Fixed-Time Synchronization of the New Single-Parameter Chaotic System

This paper advances a new single-parameter chaotic system with a simple structure, and some basic dynamic behavior of the new single-parameter system is discussed, such as equilibria, dissipativity, the existence of an attractor, Lyapunov exponent, Poincaré map, spectrum map, and bifurcation diagram. Moreover, a new fixed-time convergence theorem is proposed for general chaotic systems based on finite-time control theory, and a fixed-time controller is also put to achieve synchronization of the new chaotic system. Simulation results are presented to show the effectiveness of the theoretical results. The conclusion of the paper is useful for the nonlinear economics and engineering application.


Introduction
After the first chaotic attractor was discovered by Lorenz, people began to search for new chaotic attractors with great enthusiasm. In recent years, for autonomous threedimensional ordinary differential equations, the search for new chaotic attractors has attracted more and more attention. For example, Lorenz system [1], Rossler system [2], Chen and Ueta system [3], Lü and Chen system [4], and Liu et al. system [5]. In a three-dimensional autonomous system, all kinds of chaotic systems with distinct features were put forward, such as chaotic systems with only five terms [6], chaotic systems with linear terms [7], and chaotic systems with only one equilibrium point [8].
In practical engineering applications, we hope that the structure of the chaotic system is as simple as possible based on the consideration implementation; for example, Lü et al. have proposed a unified chaotic system with a single parameter with a simple structure, and it has been well applied in practice [9,10]. erefore, how to construct a simple chaotic system is still a meaningful work in engineering.
Furthermore, research on synchronization and control of the chaos system is particularly topical for decades [11][12][13]. And, people came to know about the value of achieving control of chaos systems in finite time in some domains of engineering. [14][15][16][17][18][19][20]. It is necessary to give the initial conditions of the network in advance, owing to that the establishment time of the finite-time control depends on the initial state of the network to a great extent. However, the initial conditions may be random, so it is not easy to solve practical problems. In order to compensate for this, Polyakov produced the fixed-time stability of the system [21], which ensures that the system is without respect to the initial conditions in a bounded time. In the past few years, a lot of fixed-time stability theories have been raised to study fixedtime stability of the system [22][23][24]. Inspired by the existing research results, we study fixed-time synchronization of the chaos system. e aim of this paper is to present a new single-parameter chaotic system with a simple structure, and some basic dynamic behavior of the new system is discussed. e synchronization of the new chaotic system is realized by using the new fixed-time controller in this paper. Compared with the existing finite-time synchronization methods of chaotic systems, the fixed-time synchronization method proposed in this paper is more general.
is paper is organized as follows: Section 2 gives the new single-parameter chaos system. Section 3 discusses some basic characters of the new chaotic system. Section 4 puts forward the fixed-time control condition for a general chaos system. Section 5 gives an illustrative example. e conclusions are given in Section 6.

The New Single-Parameter Chaos System
Consider the following system: which is chaotic for 0.16 < a < 4.8 (see Figures 1-4 for a � 0.5).

Some Basic Dynamic Properties of New Chaotic Systems
In this section, some basic characteristics of the new chaotic system are analyzed.

Equilibria. Let
When a > 0, the two equilibrium points of the system can be expressed as , 0). Let a � 0.5, and it is easy to obtain the following form of system equilibrium (1) is linearized, and the Jacobian matrix is defined as J � According to equations (3) and (4), it can be determined that both E 1 and E 2 are saddle focus nodes.

Dissipativity and the Existence of Attractor.
For dynamical system (1), we can obtain where −1 is a negative value. System (1) is one dissipative system, and an exponential contraction of system (1) is is means that every volume containing the trajectory of the dynamic system will be reduced to zero as t ⟶ ∞ at the exponential rate −1, and the asymptotic motion is located on the attractor of system (1).

Lyapunov Exponent.
It is generally true that a system is chaotic as long as at least one of its Lyapunov exponents is positive. When the initial value is (1, 1, 1), the Lyapunov exponent of system (1) is calculated to be L 1 � 0.7125, L 2 � 0.0082, and L 3 � −0.4486, and its dimension is It implies that system (1) is a dissipative system. Moreover, its Lyapunov dimension is fractional. e fractal character of the attractor means that the system has aperiodic orbits, and the orbits near it are divergent.

Poincaré Map, Spectrum Map, and Bifurcation Diagram.
In Figure 5, the continuous broadband feature is exhibited for the spectrum of system (1). For z � 0, x � 0, and y � 0, Figures 6(a)-6(c)visualize the Poincaré map in planes, respectively, and several sheets of the attractors are displayed.
is can better sum up all the possible behaviors by the bifurcation graph when the parameters on a graph are changing. For 0 ≤ a ≤ 8, Figure 7 is a bifurcation diagram of system (1), which shows its complex bifurcation phenomena.

Fixed-Time Synchronization of the General Chaotic System
Considering the following chaotic system in a general form: where x � (x 1 , . . . , x n ) T and f(x): R m ⟶ R m is a continuous function. So, the controlled chaos system is given by where y � (y 1 , . . . , y n ) T and r(x, y, t) is the controller. en, the error system can be Assumptions 1. We also assume that f is Lipschitz with respect to its argument, i.e., where e � (e 1 , e 2 , . . . , e n ) T � ( Lemma 1 (see [21]). For δ i ≥ 0, i � 1, . . . , n, 0 < ς ≤ 1, and θ > 1, then Lemma 2 (see [21]). Assume that a continuous and a positive-definite function υ(t) satisfies

Complexity
where μ > 0 and 0 < p < 1 are all constants. en, for any given t 0 , υ(t) satisfies with T 1 given by In the following, the new fixed-time controller based on Lemma 2 is proposed.

Remark 1.
In [12][13][14], different finite-time controllers are designed for different chaotic systems. e fixed-time controllers designed in this paper are more general and can be used in other different chaotic systems.

Remark 2.
For the complex dynamic network satisfying Assumption 1, the method of eorem 1 can be used to control the complex network.

Simulation and Results
We assume that the new chaotic system is the drive system, that is, And, the response system is e initial conditions of the master and slave systems are (1, 2, 1) and (2, 1, 3), respectively. rough simulation calculation, η ≈ 58.2516, and let k 1 � 117, k 2 � 118, and k 3 � 0.5. Synchronization of two chaotic systems cannot be achieved without a controller as shown in Figure 8. 6 Complexity synchronization evolution of two chaos systems is shown in Figure 9. Numerical simulations illustrate the validity of eorem 1.

Conclusion
e paper has presented a new chaotic system with a single parameter and discussed some basic dynamic behavior of the new system. Base on finite-time control theory, the fixedtime synchronization criterion of the new chaotic system has also been obtained. Finally, the result of numerical simulation proves that they are feasible. In the future, we will discuss the fixed-time synchronization of general complex networks.

Data Availability
No data were used to support this study.