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A general model for an array of discrete-time neural networks with hybrid coupling is proposed, which is composed of nonlinear coupling and time-varying delays. The coupling terms are described in terms of Lipchitz-type conditions that reflect more realistic dynamical behaviors of coupled systems in practice. The properties of Kronecker product are employed in order to pursue mathematical simplicity of dynamics analysis. On the basis of Lyapunov stability theory, an effective matrix functional is utilized to establish sufficient conditions under which the considered neural networks are globally synchronized. These conditions, which are dependent on the lower bound and the upper bound of the time-varying time delays, are expressed in terms of several linear matrix inequalities (LMIs), and therefore can be easily verified by utilizing the numerically efficient Matlab LMI toolbox. One illustrative example is given to justify the validity and feasibility of the proposed synchronization scheme.

In this paper, various kinds of spontaneous dynamic patterns are investigated based on a two-layer nonlinearly coupled Brusselator model. It is found that, when the Hopf mode or supercritical Turing mode respectively plays major role in the short or long wavelength mode layer, the dynamic patterns appear under the action of nonlinearly coupling interactions in the reaction–diffusion system. The stripe pattern can change its symmetrical structure and form other graphics when influenced by small perturbations sourced from other modes. If two supercritical Turing modes are nonlinearly coupled together, the transition from Turing instability to Hopf instability may appear in the short wavelength mode layer, and the twinkling-eye square pattern, traveling and rotating pattern will be obtained in the two subsystems. If Turing mode and subharmonic Turing mode satisfy the three-mode resonance relation, twinkling-eye patterns are generated, and oscillating spots are arranged as square lattice in the two-dimensional space. When the subharmonic Turing mode satisfies the spatio-temporal phase matching condition, the traveling patterns, including the rhombus, hexagon and square patterns are obtained, which presents different moving velocities. It is found that the wave intensity plays an important role in pattern formation and pattern selection.

In this paper, we discuss exponential synchronization of nonlinear coupled dynamical networks. Sufficient conditions for both local and global exponential synchronization are given. These conditions indicate that the left and right eigenvectors corresponding to eigenvalue zero of the coupling matrix play key roles in the stability analysis of the synchronization manifold.

A study of coupling characteristics between two optically nonlinear planar waveguides has been performed in terms of their individual analytic solutions. It is shown that with an appropriate choice of guide separation and proper redefinition of effective propagation constants, the commonly adopted dual waveguide coupled equations can be formally retained even when the optical nonlinearity in each guide is fully taken into account. The coupling coefficients are determined explicitly in terms of the individual mode fields and the associated intensity-dependent effective index of refraction. A specific numeric illustration of the power flow pattern was given on the basis of its analytical expression derived from the coupled equations. The result describes the detailed coupling characteristics and its variation with respect to input optical power, demonstrating its viability for active optical device applications.

Based on the modified third-order shear deformation theory, the harmonic balance method, and the pseudo-arclength continuation method with two-point prediction, the nonlinear forced vibration response of incompressible hyperelastic moderately thick cylindrical shells subjected to a concentrated harmonic load at mid-span and simply supported boundary conditions at both ends is investigated. The algorithmic procedure for solving steady-state periodic solutions of strongly nonlinear systems of differential equations is presented. The structural response characteristics of shells under different excitation amplitudes and structural parameters are analyzed. The numerical results indicate that the aspect ratio of moderately thick hyperelastic cylindrical shells has a significant effect on the natural frequency ratio. Different frequency ratios lead to varying nonlinear mode coupling effects. The coupling effects among modes result in complex nonlinear behavior in the vibration response of each mode, leading to abundant multi-valued phenomena in the response curve.