—Chaos synchronization and generalized chaos synchronization (GCS) are of essential importance for many physical, circuit, biological, and engineering systems. This paper introduces the definitions of generalized stability (GST) in bidirectional discrete and differentiable systems, which are the extensions for the definitions of chaos generalized synchronization of corresponding bidirectional discrete and differentiable chaos systems. Two constructive generalized stability (GST) theorems for bidirectional discrete systems and bidirectional differential equations (BDS and BDE) are introduced, which give general representations for GST BDS and GST BDE. Using the two theorems, one can easily construct new chaos systems to make the system variables be in GST. Two 8-dimensional GST systems are presented to illustrate the effectiveness of the theoretical results. By combining the 8-dimensional systems with the GCS theorem, two 12- dimensional GCS systems are designed. Numerical simulations verify the chaotic dynamics of such discrete systems and differential equations. Using the two 12-dimensional GCS systems designs two chaotic pseudorandom number generators (CPRNGs). The FIPS 140-2/SP800-22 test suite are used to test the randomness of the four 1,000/100 key-streams consisting of 20,000 bits generated by our CPRNGs, the RC4 algorithm, the ZUC algorithm, respectively. The results show that the randomness performances of our CPRNGs are promising. In addition, theoretically the key space of the each CPRNG is larger than 21196.
—Generalized stability, bidirectional systems, numerical simulation, RANDOMNESS test.
The authors are with the School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, PR China (e-mail: email@example.com, firstname.lastname@example.org, Zhangmei_math@163.com).
Cite: Xiuping Yang, Lequan Min, and Mei Zhang, "Generalized Stability Theorems for Bidirectional Discrete Systems and Differential Equations with Application," International Journal of Modeling and Optimization vol. 5, no. 4, pp. 257-267, 2015.