Based on the stability theory of fractional-order systems, the dynamic behaviors of fractional-order Liu system are studied theoretically. Coupling synchronization of two identical fractional-order chaotic systems is also studied and a simple criterion is presented. Numerical simulations show the effectiveness of our methods.

A hydroxyl-terminated hyperbranched polyimide was synthesized via the A 2 + B 3 reaction between dianhydride and triamine monomers. The hydroxyl groups at the peripheral positions were then introduced by modification of the anhydride end groups via a reaction with 4-aminophenol. Based on the hydroxyl-terminated hyperbranched polyimide, HBPI BPADA-TAP(OH) , we successfully fabricated hybrid ternary composites, which were comprised of a linear polyimide (PI 6FDA-APB ), HBPI BPADA-TAP(OH) , and an inorganic SiO 2 component. The material was designed to satisfy the requirement for cutting-edge insulators with a low dielectric constant and a high thermal stability. Because of the appropriate choice of the hybrid ternary composite systems with HBPI BPADA-TAP(OH) and inorganic silica, it is sensible to improve the dielectric properties and thermal resistant properties of unary systems or improve the disadvantages of the dielectric and optical properties of binary systems. For an optimized composition, the dielectric constant (D k ) of the PI 6FDA-APB -HBPI BPADA-TAP(OH) -30%-SiO 2 -20% composite reaches the lowest value of 2.24 at 100 kHz. Research also showed that the optical transparency is significantly improved with the increase of the HBPI BPADA-TAP(OH) content in the composite. Compared with the binary linear PI 6FDA-APB -SiO 2 composite, the transmittance increases from 1% to 75% at the wavelength of 450 nm. The incorporation of SiO 2 can preserve the good thermal properties of the hybrid composites containing HBPI BPADA-TAP(OH) . By adding 10% of HBPI BPADA-TAP(OH) to the PI 6FDA-APB -SiO 2 -20% system, the coefficient of thermal expansion of the hybrid ternary composite is 20.9 ppm C À1 in the temperature range from 100 to 150 C, which is significantly lower than that of the linear polyimide (37.1 ppm C À1 for PI 6FDA-APB ). Because of these optimized properties, hybrid ternary composites have the potential for use in applications in the micro-electronic insulator fields, such as interlayer dielectrics of advanced electronic devices.

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The contribution of this work is to study the control of unknown chaotic systems with input saturation, and the backstepping-based an adaptive fuzzy neural controller (AFNC) is proposed. In many practical dynamic systems, physical input saturation on hardware dictates that the magnitude of the control signal is always constrained. Saturation is a potential problem for actuators of control systems. It often severely limits system performance, giving rise to undesirable inaccuracy or leading instability. To deal with saturation, we construct a new system with the same order as that of the plant. With the error between the control input and saturation input as the input of the constructed system, a number of signals are generated to compensate the effect of saturation. Finally, simulation results show that the AFNC can achieve favorable tracking performances.