Synchronizing chaotic PDE system using backstepping and its application to image encryption

This paper is concerned with the observer design problem for a nonlinear hyperbolic system with a modified van der Pol boundary condition in which an integral term is included. The observer plays a role of a chaotic synchronizing system when applying it to secure communication. In our previous work (Sano et al. Secure communication systems using distributed parameter chaotic synchronization. SICE Trans. 2021;57(2):78–85), a nonlinear hyperbolic system without the integral term was treated and a simple synchronizing system was constructed, where the constant coefficients of system played a role of encryption keys. But the keys were vulnerable from a safety standpoint. On the other hand, in the case where the integral term is included in the boundary condition, we need to construct observers. The weighted function contained in the integral term gives a new encryption key of distributed type. An application to image encryption is also discussed and numerical simulation results are given.