Abstract
In this paper, a class of n-dimensional discrete chaotic systems with modular operations is studied. Sufficient conditions for transforming this kind of discrete mapping into a chaotic mapping are given, and they are proven by the Marotto theorem. Furthermore, several special systems satisfying the criterion are given, the basic dynamic properties of the solution, such as the trace diagram and Lyapunov exponent spectrum, are analyzed, and the correctness of the chaos criterion is verified by numerical simulations.
Highlights
Chaotic systems are mainly characterized by their sensitivity to initial values and system parameters, local instability, boundedness, ergodicity, unpredictability and the fractal structure of their chaotic orbits
With the wide application of chaotic systems in secure communications, the structures of chaotic system problems have attracted increasing attention from scholars [4,5,6], and a new chaotic system can provide a new pseudorandom number generator, which can be further applied to the design of cryptosystems [4,6,7,8]
In reference [6], a class of four-dimensional discrete chaotic systems without an equilibrium point was proposed, the sufficient conditions for this kind of systems to have no equilibrium point are given and the conclusion drawn that the maximum Lyapunov exponent of a given system is positive
Summary
Chaotic systems are mainly characterized by their sensitivity to initial values and system parameters, local instability, boundedness, ergodicity, unpredictability and the fractal structure of their chaotic orbits. In the study of chaos control, reference [17] proposed the Chen–Lai algorithm by considering the following one-dimensional mapping: xk+1 = f (xk) + (N + ec)xk(mod). In reference [18], the linear control term of the Chen–Lai algorithm was not considered; only systems of the structural form xk+1 = f (xk)(mod1) were analyzed. The authors proposed sufficient conditions for this form to be a chaotic map, provided the corresponding discrimination theorem, and proved the existence of chaos in the sense of Li–Yorke by using the snap-back repeller and the Marotto chaos criterion. In view of two cases where g(xk) is a high-dimensional linear system and a highdimensional polynomial system, the determination theorems for the chaotic properties of these two systems are presented, and their chaotic properties in the Li–Yorke sense are proven via the snap-back repeller and the Marotto chaos theorem. The structure of this paper is as follows: Section 2 gives some necessary concepts and lemmas; Section 3 provides the discriminant theorem for a high-dimensional discrete chaotic system and proves it; Section 4 gives examples and conducts a numerical simulation according to the theory in Section 3; and Section 5 is the conclusion of this paper
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