Abstract and Applied Analysis
Volume 2013 (2013), Article ID 909721, 10 pages
http://dx.doi.org/10.1155/2013/909721
Research Article

Implementation on Electronic Circuits and RTR Pragmatical Adaptive Synchronization: Time-Reversed Uncertain Dynamical Systems' Analysis and Applications

Shih-Yu Li,1,2 Cheng-Hsiung Yang,3 Li-Wei Ko,1,2 Chin-Teng Lin,2,4 and Zheng-Ming Ge5

1Department of Biological Science and Technology, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan
2Brain Research Center, National Chiao Tung University, Hsinchu, Taiwan
3Graduate Institute of Automation and Control, National Taiwan University of Science and Technology, Taipei, Taiwan
4Institute of Electrical Control Engineering, National Chiao Tung University, Hsinchu, Taiwan
5Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan

Received 18 October 2012; Accepted 27 January 2013

Academic Editor: Gani Stamov

Copyright © 2013 Shih-Yu Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We expose the chaotic attractors of time-reversed nonlinear system, further implement its behavior on electronic circuit, and apply the pragmatical asymptotically stability theory to strictly prove that the adaptive synchronization of given master and slave systems with uncertain parameters can be achieved. In this paper, the variety chaotic motions of time-reversed Lorentz system are investigated through Lyapunov exponents, phase portraits, and bifurcation diagrams. For further applying the complex signal in secure communication and file encryption, we construct the circuit to show the similar chaotic signal of time-reversed Lorentz system. In addition, pragmatical asymptotically stability theorem and an assumption of equal probability for ergodic initial conditions (Ge et al., 1999, Ge and Yu, 2000, and Matsushima, 1972) are proposed to strictly prove that adaptive control can be accomplished successfully. The current scheme of adaptive control—by traditional Lyapunov stability theorem and Barbalat lemma, which are used to prove the error vector—approaches zero, as time approaches infinity. However, the core question—why the estimated or given parameters also approach to the uncertain parameters—remains without answer. By the new stability theory, those estimated parameters can be proved approaching the uncertain values strictly, and the simulation results are shown in this paper.