A NEW THREE-DIMENSIONAL CHAOTIC SYSTEM WITHOUT EQUILIBRIUM POINTS , ITS DYNAMICAL ANALYSES AND ELECTRONIC CIRCUIT APPLICATION

Original scientific paper In this paper, a new three-dimensional chaotic system without equilibrium points is introduced and analysed. Basic dynamical analysis of this new chaotic system without equilibrium points is carried out by means of system equilibria, phase portraits, sensitivity to initial conditions, fractal dimension and chaotic behaviours. In addition, in this paper Lyapunov exponents spectrum and bifurcation analysis of the proposed chaotic system have been executed by means of selected parameters. The chaotic system without equilibrium points has been executed by detailed theoretical analysis as well as simulations with designed electronical circuit. A chaotic system without equilibrium points is also known as chaotic system with hidden attractor and there are very few researches in the literature. Since they cannot have homoclinic and heteroclinic orbits, Shilnikov method cannot be applied to find whether the system is chaotic or not. Therefore, it can be useful in many engineering applications, especially in chaos based cryptology and coding information. Furthermore, introduced chaotic system without equilibrium points in this paper can have many unknown dynamical behaviours. These behaviours of the strange chaotic attractors deserve further investigation.


Introduction
Chaotic dynamical equation research is an important problem in nonlinear science.Chaotic systems have been widely studied within scientific and engineering environments in the last years [1÷23].It is possible to classify some existing 3D autonomous chaotic systems with standard parameter values.Case 1 covers chaotic attractors with 1 saddle and 2 unstable saddle-foci, like the Lu system [6], and Chen system [7], and Case 2 covers attractors having 2 unstable saddle-foci, like DLS [8], the Rössler system [9], and the Burke-Shaw system [10,19].Case 3 covers chaotic attractors with 1 saddle and 2 stable node-foci.For instance, another such 3D chaotic system was introduced by Yang and Chen [11].Pehlivan and Uyaroglu [12] and Yang et al. [13] recently designed novel chaotic systems with 2 stable node-foci.
Chaotic systems with no equilibria have been investigated firstly in Ref. [24÷26].In Ref. [24], a systematic search to find 3D chaotic systems with quadratic nonlinearities and no equilibria was performed.Since they cannot have homoclinic and heteroclinic orbits, the Shilnikov method [27] cannot be applied to no equilibria systems.Chaotic system without equilibrium points is named as chaotic system with hidden attractor [28,29].
In this study, a new three-dimensional chaotic system without equilibrium points is discussed.For the new system, dynamical analyses such as equilibria, phase portraits, Lyapunov exponent spectrum, fractal dimension and bifurcation analysis were performed with Matlab programme in the first part.In the second part, electronic circuit of the new chaotic system was designed in OrCAD-PSpice programme and phase portraits were observed.The last part includes the results and evaluations.

The new chaotic system and its properties
The new chaotic system without equilibrium points is described by the following differential equation 1: (2)

System equilibria
The equilibria of the system are found by settings 0 .
Since there are not real equilibrium points, the four equilibrium points E 1 , E 2 , E 3 and E 4 are complex.Because the new chaotic system does not have equilibrium points, the new system is without equilibrium points.The local behaviour of the system around these complex number equilibrium points can be investigated by using the following Jacobian matrix: For the complex number equilibrium point E 1 , the Jacobian matrix becomes the following Eq.( 5): To obtain its eigenvalues, let det(λ•I − J(E 1 )) = 0.Then, the characteristic equation has the following form:

Sensitivity to initial conditions
Sensitivity to initial conditions means that any arbitrarily small perturbations in the initial state of the system can lead to the significantly distant behaviour of the future state of the chaotic system.This dependence of initial conditions in the system makes the long term prediction impossible.Obtained time series using ode45 function for x1(0) = 0 and x2(0) = 0,001 are given in Fig. 1.It shows that the evolution of the chaos trajectories is very sensitive to initial conditions.

Phase portraits
The initial values of the new chaotic system without equilibrium points are given as (0, 0, 0) and its simulations have been realized for the phase portraits.This new chaotic system exhibits rich dynamic behaviors.We show the y-z, x-y, x-z and x-y-z phase portraits of the new chaotic system using Matlab ode45 function in Fig. 2.

