Tipping point prediction and mechanism analysis of malware spreading in cyber-physical systems

Malware spreading, characterized by differential equations, in networked systems has become a common topic of study in the scientific literature. In this paper, we focus primarily on the propagation dynamics of malware in cyber-physical systems (CPSs). Although some work prior has been done to understand how the malware dynamics evolves in complex networks, these studies have restricted their attention to stability, bifurcation and oscillation based on ordinary differential equation models. Here, we address the question "how does one detect tipping phenomena which result in a total collapse of CPSs if the spread of malware is related to both time and space?" To this end, we use a reaction-diffusion equation to model spatio-temporal dynamic evolutions of malware in CPSs and accurately predict the emergence of tipping points due to the Turing instability and Hopf bifurcation for the first time. In order to further reveal the mechanism of tipping, the specific conditions of the Turing instability and Hopf bifurcation are given. It is confirmed that the introduction of space factor and transmission delay causes CPSs to change from stable to unstable, and then the tipping occurs. Moreover, the delay-dependent stability is discussed and the stability and direction of Hopf bifurcation are explored. These studies provide a strategic guidance for the prediction and control of malicious virus in CPSs. Finally, some numerical simulation examples are provided to substantiate the theoretical results.