Abstract—A new nonlinear dynamical model with three

quadratic nonlinear terms is introduced in this work. Lyapunov exponents are calculated to discover properties such as dissipativity and chaotic behavior. It is also established that there is no rest point for the proposed nonlinear plant signifying that the dynamical model undergoes hidden chaotic behavior. The proposed hidden chaotic system is implemented using electronic devices, which simulation results are in good agreement with MATLAB simulations.

Index Terms—Chaos, chaotic systems, hidden attractor, nonlinear plant, dynamical systems, circuit model.

I. INTRODUCTION Nowadays, chaotic systems have a new classification

that can be performed according to their nonlinear dynamics, so that one can distinguish two kinds of attractors: self-excited attractor and hidden attractor. In the former case, the attractor has a basin of attraction that is excited from unstable equilibrium point. As already mentioned in [1], the classical systems under this category are the classical nonlinear systems such as Lorenz’s, Rössler’s, Chen’s, Lü’s, or Sprott’s systems. However, nowadays systems with hidden attractors have received great attention from both the theoretical and practical point of views. Self-excited attractors can be localized straightforwardly yet applying manual calculations. In contrast, one must develop specific computational procedures to identify a hidden attractor because the evaluation of their equilibrium points is difficult and they do not help in their localization.

Chaotic systems have a variety of engineering applications, as in the transmission of data. For example, Liu in [2] is proposing a direct acquisition algorithm using chaotic sequences to improve the communication systems based on chaotic DSSS signals, which aids in overcoming difficulties in different acquisition problems. Gohari et al. [3] proposed an algorithm for 3-D planning using chaotic maps for the motion planning and

regulation of a quadrotor for boundary surveillance applications. For secure systems, Wang and Li [4] devised a color image encryption method which is constructed using Hopfield chaotic neural network. Hua et al. [5] dealt with the image encryption problem by proposing a cosine-transform-based chaotic system, which is useful in cryptography. Naseer et al. [6] proposed a new approach to improve the security of multimedia systems using a chaotic map. Zarebnia et al. [7] investigated the image encryption problem for gray scale images with a numerical algorithm based on hybrid chaotic dynamical systems. Nasr et al. [8] investigated the problem of workspace coverage for mobile robotic motion with a flatness controller and multi-scroll chaotic attractor. Karakaya et al. [9] proposed a memristive chaotic circuit and discussed also FPGA implementation and TRBG based on it. Wang and Dong [10] discussed a 4-D autonomous quadratic hyperchaotic system from the classical Lorenz system and built an electronic circuit design. As one sees, the development of new chaotic systems opens the possibility of improving such kinds of applications.

This research work introduces a new nonlinear dynamical model with three nonlinear terms of quadratic polynomial type. To ensure that the proposed dynamical model is chaotic, Lyapunov exponents are calculated using time-series of the solutions, which also help to discover properties such as dissipativity and chaoticity. The dynamical analysis shows that there is no rest point for the proposed nonlinear plant, in this sense one can say that this new nonlinear dynamical model undergoes hidden chaotic behavior [1].

Section II introduces the new nonlinear dynamical model, which dynamical analysis confirms that the attractor is hidden. Numerical simulations are performed to observe the corresponding attractor in different phase-space portraits. Section III shows the scaling of the original new nonlinear dynamical model, in order to perform a circuit simulation using MultiSim. Finally, the conclusions are given in Section IV.

A New Nonlinear Dynamical Model with Three Quadratic Nonlinear Terms and Hidden Chaos

1Sundarapandian Vaidyanathan, 2Esteban Tlelo-Cuautle, 3Aceng Sambas, 4Francesco Grasso

1Research and Development Centre, Vel Tech University, 400 Feet Outer Ring Road, Avadi, Chennai-600062, Tamil Nadu, India (e-mail: sundarcontrol@gmail.com).

2Department of Electronics, INAOE, Tonantzintla Puebla 72840, Mexico (e-mail: etelo@inaoep.mx). 3A. Sambas is with the Department of Mechanical Engineering, Universitas Muhammadiyah

Tasikmalaya, Indonesia (e-mail: acengs@umtas.ac.id ). 4Università di Firenze, Dept. of Information Engineering (DINFO). 50139 Firenze, Italy.

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II. A NEW NONLINEAR HIDDEN CHAOS MODEL The new nonlinear dynamical model with three

quadratic nonlinear terms is given in (1). It can be appreciated that is has three state variables 1 2 3, ,ξ ξ ξ .

!ξ1 = ξ2!ξ2 = aξ1ξ3 +bξ2

2 − cξ2 −ξ2ξ3!ξ3 = ξ2

2 −1

⎧

⎨⎪⎪

⎩⎪⎪

(1)

In the plant modeled by the dynamical equations from

(1) one can see three coefficients ,a b and .c The appropriate values of these parameters to produce a positive Lyapunov exponent to confirm chaotic behavior are ( , , ) (0.1,0.1,0.15)a b c = . In this manner, the three

Lyapunov exponents of (1) are estimated in MATLAB after 1 5T E= seconds for and (0) (0.2,0.2,0.2).ξ = It

results in 1 2 3( , , ) (0.0536,0, 0.1829).LE LE LE = − Therefore, in view of the signs ( ,0, )+ − of the Lyapunov exponents and having a negative sum, it follows naturally that this new nonlinear dynamical model exhibits dissipative chaotic motion.

