Construction of a Class of High-Dimensional Discrete Chaotic Systems

: 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.


Introduction
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. As a complex nonlinear dynamical system and due to its significant dynamic characteristics, chaos has become a new research hotspot and has been widely used in secure communications and other research fields [1][2][3]. 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]. Continuous system discretization is needed for actual use cases, and the direct use of discrete chaos is a very good choice. There are some commonly used chaos criteria for determining whether there is chaos in a discrete dynamical system. For the determination of chaos of a discrete mapping, Li-Yorke proposed three periodic chaos implication theorems; namely, the three periodic theorems [9]: the Marotto chaos determination theorem [10] and its improved Marotto theorem [11], heteroclinic repellent theory [12], and coupled expansion theory [13]. Onedimensional chaotic systems, such as logistic mappings, have the characteristics of simple structures and good performance, and they are one of the most widely used chaotic system types [14,15]. A logistic mapping has a quadratic polynomial form. In reference [4], a judgment theorem for a special cubic polynomial chaotic system was proposed, and the proof was given by using the periodic three theorem. 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. In addition to the direct use of the discrimination theorem and the above method for verifying that the maximum Lyapunov exponent of a system is positive, there is also an indirect method for proving the chaos property of a system by using its topological conjugate relationship with a known chaotic system, such as the method proposed by reference [16].
In the study of chaos control, reference [17] proposed the Chen-Lai algorithm by considering the following one-dimensional mapping: The problem by which a system can become a chaotic system under the control of the linear control term "    c k N e x " was studied. In reference [18], the linear control term of the Chen-Lai algorithm was not considered; only systems of the structural form k k x f x 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. Reference [19] used a continuous sawtooth function instead of a modular function to complete the iterative process. In general, in the process of designing chaotic cryptosystems, high-dimensional systems have attracted increasing interest from researchers because of their highly complex dynamic properties, but the constructions and theoretical proofs of high-dimensional discrete chaotic systems are always difficult. Most of the existing high-dimensional discrete chaotic systems are designed based on the necessary conditions of the chaotic systems, such as having a positive Laypunov exponent [5][6][7]. According to the Marotto theorem, sufficient conditions that an n-dimensional discrete system is a chaotic system can be given.
This article does not consider the Chen-Lai algorithm for linear control and only considers a high-dimensional modular system, such as In view of two cases where   k g x 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.

Basic Concepts and Lemmas
To study the theoretical structure of high-dimensional discrete chaos, some concepts are agreed, and some lemmas are given. For an over-one diagonally dominant matrix and a strictly over-one diagonally dominant matrix, there are some properties as follows. First, from definitions 1 and 2, it can be seen that an over-1 diagonally dominant matrix must be a diagonally dominant matrix. Additionally, a diagonally dominant matrix must be an invertible matrix [20], so if there has ever been a diagonally dominant matrix, it must have been a reversible matrix; see Lemma 1 below.

Matrix Theoretical Analysis
For a strictly over-one diagonally dominant matrix, Lemma 2 follows. (1) Proof of Lemma 3. If A is a strictly over-one diagonally dominant matrix, then A is a strictly diagonally dominant matrix. According to Lemma 1,  A 1 exists. From Lemma 2, we know that there exists , and we can con- Let us expand the left-hand side of the identity according to    Taking the infinite norm of both sides of the above equation and according to the formula If we rearrange the formula, we obtain Because A is a strictly over-one diagonally dominant matrix, In conclusion,

Marotto Theorem
The following is the definition of the snap-back repeller in the dynamical system and Marotto theorem.

Definition 4 [10] (snap-back repeller). Note that
x is a closed sphere whose center is the point * x and whose radius is r . Assume that the fixed point * x of a differentiable mapping g in n R satisfies the following conditions: Then, we call the fixed point * x a snap-back repeller of mapping g .
On the basis of the definition of a snap-back repeller, we have the following Marotto theorem.

