This paper establishes two theorems for two time-delayed (chaotic) discrete systems to achieve timedelayed generalized synchronization (TDGS). These two theorems uncover the general forms of two TDGS systems via a prescribed transformation. As examples, we convert the Lorenz three-dimensional chaotic map to an equal time-delayed system as the driving system, and construct the TDGS driven systems according to the Theorems 1 and 2. Numerical simulations demonstrate the effectiveness of the proposed theorems.

We establish two theorems for two time-delayed (chaotic) differential equation systems to achieve time-delayed generalized synchronization (TDGS). The theorems uncover general forms of two TDGS systems via a prescribed differentiable transformation. Based on the theorems, we use two-coupled Ikeda equations as the driving system to construct TDGS driven systems via two prescribed transformations. Numerical simulations demonstrate the effectiveness of the proposed theorems. It may be expected that our theorems provide new tools for understanding and studying TDGS phenomena.

This paper presents a theorem for designing con troller to make two independent (chaotic) discrete systems realize directional partial generalized synchronization (pGS). A non symmetric encryption scheme is introduced based on the theorem.One numerical simulation example is given to illustrate the effectiveness of the proposed theorem and the encryption scheme.The scheme is able successfully to encrypt and decrypt original information without any loss. The analysis of the key space, sensitivity of key parameters show that this scheme has sound security. The key space of the encryption scheme is larger than 1068• It can be expected that our theorem and encryption scheme provide new tools for understanding and studying PGS phenomena and information encryptions.