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.
Highlights
Nonlinear dynamics, commonly called the chaos theory, changes the scientific way of looking at the dynamics of natural and social systems, which has been intensively studied over the past several decades [1–11]
Let us review the classical Lorentz system [21], which is an extraordinary three-dimensional nonlinear system proposed by the mathematical meteorologist Lorenz
The phase diagrams of the two simulation results given below show that the chaotic signals generated by the electronic circuits can perform high similarity with the original one generated by the ideal simulation tools
Summary
Commonly called the chaos theory, changes the scientific way of looking at the dynamics of natural and social systems, which has been intensively studied over the past several decades [1–11]. More and more applications of chaos synchronization in secure communication have made it much more important to synchronize two different dynamics systems with uncertain parameters in recent years. In this regard, some works on synchronization of two different dynamical systems with uncertain parameters have been performed [33–37]. Pragmatical asymptotically stability theorem and an assumption of equal probability for ergodic initial conditions [38–40] are used to prove strictly that those estimated parameters approach the uncertain values. 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.
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