Lyapunov exponents spectrum and fractional dimension
Fig. 3 shows the Lyapunov exponents spectrum of the new system without equilibrium points for a varying parameter b.As can be seen from the Lyapunov exponents spectrum, b is changed in the range of 0 ÷ 1.The system is chaotic when the first Lyapunov exponent (blue line) is positive, the second Lyapunov exponent (green line) is zero and the third Lyapunov exponent (red line) is negative.
The Lyapunov exponent is a way for analyzing the nonlinear behaviour of the dynamical system.If there is at least one positive Lyapunov exponent, the system is chaotic.The Lyapunov exponents of the system are computed numerically with the parameter values a = 2,8, b = 0,1, c = 1,4, d = 1, e = 10 and f = 2 and the initial conditions chosen as x(0) = (0, 0, 0).The corresponding values for the Lyapunov exponents are L 1 = 0,1403, L 2 =0, L 3 = −2,1515.Eq. ( 10) means the system is really a threedimensional chaotic system and the Lyapunov dimensions of the system are fractional.Having a strange attractor and positive Lyapunov exponent, it is obvious that the system is chaotic.
In Fig. 6, detailed chaotic status in bifurcation diagram when 'b' is changed in the range 0,05 ÷ 0,47 is given.

Circuit implementation
The electronic circuit of the new chaotic system was designed in OrCAD-PSpice programme.The electronic circuit schematic of the new system is seen in Fig. 7.The circuit includes simple electronic elements such as resistors, operational amplifiers, and is easy to construct.
Experimental electronic circuit of the new chaotic system was designed for parameter a = 2,8, b = 0,2, c = 1,4, d = 1, e = 10, f = 2, and initial conditions x(0) = 0, y(0) = 0, z(0) = 0. Since it does not require the initial condition voltages for executing the circuit, realization of the new circuit in real time is easy.
The OrCAD-PSpice simulation oscilloscope outputs (phase portraits) of circuitry of the new system without equilibrium points, for parameters a = 2,8, b = 0,2, c = 1,4, d = 1, e = 10 and f = 2, are seen in Fig. 8 ÷ 10.As they can be seen from the Matlab ode45 function simulation outputs in Fig. 2 and the OrCAD-PSpice simulation outputs in Fig. 8, the results are the same.The studies about chaotic systems without equilibrium points are very few in the literature.In this paper, a new three-dimensional chaotic system without equilibrium points is discussed.Dynamical properties of the new system were analysed by means of equilibrium points, phase portraits, sensitivity to initial conditions, fractal dimension and chaotic behaviours.Moreover, Lyapunov spectrum and bifurcation analysis have been executed by means of selected parameters.In all analysis, MATLAB programme was used.The results of analysis show that the new chaotic system has no equilibria.So, the new system is a chaotic system without equilibrium points.
Also, it is a new hidden chaotic attractor, too.Since they cannot have homoclinic and heteroclinic orbits, Shilnikov method cannot be applied to find whether the system is chaotic or not.Because of the hidden attractors in the proposed new chaotic system, they can be useful in many engineering applications such as chaos based cryptology, coding information, information compression, random number generator, chaos based music and image generator, etc. in the near future.
In this paper, in addition to basic dynamical analysis, the new chaotic system without equilibrium points has been executed for its simulations using the designed electronical circuit in OrCADPSpice programme.The new chaotic system without equilibrium points is confirmed through electronic circuit design.The simulation results of the Matlab and the OrCAD-PSpice programs were obtained for the same shaped phase portraits.
the new chaotic system are x(0) = 0, y(0) = 0, z(0) = 0 and ten terms and six parameters (a, b, c, d, e and f).Typical parameters are a = 2,8, b = 0,2, c = 1,4, d = 1, e = 10 and f = 2.The new system can be described using typical parameters by the following Eq.(2): Solving the above characteristic equation, the eigenvalues are found as the equilibrium points E 2 , E 3 and E 4 , we have the corresponding eigenvalues, respectively; for E 2 :

Figure 7
Figure 7The electronic circuit schematic of the new chaotic system without equilibrium points