Abstract
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.
1. 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 three-dimensional 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]. Therefore, 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 [11–13]. And, people came to know about the value of achieving control of chaos systems in finite time in some domains of engineering. [14–20]. 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 fixed-time stability of the system [22–24]. Inspired by the existing research results, we study fixed-time synchronization of the chaos system. The 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. The 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.
This 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. The conclusions are given in Section 6.
2. The New Single-Parameter Chaos System
Consider the following system:which is chaotic for (see Figures 1–4 for ).
3. Some Basic Dynamic Properties of New Chaotic Systems
In this section, some basic characteristics of the new chaotic system are analyzed.
3.1. Equilibria
Let
When , the two equilibrium points of the system can be expressed as and .
Let , and it is easy to obtain the following form of system equilibrium and .
For , system (1) is linearized, and the Jacobian matrix is defined as .
Let to get the eigenvalues of matrix .
Therefore, when , the eigenvalues of matrix are
Similarly, when , the eigenvalues of matrix are
According to equations (3) and (4), it can be determined that both and are saddle focus nodes.
3.2. Dissipativity and the Existence of Attractor
For dynamical system (1), we can obtainwhere is a negative value. System (1) is one dissipative system, and an exponential contraction of system (1) is . This means that every volume containing the trajectory of the dynamic system will be reduced to zero as at the exponential rate , and the asymptotic motion is located on the attractor of system (1).
3.3. 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 , , and , and its dimension is
It implies that system (1) is a dissipative system. Moreover, its Lyapunov dimension is fractional. The fractal character of the attractor means that the system has aperiodic orbits, and the orbits near it are divergent.
3.4. 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. This can better sum up all the possible behaviors by the bifurcation graph when the parameters on a graph are changing. For , Figure 7 is a bifurcation diagram of system (1), which shows its complex bifurcation phenomena.
(a)
(b)
(c)
4. Fixed-Time Synchronization of the General Chaotic System
Considering the following chaotic system in a general form:where and is a continuous function.
So, the controlled chaos system is given bywhere and is the controller.
Then, the error system can be
Assumptions 1. We also assume that is Lipschitz with respect to its argument, i.e.,where .
Lemma 1 (see [21]). For , , and , then
Lemma 2 (see [21]). Assume that a continuous and a positive-definite function satisfieswhere and are all constants. Then, for any given , satisfieswith given by
In the following, the new fixed-time controller based on Lemma 2 is proposed.
Theorem 1. Assuming that hypothesis 1 holds, two chaotic systems (7) and (8) can be synchronized by the following fixed-time controllers:and the time is bounded aswhere , and .
Proof. LetThen,Obviously,So,Let , thenWhen , from Lemma 2, we haveBy (23)-(24),If , and it is known from Lemma 2 that the zero solution of system (9) is fixed-time stable.
From (21),So,When , , the settling time can be estimated asWhen , , soThe proof is completed.
Remark 1. In [12–14], different finite-time controllers are designed for different chaotic systems. The 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 Theorem 1 can be used to control the complex network.
5. Simulation and Results
We assume that the new chaotic system is the drive system, that is,
And, the response system is
The initial conditions of the master and slave systems are (1, 2, 1) and (2, 1, 3), respectively. Through simulation calculation, , and let , , and . Synchronization of two chaotic systems cannot be achieved without a controller as shown in Figure 8. The synchronization evolution of two chaos systems is shown in Figure 9. Numerical simulations illustrate the validity of Theorem 1.
6. Conclusion
The 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 fixed-time 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.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (61673221); the Natural Science Foundation of Jiangsu Province (BK20181418); the Natural Science Foundation of Jiangsu Higher Education Institutions (19KJB120007); the Six Talent Peaks Project in Jiangsu Province (DZXX-019).