Chaos theory describes nonlinear dynamical systems that are very sensitive to initial conditions. Since the experimental discovery of a chaotic system by Lorenz, chaos theory has found applications in several areas in science and engineering . Chaos is a phenomenon caused by the sensitivity of perturbation structural parameters and the initial conditions of some categories of dynamic systems  .
In last decades, chaotic circuit has received considerable interest in the researches due to the fact that they have been applied in abundant areas like in secure communications, simulating economical models, design of electronic circuits, robotics, image processing, and neural networks  . For this reason, a large number of chaotic systems, which are implemented in circuit, are reported in the literature .
Electrical laws are necessary to analyze any electrical circuit effectively and efficiently by determining different circuit parameters such as current, voltage power and resistance. These laws include Ohms law, Kirchhoff’s current and voltage laws etc. .
This paper contains: Section 2, we description 3-D chaotic system; it is mainly consisted of six simple terms including two nonlinear terms. Section 3, an electronic circuit is designed to implement chaotic system (1). Section 4, we simulated the designed circuit by electronic simulation MultiSIM 10 program. And Section 5, we presented the conclusions.
2. Description of Chaotic System with Two Equilibrium Points
An autonomous 3-D dynamical system  with two quadratic cross-product nonlinear terms
where , are the quadratic cross-product nonlinear terms in the dynamical system. The system (1) is chaotic when the parameters values ( and ) are taken as
So for the given values of parameters, the Lyapunov exponents of system (1) are determined as
, and .
Also, Kaplan-Yorke dimension of system (1) is calculated as
which shows hyper-chaotisity of system (1).
Therefore, the hyperchaotic system (1) has a strange attractor. For graphical results, we used Matlab and take initial states .
Figure 1 shows chaotic attractor for system (1) in .
Figures 2(a)-(c) show the system (1) exhibit chaotic attractors in (a): plane, (b): plane, (c): plane.
Now, the wave-form for the system (1) has a non-periodic shape, shown in Figures 3(a)-(c) which is one of the basic characteristic behaviors of chaotic dynamical system.
3. The Proposed Circuit of 3-D Chaotic System (1)
An electronic circuit is designed to implement chaotic system (1). The circuit consists of electronic elements: capacitors, multipliers, resistors and operational amplifiers TL032CN.
Figure 1. Chaotic attractor of system (1) with (ρ, a, δ, ϕ) = (10, 296.5, 40, 10) in (x1-x2-x3) space.
Figure 2. Chaotic attractor of system (1) with (ρ, a, δ, ϕ) = (10, 296.5, 40, 10) in (a) (x1-x2) plane, (b) (x2-x3) plane, (c) (x1-x3) plane.
Figure 3. The wave-form of system (1). (a): Time versus x1; (b): Time versus x2; (c): Time versus x3.
By applying Kirchhoff’s laws , the corresponding circuit equations described as:
where are the output voltages and is the fixed multipliers constant, hence the outputs are and .
Voltages and time normalized by dimensionless states variables
Substitute (4) in equations of system (3) we get:
Comparing system (1) with system (5) gives following conditions:
Take convenient values for capacitances and resistances as
We obtained the experimental electronic circuit (5) for system (1) with parameters , , , .
3.1. Equilibrium Points
To find equilibrium points we need to solve the nonlinear equations as follows:
We get two equilibrium points
3.2. Stability Analysis
3.2.1. Characteristic Equation Roots
The Jacobian matrix of system (5) is:
Now, find characteristic equation by setting , we get the same equation at and :
We obtain the same eigenvalues at equilibrium points and :
Since, there are positive eigenvalues, so system (5) is unstable.
3.2.2. Routh Stability Criteria
From characteristic Equation (9) we get
Since, there is a negative element in the first column of Table 1. Therefore, System (5) is unstable.
3.2.3. Hurwitz Stability Criteria
We use determinants formed from coefficients of the characteristic Equation (9) we get:
Since there is values of minors are less than zero, so the system (5) is unstable.
Table 1. Routh array of system (5).
3.2.4. Lapiynuov Function
Lapiynuov function and its derivatives for system (5) yield (10) & (11).
We assume that
By substituting (5)
Since , therefore the system (5) is unstable.
3.3. Lapiynuov Exponent and Lapiynuov Dimension
The values of Lapiynuov exponents are:
Therefore, the Lapiynuov dimension “Kaplan-Yοrke dimension” is:
So the system is hyperchaotic system, as shown in Figure 4.
4. The Simulation Results
In this section, the designed circuit to implement the chaotic system (1) was simulated by electronic simulation MultiSIM 10 program; Figure 5 shows sketch of the designed circuit of chaotic system (1)
The outputs voltages signals versus time, and phase portraits of the attractors are presented, in Figure 6 which shows the chaotic behaviors of the circuit.
Figure 4. Lapiynuov exponent of system (5).
Figure 5. Sketch of the designed circuit of chaotic system (1).
Figure 6. The left side shows the output voltage signals in (a) (V1 versus time), (b) (V2 versus time), (c) (V3 versus time); While right side shows the phase portraits of the designed electronic circuit in (d) plane, (e) plane, (f) plane.
A three-dimensional chaotic system with two equilibrium points is analyzed. The Lyapunov dimension of the chaotic system is computed as DL = 2.223495, which shows that the system is hyperchaotic system. An electronic circuit is designed to implement chaotic system (1). Then, the basic characteristics of the proposed circuit model were analyzed by equilibrium points, stability analysis methods (such as characteristic equation roots, Routh criterion and Lapiynuov function); all these methods proved the instability of the new designed circuit (3), Lyapunov exponents and Kaplan-Yorke dimension that shows chaotisity of the designed circuit. The designed circuit is simulated by electronic simulation MultiSIM 10 program; the results show that there is a well qualitative agreement between the experimental achievements and numerical simulation which is obtained by using Matlab.
The authors are very grateful to the University of Mosul/College of Computer Sciences and Mathematics for their provided facilities, which helped to improve the quality of this work.
 Aziz, M.M. and Hamid, M.A. (2019) The possibility of Increasing the Predictability Indices after Control of 3-D Continuous-Time system. International Conference on Computing and Information Science and Technology and Their Applications, Kirkuk, 3-5 March 2019, 1-5.
 Lamamra, K., Vaidyanathan, S., Putra, W.T., Darnila, E., Sambas, A. and Mujiarto (2020) A New 3-D Chaotic Jerk System with Four Nonlinear Terms, Its Backstepping Synchronization and Circuit Simulation. Journal of Physics: Conference Series, 1477, Article ID: 022017.