Realization of synchronization between hyperchaotic systems by using a scheme of intermittent linear coupling

Abstract

Based on the Lyapunov stability theory, it is confirmed that complete synchronization can be realized under intermittent linear coupling. The linear controller is selected as ‘stop’ or ‘on control’ by using a switch function; while the first switch function is realized by using a rectangular wave with the same amplitude, and the controller turns on/off in the peiod Ta, Tb alternately. The second switch function is adjusted by a square wave with the same amplitude, and the interval period is marked as T0. At first, a class of exponential Lyapunov function is designed to discuss the reliability and possibility of complete synchronization induced by indirectional linear coupling when the controller is adjusted by two types of switch function. The averaged power consumption of controller within a transient period is defined to measure the cost and efficiency of this scheme. In numerical studies, for the case of first switch function (rectangular wave), the distribution of the largest conditional Lyapunov function for the response system is calculated in the two-parameter space for interval period Ta vs. Tb, the synchronization area vs. nonsynchronization area, the distribution of averaged power consumption in the parameter space Ta vs. Tb. It is also confirmed that complete synchronization can be reached at appropriate Ta, Tb, and coupling intensity. In the case of the second switch function, the distribution of the largest conditional Lyapunov function for the response system is calculated in the two-parameter space for coupling intensity k vs. interval period T0, and the series of error function and averaged power consumption. It is found that complete synchronization can be realized at appropriate coupling intensity and interval period T0. It is also found that the averaged power consumption of controller within a transient period can reach a smallest value at an appropriate coupling intensity. Numerical results are consistent with the theoretical analysis.