cases effortlessly. 4) Simulate a higher-dimensional model with emphasis on their rich dynamics. 5) Study a class of continuous-time hybrid systems as the counterparts of the discrete-time systems discussed above. 6) Propose new controller design methods based on this network data transmission strategy to improve the performance of each individual system and the whole network. We hope that this research and the problems posed here will rouse interests of researchers in such fields as control, dynamical systems and numerical analysis.
SUMMARYThe first period-doubling bifurcation of a dc-dc buck converter controlled by a zero-average dynamic strategy is studied in detail. Owing to the saturation of the duty cycle, this bifurcation is followed by a border-collision bifurcation, which is the main mechanism to introduce instability and chaos in the circuit. The multiparameter analysis presented here leads to a complete knowledge of the relatioship between these two bifurcations. The results are obtained by using a frequency-domain approach for the study of period-two oscillations in maps.
The application of the graphical Hopf theorem (GHT) as a tool for detecting invariant cycles in maps is presented. The invariant cycle emerging from the bifurcation is approximated using an analogous version of the GHT for continuous-time systems. This technique is formulated in the so-called frequency domain and it involves the use of the Nyquist stability criterion and the harmonic balance method. Some examples are included for illustration.
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