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This paper investigates the synchronized stationary distribution of stochastic coupled networks with Markovian switching (CNMS) and the exponential synchronization of CNMS. Except for the Lyapunov method, the periodically intermittent control strategy and the technique of graph theory are applied to the study of synchronized stationary distribution and exponential synchronization. Some sufficient conditions to achieve synchronized stationary distribution and exponential synchronization are presented. The results show that the existing domain of synchronized stationary distribution is affected by control rate and the realization of exponential synchronization is also influenced by intensity of control, control rate. In addition, the theoretical results are applied to stochastic coupled oscillators and coupled Chua’s circuits with Markovian switching. Finally, two examples with numerical simulations are provided to illustrate the effectiveness of the results developed.

Inspired by the supramolcular chemistry assembly theory, carbon-coated MnO2 nanospheres (C@MnO2) were successfully synthesized by a two-step method. First, the supramolecular solution was prepared by β-cyclodextrin inclusion of manganese acetate, which was then treated by a hydrothermal reaction and the calcination in the following. The scan electron microscopy (SEM) and transmission electron microscopy (TEM) images present that the C@MnO2 composites have the nanosphere morphology, in which the carbon layer originated from the carbonization of β-cyclodextrin was uniformly coated on the MnO2 nanoparticles. The electrochemical characteristics were examined by galvanostatic charge-discharge (GCD), cyclic voltammetry (CV), and electrochemical impedance spectroscopy (EIS). The results demonstrated that C@MnO2 exhibited high specific capacitance (396.4 F g−1 at 2 A g−1) and good rate capability. In addition, 84.1% of the initial capacitance was maintained after 10000 cycles, indicating the excellent electrochemical stability. The superior electrochemical performance of the C@MnO2 was attributed to the outstanding uniform nanospheres structure that facilitates ion mobility and the synergistic effect between conductive carbon and pseudocapacitive MnO2. Hence, C@MnO2 can be a potential candidate material for the applications in supercapacitors.

A new type of sandwiched nanosheets 3D MoS2/C@RGO is synthesized by supramolecular self-assembly methods, as a novel and excellence electrode material. The surface characterization techniques show that the sandwiched 3D MoS2/C@RGO possesses rich porosity and large specific area, which is convenient for electrolyte ion transportation and beneficial for the improved electrical conductivity. An outstanding specific capacitance of the prepared MoS2/C@RGO of 340.0 F/g at 1 A g−1 is achieved with an excellent cycle stability of 90% after 1000 cycles at the scanning rate of 40 mV/s. The sandwiched 3D MoS2/C@RGO is suggested to be an ideal candidate for supercapacitors.

The remarkable electrochemical performance of graphene-based materials has drawn a tremendous amount of attention for their application in supercapacitors. Inspired by supramolecular chemistry, the supramolecular hydrogel is prepared by linking β-cyclodextrin to graphene oxide (GO). The carbon nanoparticles-anchored graphene nanosheets are then assembled after the hydrothermal reduction and carbonization of the supramolecular hydrogels; here, the β-cyclodextrin is carbonized to carbon nanoparticles that are uniformly anchored on the graphene nanosheets. Transmission electron microscopy reveals that carbon nanoparticles with several nanometers are uniformly anchored on both sides of graphene nanosheets, and X-ray diffraction spectra demonstrate that the interlayer spacing of graphene is enlarged due to the anchored nanoparticles among the graphene nanosheets. The as-prepared carbon nanoparticles-anchored graphene nanosheets material (C/r-GO-1:3) possesses a high specific capacitance (310.8 F g–1, 0.5 A g–1...