A three-dimensional chaotic system generating single-wing and two-wing chaotic attractors

In this paper, a three-dimensional chaotic system is proposed based on a simple 3-D autonomous system by adding a linear piecewise function. It is very interesting that the new three-dimensional chaotic system can generate single-wing or two-wing chaotic attractor with variation of parameter. Several basic characteristics of the system, such as bifurcation diagram, phase orbits, and Lyapunov exponents are given to investigate different chaotic motions for the new system. The new system is found to be chaotic in a wide parameter range, and to show many complex dynamical behaviors. That is, the results obviously show the system is chaotic and its dynamics are very complex.


Introduction
In recent decades, many chaotic systems have been proposed since Lorenz discovered a simple three-dimensional smooth autonomous chaotic system in 1963 [1].It is well known that chaotic systems can show complex dynamic characteristics that can be used in many application fields such as engineering [2], image encryption [3], secure communications [4] etc. Therefore, many scholars and researchers proposed and studied some chaotic systems, such as Chen system [5], Lü system [6], Qi system [7], and so on [8,9].
Generally, the structure and characteristics of the chaotic systems can be changed when introducing linear piecewise function.Therefore, by the method of introducing linear piecewise function, researchers have designed many complex chaotic systems.For example, in [10], the paper introduced the method of designing chaotic system by using piecewise linear function, and discussed the computer simulation and circuit implementation in detail.In addition, the application of the nondominated sorting genetic algorithm to optimize two characteristics of multi-scroll chaotic oscillators was introduced in [11].Where the linear piecewise function is used to generate the two multi-scroll chaotic oscillators.
In this paper, a new chaotic system is designed by adding a linear piecewise function to the second equation of a simple 3-D autonomous system [12].This system can show some very complex dynamic characteristics generating single-wing or two-wing chaotic attractor when varying the system parameters.In the following sections, bifurcation diagram, phase orbits, and Lyapunov exponents of the system are calculated to analyze the dynamic characteristics of the system.The numerical results analysis clearly show that this system is chaotic and its dynamics are very complex.

The 3-D chaotic system
Considering the simple three-dimensional autonomous system [12]:

P -209
Where the parameters a, b, c, d and h are real constants.
Here, a linear piecewise function is introduced to the three-dimensional autonomous system above.This piecewise function will substitute for the linear term z of the second equation, and the new chaotic system is obtained as following.
̇= − Where (z) is the linear piecewise function described as formula (3): 4) is the turning points formula of the linear piecewise function.Here, we set = 1, 1, = 0.5 , so the linear piecewise function can be simplified as equation (5).

Dynamical characteristics of the chaotic system
For the chaotic system (2), when the parameters a, b, c, d, h and are taken different values, the system can show different kinds of chaotic states.By analyzing, the system is found to show a single-wing chaotic attractor when the parameters 37, 25, = 13, = 6, ℎ = 8, = 2 and two-wing chaotic attractor when the parameters 37 , 25 , = 16 , = 6 , ℎ = 8 and = 0.5, respectively, as shown in Fig. 1 and Fig. 2.
In the following section, the Lyapunov exponents and the corresponding bifurcation diagram of system (2) are given to analysis the complex chaotic dynamics.3), the system (2) exhibits periodic dynamics with the biggest Lyapunov exponent of the system (2) being zero.when ∈ [9.3, 25.2), the biggest Lyapunov exponent of the system (2) is found to positive, implying that the system is chaotic.
In addition, by analyzing the corresponding bifurcation diagram in Fig. 3 (b), the system (2) also exhibits periodic dynamics when ∈ [1.8, 9.3), here, a period-2 orbit is given to show its periodic dynamics when = 8, as shown in Fig. 4 (a) and Fig. 4 (b).And this system also becomes chaotic when ∈ [9.3, 25.2), and its dynamics very complex.A single-wing chaotic attractor is found when the parameter = 16, as shown in Fig. 4 (c) and Fig. 4 (d).And a double-wing chaotic attractor is observed when the parameter = 20, as shown in Fig. 4 (e) and

Lyapunov exponents and bifurcation diagram
analysis when = .
When varying parameter c, the Lyapunov exponents the corresponding bifurcation diagram of the system (2) are shown in Fig. 5(a

Conclusion
In this paper, a three-dimensional chaotic system is designed by introducing a linear piecewise function.It is found that the system can generate a single-wing chaotic attractor or two-wing chaotic attractor by varying parameter .Some basic properties of the chaotic system were analyzed by means of Lyapunov exponents and bifurcation diagrams analysis in detail.The numerical results confirm that this chaotic system is chaotic and it can show complex dynamic characteristics.