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In this paper, new electronic circuit was designed with two-equilibrium as an engineering application on a three-dimensional chaotic system. The circuit consists of resistors, capacitors, voltages and operational amplifiers TL032CN. The adopted continuous-time chaotic dynamical system is with quadratic cross-product nonlinear terms and parameters. The basic characteristics of the proposed circuit model were analyzed in detail by equilibrium points, stability analysis, Lyapunov exponents and Kaplan-Yorke dimension. The results were simulated theoretically using MultiSIM 10, and it was well consistent with the results obtained from the Matlab program.

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 [

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 [

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.

An autonomous 3-D dynamical system [

x ˙ 1 = ρ ( x 2 − x 1 ) x ˙ 2 = a x 1 − δ x 1 x 3 x ˙ 3 = φ x 1 x 2 − x 3 (1)

where x 1 x 3 , x 1 x 2 are the quadratic cross-product nonlinear terms in the dynamical system. The system (1) is chaotic when the parameters values ( ρ , a , δ and φ ) are taken as

ρ = 10 , δ = 40 , a = 296.5 , φ = 10 (2)

So for the given values of parameters, the Lyapunov exponents of system (1) are determined as

L 1 = 2.509426 , L 2 = 0.132019 and L 3 = − 11.818787 .

Also, Kaplan-Yorke dimension of system (1) is calculated as

D K Y = 2 + L 1 + L 2 | L 3 | = 2.22349544

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 x | x 1 ( 0 ) , x 2 ( 0 ) , x 3 ( 0 ) = [ − 2 , 7 , 12 ] .

Figures 2(a)-(c) show the system (1) exhibit chaotic attractors in (a): ( x 1 , x 2 ) plane, (b): ( x 2 , x 3 ) plane, (c): ( x 1 , x 3 ) plane.

Now, the wave-form x 1 ( t ) , x 2 ( t ) , x 3 ( t ) 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.

An electronic circuit is designed to implement chaotic system (1). The circuit consists of electronic elements: capacitors, multipliers, resistors and operational amplifiers TL032CN.

By applying Kirchhoff’s laws [

d V x 1 d t = 1 R 1 C 1 ( V x 2 − V x 1 ) d V x 2 d t = 1 R 2 C 2 V x 1 − 1 R 3 C 2 V x 1 V x 3 d V x 3 d t = 1 R 4 C 3 V x 1 V x 2 − 1 R 5 C 3 V x 3 (3)

where V x 1 , V x 2 , V x 3 are the output voltages and k m = 10 V is the fixed multipliers constant, hence the outputs are V x 1 x 3 = V x 1 V x 3 / k m and V x 1 x 2 = V x 1 V x 2 / k m .

Voltages and time normalized by dimensionless states variables

V x 1 = 1 V ⋅ x 1 , V x 2 = 1 V ⋅ x 2 , V x 3 = 1 V ⋅ x 3 , t ′ = τ ⋅ t = 100 μ s ⋅ t (4)

Substitute (4) in equations of system (3) we get:

d x 1 d t ′ = τ R 1 C 1 ( x 2 − x 1 ) d x 2 d t ′ = τ R 2 C 2 x 1 − τ R 3 C 2 x 1 x 3 d x 3 d t ′ = τ R 4 C 3 x 1 x 2 − τ R 5 C 3 x 3 (5)

Comparing system (1) with system (5) gives following conditions:

τ R 1 C 1 = ρ , τ R 2 C 2 = a , τ R 3 C 2 = δ , τ R 4 C 3 = φ , τ R 5 C 3 = 1 (6)

Take convenient values for capacitances and resistances as

C 1 = C 2 = C 3 = 1 mF , R 1 = R 4 = 10 Ω , R 2 = 337.268 m Ω , R 3 = 2.5 Ω , R 5 = 100 Ω . (7)

We obtained the experimental electronic circuit (5) for system (1) with parameters ρ = 10 , a = 296.5 , δ = 40 , φ = 10 .

