Abstract

In this study the issue of cyber-security in complex multi-network systems is theoretically investigated in relation to controlled synchronization of multi-networks (e.g. such as IoTs or military CCCs or Corporate Management) in possible events of infliction/occurrence permanent fault and/or recoverable fault in coupled dynamical node systems among different networks or coupling terms in the same network. The representation models of such multi-networks and coupling terms among different networks and within same network are described separately in the mathematic modeling consideration in order to be clear thus enabling envisaging multi-network functioning/malfunctioning phenomenology. This study emanates essentially from exploiting insights gained via graphs theory and dynamics in graphs.  The dimension-transformation matrix is used to deal with the mismatched nodes dimension in the different networks whenever needed. An innovated synchronization controller is designed on the grounds of pinning control theory and employing a pinning control scheme. Based on Lyapunov stability theory, slightly adjusted to networks of nonlinear dynamical systems, a sufficient stability condition and a pinning control scheme are presented by means of which the nodes in the same network can achieve synchronization to a possibly limit-cycle and rarely equilibrium steady state, and thus enhancing the cyber-security of multi-network systems. An simulation example of reasonably complicated multi-network system coupled by three networks, having dramatic nonlinear nodes such as oscillatory Chen, hyper-oscillatory Lorenz and oscillatory Rossler dynamical systems, numerical simulation is was carried out and results showed the proposed controller can achieve synchronization for the multi-network system as a whole regardless what inflicted attack has caused disturbance of normal nominal operation. In addition, it has been demonstrated that the specifically conceived pinning control scheme is more effective than the randomly pinning scheme thus even further enhancing cyber-security resilience by controlled operating synchronization. Due to the influence of environment and human factors, the coupling strength of nodes in a complex network may change randomly with time. Recoverable failure is the random and unpredictable whether is an interference, an deliberatly inflicted attack on dynamical node system in some networ of the multi-network system or among and between networks. In the current research random variables are used to describe the random variation of coupling strength, and a class of neural network synchronization control problem is studied. In some of background studies, the problem of complex network synchronization with nonlinear stochastic distribution is studied. The stochastic variation of nodes is represented by the introduction of random variables with Bernoulli distribution, but the connections among nodes are not considered. In other studies, the cascade failures caused by the attack on complex network are studied, and the research is carried out from the occurrence to the propagation process, but it is not studied from the control point and does not consider the recoverability of node connection failures. In complex dynamic multi-networks cases of permanent and recoverable fault or in the presense of both are investigated. The nodes in the network are assumed randomly switched between all connections and all failures. The network failure time ratio is designed to synchronize the network. Existing literature does not consider the coupling interconnection is subject to interference, failure, or even attacked and yet it is the most vulnerable oe. Therefore, considering the recoverability fault and permanent fault of the nodes, it is more practical to study the synchronization control problem of multi-network. Present author and his collaborators have participated in considerable many of these studies.   Bog.i.SvP.!