A new approach for solving multi-pantograph type delay differential equations

<p>In this paper, a modified procedure based on the residual power series method (RPSM) was implemented to achieve approximate solution with high degree of accuracy for a system of multi-pantograph type delay differential equations (DDEs). This modified procedure is considered as a hybrid technique used to improve the curacy of the standard RPSM by combining the RPSM, Laplace transform and Pade approximant to be a powerful technique that can be solve the problems directly without large computational work, also even enlarge domain and leads to very accurate solutions or gives the exact solutions which is consider the best advantage of this technique. Some numerical applications are illustrated and numerical results are provided to prove the validity and the ability of this technique for this type of important differential equation that appears in different applications in engineering and control system.</p>

Resonating delay equation

We propose here a delay differential equation that exhibits a new type of resonating oscillatory dynamics. The oscillatory transient dynamics appear and disappear as the delay is increased between zero to asymptotically large delay. The optimal height of the power spectrum of the dynamical trajectory is observed with the suitably tuned delay. This resonant behavior contrasts itself against the general behaviors where an increase of delay parameter leads to the persistence of oscillations or more complex dynamics.

Stability Control Method of Hybrid Impulsive System Based on M-Matrix

Aiming at the problems of poor control effect and poor stability of the mixed pulse system with the traditional method, this paper introduces the M-matrix to establish the pulse delay differential indefinite formula and realize stability control of the mixed pulse system. The synchronization problem of mixed-pulse systems in complex networks is analyzed using M matrix. The local coupling strength of the impulsive system is controlled according to the adaptive method. A class of Multi-Lyapunov functions is constructed for stability control of hybrid impulsive systems. The proposed method is proved to have better control effect through experiments.