To find equilibrium points we need to solve the nonlinear equations as follows:

τ R 1 C 1 ( x 2 − x 1 ) = 0 τ R 2 C 2 x 1 − τ R 3 C 2 x 1 x 3 = 0 τ R 4 C 3 x 1 x 2 − τ R 5 C 3 x 3 = 0 (8)

We get two equilibrium points

E 1 = ( 593 2 20 , 593 2 20 , 593 80 ) , E 2 = ( − 593 2 20 , − 593 2 20 , 593 80 )

The Jacobian matrix of system (5) is:

J = [ − τ R 1 C 1 τ R 1 C 1 0 τ R 2 C 2 − τ R 3 C 2 x 3 0 − τ R 3 C 2 x 1 τ R 4 C 3 x 2 τ R 4 C 3 x 1 − τ R 5 C 3 ]

J E 1 = [ − 10 − λ 10 0 266.85 − λ − 34.43835072 8.609587679 8.609587679 − 1 − λ ]

And

J E 2 = [ − 10 − λ 10 0 266.85 − λ 34.43835072 − 8.609587679 − 8.609587679 − 1 − λ ]

Now, find characteristic equation by setting det ( J − λ I ) = 0 , we get the same equation at E 1 and E 2 :

λ 3 + 11 λ 2 − 2362 λ + 3261.5 = 0 (9)

We obtain the same eigenvalues at equilibrium points E 1 and E 2 :

λ 1 = 1.39097 , λ 2 = 42.622 , λ 3 = − 55.013

Since, there are positive eigenvalues, so system (5) is unstable.

From characteristic Equation (9) we get

a 0 = 3261.5

a 1 = − 2362

a 2 = 11

a 3 = 1

b 1 = a 1 − a 3 a 0 a 2 = − 5634.5

Since, there is a negative element in the first column of

We use determinants formed from coefficients of the characteristic Equation (9) we get:

Δ 1 = a 2 = 11 > 0

Δ 2 = | a 2 a 0 a 3 a 1 | = − 29243.5 < 0

Δ 3 = | a 2 a 0 0 a 3 a 1 0 0 a 2 a 0 | = − 95377675.25 < 0

Since there is values of minors are less than zero, so the system (5) is unstable.

λ 3 | 1 | −2362 |
---|---|---|

λ 2 | 11 | 3261.5 |

λ 1 | −5634.5 | 0 |

λ 0 | 3261.5 | 0 |

Lapiynuov function and its derivatives for system (5) yield (10) & (11).

We assume that

V ( x 1 , x 2 , x 3 ) = 1 2 ( x 1 2 + x 2 2 + x 3 2 ) (10)

V ˙ ( x 1 , x 2 , x 3 ) = ∂ v ∂ x 1 d x 1 d t ′ + ∂ v ∂ x 2 d x 2 d t ′ + ∂ v ∂ x 3 d x 3 d t ′

V ˙ ( x 1 , x 2 , x 3 ) = x 1 x ˙ 1 + x 2 x ˙ 2 + x 3 x ˙ 3 (11)

By substituting (5) in Equation (11) we get:

V ˙ ( x 1 , x 2 , x 3 ) = − 10 x 1 2 + 306.5 x 1 x 2 − 30 x 1 x 2 x 3 − x 3 2

Since V ˙ ( x 1 , x 2 , x 3 ) > 0 , therefore the system (5) is unstable.

The values of Lapiynuov exponents are:

L 1 = 2.509426 , L 2 = 0.132019 , L 3 = − 11.818787

Therefore, the Lapiynuov dimension “Kaplan-Yοrke dimension” is:

D L = 2 + L 1 + L 2 | L 3 | = 2.22349544

So the system is hyperchaotic system, as shown in

In this section, the designed circuit to implement the chaotic system (1) was simulated by electronic simulation MultiSIM 10 program;

The outputs voltages signals V x 1 , V x 2 , V x 3 versus time, and phase portraits of the attractors are presented, in

By comparing

A three-dimensional chaotic system with two equilibrium points is analyzed. The Lyapunov dimension of the chaotic system is computed as D_{L} = 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.

The authors declare no conflicts of interest regarding the publication of this paper.

Aziz, M.M. and Merie, D.M. (2020) A Three-Dimensional Chaotic System and Its New Proposed Electronic Circuit. Open Access Library Journal, 7: e6555. https://doi.org/10.4236/oalib.1106555