Lemma 6 [10] (Marotto theorem).
If an n-dimensional differentiable mapping  : n n g R R has a snap-back repeller, mapping g has chaotic properties in the Li-Yorke sense.
Based on some concepts and lemmas given in this section, the decision theorem of high-dimensional discrete chaos is given below.

Construction of High-Dimensional Linear Chaos Theory
The n-dimensional discrete system is expressed as The following discussion is provided for the cases in which   x k g is a linear system and a multino- For the case where   x k g is a linear system, the following theorem is given.
ii a i n in A , and A is a strictly over-one diagonally dominant matrix. Then, system (2) is chaotic in the sense of Li-Yorke.
It is easy to see that when , and we only need to prove that  x 0 * is the snap-back repeller. Because A is a strictly over-one diagonally dominant matrix, we know from Lemma 1 If we substitute this back into the original system, we obtain that We know from Lemma 3 that From this, we know that becomes the exclusion domain including x * . The Jacobian matrix In view of the fact that   x k g is a multinomial system, the following theorems are given.

Theorem 2.
Consider an n-dimensional discrete nonlinear control system: in the above formula, where   x k g is a polynomial system of many variables that is described as   x i k f is obviously a multivariate polynomial system: Next, let us prove that there is a point Step 1. We first prove that there is a point . This is equivalent to The following step is used to prove that there exists a point Two functions are defined for this purpose: Thus, Define a scalar function as follows: According to the above formula, we can infer that   ( ) Let us consider the value range of z . If   and as a result, , so the following inequality can be obtained: Therefore, from It can be concluded that there exists a point z such that Step 2. Next, we prove that there exists a point Instead of using ρ in step 1 as ρ , let 2 ρ ρ   ; then, two functions are defined as follows: . Similar to step 1, it can be concluded that According to Equations (11) and (12), there exists a point (2) There exists  n y R in   In conclusion, we know that  x 0 * is a snap-back repeller. The proof is completed. Then, the modular polynomial operation system (3) is chaotic in the sense of Li-Yorke. □

Numerical Simulation of System (2)
In system (2) where a is the system parameter and  x ( , , ) T k x y z . When we choose  2 a , we obtain , that is, matrix A is strictly over-1 diagonally dominant. Therefore, according to theorem 1, system (2) represents chaos.
MATLAB is used to simulate system (2). The following parameters are selected: . The phase diagram distribution is shown in Figure   1.    Figure 3a,c, it can be found that the orbit produced by the small change in the initial value is obviously different, while the differences between the two orbits before and after the perturbation are shown in Figure 3b,d, respectively.   The following is a numerical simulation of a five-dimensional system such that

Conclusions of this Paper
In this paper, sufficient conditions for transforming a class of high-dimensional discrete systems into chaotic systems based on modular operations are given, and the proofs are given by the Marotto theorem. Based on the above conditions, two kinds of systems with high-dimensional linear modular operations and high-dimensional polynomial modular operations are constructed, and the given chaotic systems are numerically simulated. The phase diagram distributions, the Lyapunov exponent spectra, the sensitivities to the chosen initial values and the chaotic bifurcation diagrams of these systems are analyzed. It is concluded that the discrete systems we constructed have chaotic dynamics characteristics, thereby verifying the correctness of the proposed theory.
The advantages of designing cryptosystems based on the chaotic systems constructed in this paper are as follows: (1) the dimension of the systems can be any positive integer n. An accepted view is that compared with the low-dimensional chaotic systems; high- dimensional chaotic systems have more complex dynamic behavior and can produce better random sequences for the design of cryptosystems; (2) the systems can design many parameters as the keys of the cryptosystems, a large number of system parameters means that the cryptosystems with large key space can be designed.
Author Contributions: Hongyan Zang carried out numerical simulation and analysis, Jianying Liu and Jiu Li studied the proof and derivation of relevant theories, Zang Hongyan and Liu Jianying wrote the paper, Zang Hongyan and Liu Jianying revised the manuscript. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.