Inverse full state hybrid projective synchronization for chaotic maps with different dimensions

A new synchronization scheme for chaotic（hyperchaotic） maps with different dimensions is presented.Specifically,given a drive system map with dimension n and a response system with dimension m,the proposed approach enables each drive system state to be synchronized with a linear response combination of the response system states.The method,based on the Lyapunov stability theory and the pole placement technique,presents some useful features：（i） it enables synchronization to be achieved for both cases of n 〈 m and n 〉 m;（ii） it is rigorous,being based on theorems;（iii） it can be readily applied to any chaotic（hyperchaotic） maps defined to date.Finally,the capability of the approach is illustrated by synchronization examples between the two-dimensional H′enon map（as the drive system） and the three-dimensional hyperchaotic Wang map（as the response system）,and the three-dimensional H′enon-like map（as the drive system） and the two-dimensional Lorenz discrete-time system（as the response system）.

Different Frequency Synchronization Theory and Its Frequency Measurement Practice Teaching Innovation Based on Lissajous Figure Method

According to the requirement of multi-parameter time and frequency measurement without frequency normalization,a different frequency synchronization theory is proposed based on Lissajous figure method and the variation lawof Lissajous figure which are used in practice teaching of frequency measurement. The theory can achieve high-precision transmission and comparison of time and frequency and precise locking and tracking of phase and frequency,improve the level of scientific research on time and frequency for postgraduate,and promote practice teaching innovation of time frequency measurement for undergraduate. Utilizing the ratio of horizontal and vertical inflection point of the Lissajous figure,the nominal frequency of the measured signal is precisely calculated.The frequency deviation between the measured frequency and its nominal frequency can be obtained by combining the turning cycle of the Lissajous figure. By observing the phase relationship between the frequency standard signal and the measured signal,the accurate measurement of the frequency is implemented. Experimental results showthat the direct measurement and comparison better than the 10-11 order of magnitude with common frequency source can be finished between any signal frequencies.The frequency measurement method based on the theory has the advantage of simple operation,quick measurement speed,small error,lownoise and high measurement precision. It plays an important role in time synchronization,communications,metrology,scientific research,educational technology practice and equipment and other fields.

Adaptive projective synchronization with different scaling factors in networks

We study projective synchronization with different scaling factors （PSDF） in N coupled chaotic systems networks. By using the adaptive linear control, some sufficient criteria for the PSDF in symmetrical and asymmetrical coupled networks are separately given based on the Lyapunov function method and the left eigenvalue theory. Numerical simulations for a generalized chaotic unified system are illustrated to verify the theoretical results.