Global Exponential Stability and Periodicity of Nonautonomous Impulsive Neural Networks with Time-Varying Delays and Reaction-Diffusion Terms

In this paper, we investigate the global exponential stability and periodicity of nonautonomous cellular neural networks with reaction-diffusion, impulses, and time-varying delays. By establishing a new differential inequality for nonautonomous systems, using the properties of M-matrix and inequality techniques, some new sufficient conditions for the global exponential stability of the system are obtained. Moreover, sufficient conditions for the periodic solutions of the system are obtained by using the Poincare mapping and the fixed point theory. The validity and superiority of the main results are verified by numerical examples and simulations.

Jacobi Multipliers in Integrability and the Inverse Problem of Mechanics

We review the general theory of the Jacobi last multipliers in geometric terms and then apply the theory to different problems in integrability and the inverse problem for one-dimensional mechanical systems. Within this unified framework, we derive the explicit form of a Lagrangian obtained by several authors for a given dynamical system in terms of known constants of the motion via a Jacobi multiplier for both autonomous and nonautonomous systems, and some examples are used to illustrate the general theory. Finally, some geometric results on Jacobi multipliers and their use in the study of Hojman symmetry are given.