On the boundedness of solutions to the Lorenz-like family of chaotic systems

Abstract

This paper deals with a class of threedimensional autonomous nonlinear systems which have potential applications in secure communications, and investigates the localization problem of compact invariant sets of a class of Lorenz-like chaotic systems which contain T system with the help of iterative theorem and Lyapunov function theorem. Since the Lorenz-like chaotic system does not have y in the second equation, the approach used to the Lorenz system cannot be applied to the Lorenz-like chaotic system. We overcome this difficulty by introducing a cross term and get an interesting result, which includes the most interesting case of the chaotic attractor of the Lorenz-like systems. Furthermore, the results obtained in this paper are applied to study complete chaos synchronization. Finally, numerical simulations show the effectiveness of the proposed scheme.

Analysis of global dynamics in an unusual 3D chaotic system

“…China e-mail: liuyongjianmaths@126.com simple nonlinearities [3][4][5][6][7][8][9][10][11][12][13]. It is very important to note that some 3D autonomous chaotic systems have three particular fixed points: one saddle and two unstable saddle-foci (for example, the Lorenz system [1], the Chen system [3], the Lü system [4], and the conjugate Lorenz-type system [14]).…”

Convergence Assessment of the Trajectories of a Bioreaction System by Using Asymmetric Truncated Vertex Functions

“…Recall that in Equation (15) the termḡ 2b is function ofX 2 , beingḡ 2b defined in Equation (17). SinceX 2 converges to Ω 2 (74), thenḡ 2b converges to a compact set satisfyinḡ…”

“…Identifying the convergence region of these systems and proving the asymptotic convergence upon arbitrarily large initial values of the state variables are regarded as important issues [1,5,12]. This stability analysis can be achieved via the following Lyapunov-function based approaches: the finite-time Lyapunov theory [8][9][10], the ultimate bound approach [13][14][15][16][17], and the Lyapunov-like function with vertex truncation approach [18,19]. For these approaches, the size of the target region is not constrained to be small, and cases with no equilibrium points can be considered.…”

“…The system trajectories converge to attractive invariant sets that are properly identified [13][14][15][16][17]. The fundamentals of the used Lyapunov based theory were originally given by Leonov at the eighties, according to [3,[15][16][17]. In these approaches, the Lyapunov function is formulated so that it appears in the right hand side of the expression of its time derivative.…”

Dynamical behaviors of the chaotic Brushless DC motors model

“…Due to the significance of scientific and engineering background of the famous Lorenz system, Leonov first studied the global boundedness of the Lorenz system and obtained many important results in [3,5]. The ultimate boundedness of the other chaotic dynamical systems, including the synchronous motor system [19], the L€ u system [20], the complex Lorenz chaotic system [21], the Lorenz-like family of chaotic systems [22], the financial chaotic system [23], a new chaotic system [24], the family of Lorenz systems [25,26], a new hyperchaotic system [27], and the complex permanent magnet synchronous motor (PMSM) system [28] was studied by many researchers and some important results were obtained. Technically, this is also a very difficult task [19][20][21][22][23][24][25][26][27][28].…”