Narrow Results

This paper addresses qualitative properties of equilibrium points in a class of delayed neural networks. We derive a sufficient condition for the local exponential stability of equilibrium points, and give an estimate on the domains of attraction of locally exponentially stable equilibrium points. Our condition and estimate are formulated in terms of the network parameters, the neurons' activation functions and the associated equilibrium point; hence, they are easily checkable. Another advantage of our results is that they neither depend on monotonicity of the activation functions nor on symmetry of the interconnection matrix. Our work has practical importance in evaluating the performance of the related associative memory. To our knowledge, this is the first time to present an estimate on the domains of attraction of equilibrium points for delayed neural networks.

In this paper, the ultimate bound and globally exponentially attractive sets of a modified Lorenz–Stenflo system are studied based on the Lyapunov stability theory and optimization theory. Comparing with the best results in the current literature, our new results include the existing results as special cases. Furthermore, the new results offer a theoretical support to studying the Hausdorff dimension of attractor of this modified Lorenz–Stenflo system. These theoretical results are also important and useful for chaos control and chaos synchronization.

This paper investigates some nonlinear dynamical behaviors about domains of attraction, bifurcations, and chaos in an axially accelerating viscoelastic beam under a time-dependent tension and a time-dependent speed. The axial speed and the axial tension are coupled to each other on the basis of a harmonic variation over constant initial values. The transverse motion of the moving beam is governed by nonlinear integro-partial-differential equations with the rheological model of the Kelvin–Voigt energy dissipation mechanism, in which the material derivative is applied to the viscoelastic constitutive relation. The fourth-order Galerkin truncation is employed to transform the governing equation to a set of nonlinear ordinary differential equations. The nonlinear phenomena of the system are numerically determined by applying the fourth-order Runge–Kutta algorithm. The tristable and bistable domains of attraction on the stable steady state solution with a three-to-one internal resonance are analyzed emphatically by means of the fourth-order Galerkin truncation and the differential quadrature method, respectively. The system parameters on the bifurcation diagrams and the maximum Lyapunov exponent diagram are demonstrated by some numerical results of the displacement and speed of the moving beam. Furthermore, chaotic motion is identified in the forms of time histories, phase-plane portraits, fast Fourier transforms, and Poincaré sections.

In this paper, the sensorless force/position control problem is investigated for a general class of dynamic contact systems with both motion sensor noise and unknown kinetic friction by designing a force observer-based controller. Firstly, in order to suppress the effect of motion sensor noise, a dead-zone extended state observer (ESO) is introduced, and the contact force is estimated. Then, based on the force estimate, a controller is designed to realize force/position tracking control, where the parameters of the observer and controller are obtained by a linear matrix inequality (LMI) method. The sufficient conditions are provided to ensure the stability of the closed-loop system in terms of LMIs. Finally, a numerical simulation is carried out to illustrate the applicability and effectiveness of the proposed method.