On differences and similarities in the analysis of Lorenz, Chen, and Lu systems

Abstract

Keywords:Lorenz-like systems Lorenz system Chen system Lu system Lyapunov exponent Chaotic analog of 16th Hilbert problem a b s t r a c t Currently it is being actively discussed the question of the equivalence of various Lorenzlike systems and the possibility of universal consideration of their behavior (Algaba et al., 2013a(Algaba et al., ,b, 2014bChen, 2013;Chen and Yang, 2013; Leonov, 2013a), in view of the possibility of reduction of such systems to the same form with the help of various transformations. In the present paper the differences and similarities in the analysis of the Lorenz, the Chen and the Lu systems are discussed. It is shown that the Chen and the Lu systems stimulate the development of new methods for the analysis of chaotic systems. Open problems are discussed.

Analytic Exact Upper Bound for the Lyapunov Dimension of the Shimizu–Morioka System

“…For the attractor of the Shimizu-Morioka system, a smooth change of coordinates is suggested and a Lyapunov function constructed, which allow one to obtain the analytic exact upper bound for the Lyapunov dimension. Similar results for some other Lorenz-like systems can be found, e.g., in [28,37,38].…”

Analysis of a Generalized Lorenz–Stenflo Equation

“…Obviously, { | , ( ) ≤ max ∈Γ , ( ) = , ∈ Γ} contains solutions of system (2). It is obvious that the set Ω , is the ultimate bound set for system (2).…”

“…holds for system (2), and thus Ω , = { | , ( ) ≤ 2 , } is the globally exponential attractive set of system (2); that is,…”

“…Since then, various complex dynamical behaviors of the Lorenz system have been studied by mathematicians, physicists, and engineers from various fields due to various applications in the fields of engineering, science, and technology [2][3][4][5][6][7][8][9][10][11][12][13][14]. In order to improve the stability or predictability of the Lorenz system, Stenflo and Leonov derived the following four-dimensional Lorenz-Stenflo system with four parameters to describe the dynamics of the atmosphere [15,16] …”

Dynamical Analysis, Synchronization, Circuit Design, and Secure Communication of a Novel Hyperchaotic System

“…The Lorenz chaotic system was proposed [1] and later the chaotic synchronization was implemented in the electronic circuit [2], which greatly inspired many scientists and accelerated the pace of chaos research [3][4][5][6][7]. A hyperchaotic system is defined as an attractor with at least two positive Lyapunov exponents and an autonomous system with phase space of dimension at least four [8].…”