PERIODIC WINDOWS OF NONLINEAR GEAR SYSTEM BASED ON SYMBOLIC DYNAMICS

The periodic window is researched by means of the symbolic dynamics and formal language. Firstly, the proper sampling period is taken and the orbital points of periodic motion are obtained through Poincar6 mapping. Secondly, according to the method of symbolic dynamics of one-dimensional discrete mapping, the symbolic sequence describing the periodic orbit is obtained. Finally, based on the symbolic sequence, the corresponding model of minimal finite automation is constructed and the entropy is obtained by calculating the maximal eigenvalue of Stefan matrix. The results show that the orbits in periodic windows can be strictly marked by using the method of symbolic dynamics, thus a foundation for control of switching between target orbits is provided.

Reconstructing Missing Hourly Real-Time Precipitation Data Using a Novel Intermittent Sliding Window Period Technique for Automatic Weather Station Data

Precipitation is the most discontinuous atmospheric parameter because of its temporal and spatial variability. Precipitation observations at automatic weather stations（AWSs） show different patterns over different time periods. This paper aims to reconstruct missing data by finding the time periods when precipitation patterns are similar, with a method called the intermittent sliding window period（ISWP） technique—a novel approach to reconstructing the majority of non-continuous missing real-time precipitation data. The ISWP technique is applied to a 1-yr precipitation dataset（January 2015 to January 2016）, with a temporal resolution of 1 h, collected at 11 AWSs run by the Indian Meteorological Department in the capital region of Delhi. The acquired dataset has missing precipitation data amounting to 13.66%, of which 90.6% are reconstructed successfully. Furthermore, some traditional estimation algorithms are applied to the reconstructed dataset to estimate the remaining missing values on an hourly basis. The results show that the interpolation of the reconstructed dataset using the ISWP technique exhibits high quality compared with interpolation of the raw dataset. By adopting the ISWP technique, the root-mean-square errors（RMSEs）in the estimation of missing rainfall data—based on the arithmetic mean, multiple linear regression, linear regression,and moving average methods—are reduced by 4.2%, 55.47%, 19.44%, and 9.64%, respectively. However, adopting the ISWP technique with the inverse distance weighted method increases the RMSE by 0.07%, due to the fact that the reconstructed data add a more diverse relation to its neighboring AWSs.

Dynamics in a Discrete-time Predator-prey System with Allee Effect

In this paper, dynamics of the discrete-time predator-prey system with Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory, and then further illustrated by numerical simulations. Chaos in the sense of Marotto is proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and rich dynamical behavior. More specifically, apart from stable dynamics, this paper presents the finding of chaos in the sense of Marotto together with a host of interesting phenomena connected to it. The analytic results and numerical simulations demostrates that the Allee constant plays a very important role for dynamical behavior. The dynamical behavior can move from complex instable states to stable states as the Allee constant increases （within a limited value）. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding of the discrete-time predator-prey with Allee effect is given.