The equilibrium or rest points of the new chaotic model given by (1) are evaluated from the following equations:

ξ2 = 0 (2a)

21 3 2 2 2 3 0a b cξ ξ ξ ξ ξ ξ+ − − = (2b)

22 1 0ξ − = (2c)

From (2a), 2 0,ξ = which contradicts (2c). In this

manner, one concludes that there is not a defined rest point for this 3-D nonlinear dynamical model, and then it signifies that the model given by (1) exhibits hidden chaos motion [1].

The different phase-space portraits 2D views of this hidden chaotic attractor are shown in Figures 1, 2, and 3, and a 3D view is given in Fig. 4. The numerical simulations can be performed using the fourth order Runge-Kutta Method in MATLAB. In all cases the simulations were performed setting the parameters ( , , ) (0.1,0.1,0.15)a b c = and using the initial conditions

(0) (0.2,0.2,0.2).ξ = The mathematical model given in (1) can be

implemented using embedded systems, as detailed in [11], or it can be implemented using analog circuits, as shown in the following Section.

Figure 1. Computer plot of the 2-D projection view of the hidden attractor (1) in 1 2( , )ξ ξ −plane.

Figure 2. 2-D phase-space projection view of the hidden attractor (1) in

2 3( , )ξ ξ −plane.

Figure 3. 2-D phase-space projection view of the hidden attractor (1) in (ξ1,ξ3)− plane.

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Figure 4. Computer plot of the 3-D view of the hidden attractor (1).

III. MULTISIM CIRCUIT MODEL OF THE NEW NONLINEAR DYNAMICAL PLANT WITH HIDDEN CHAOS

The new nonlinear dynamical model given in (1), which dynamical analysis confirms that the attractor is hidden, can be simulated using commercially available electronic devices that can be modeled in the circuit simulator called MultiSim. In this manner, this Section shows the scaling of (1), as given by (3), and these equations are synthesized by multipliers, amplifiers and passive resistors and capacitors. It leads us to the circuit sketched in Fig. 5.

!ξ1 = ξ2

!ξ2 =aξ1ξ34

+bξ2

2

2− cξ2 −

ξ2ξ34

!ξ3 = ξ22 − 4

⎧

⎨

⎪⎪

⎩

⎪⎪

(3)

Applying Kirchhoff’s current and voltage laws to Fig. 5 one gets (4), which associates the equations given in (3). Still ξ1, ξ2, ξ3 are the states variables but they are representing the voltages across the capacitors C1, C2 and C3, respectively.

!ξ1 =1R1C1

ξ2

!ξ2 =1R2C2

ξ1ξ3 +1R3C2

ξ22 −

1R4C2

ξ2 −1R5C2

ξ2ξ3

!ξ3 =1R6C3

ξ22 −

1R7C3

V1

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

(4)

The values of the circuit elements in Fig. 5 are selected as: R1 = R6 = 400 kΩ, R2 = 16 MΩ, R3 = 8 MΩ, R4 = 2.7 MΩ, R5 = 1.6 MΩ, R7 = R8 = R9 = R10 = R11 = 100 kΩ, V1 = 1 VDC, C1 = C2 = C3 = 1 nF. The multiplier and amplifier are commercially available, and the power supplies of all active devices are set to ±15 Volts.

The simulation results of the chaotic attractors using MultiSim are shown in Figures 6, 7 and 8, which are in good agreement with the simualtion results shown in Figures 1, 2 and 3, respectively.

Figure 5. MultiSim circuit of the new dynamical model with hidden chaos.

Figure 6. MultiSim circuit simulation result of the new dynamical model

with hidden chaos in ξ1- ξ2 plane.

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Figure 7. MultiSim circuit simulation result of the new dynamical model

with hidden chaos in ξ2- ξ3 plane.

Figure 8. MultiSim circuit simulation result of the new dynamical model

with hidden chaos in ξ1- ξ3 plane.

IV. CONCLUSION We have shown the numerical and circuit simulation results of a new 3D dynamical model with hidden chaos. Lyapunov exponents were calculated to confirm that the system exhibits chaotic behavior, and the dynamical analysis showed the hidden attractor characteristic. The significant research contribution of this paper is the discovery of a new nonlinear dynamical model with three quadratic nonlinear terms exhibiting dissipativity and hidden chaoticity. The good agreement of the results between MATLAB and MultiSim confirmed the generation of hidden chaos.

REFERENCES [1] V.T. Pham, S. Vaidyanathan, C. Volos and T. Kapitaniak,

Nonlinear Dynamical Systems with Self-Excited and Hidden Attractors, Springer, Berlin, Germany, 2018.

[2] F. Liu, “Unconventional direct acquisition method for chaotic DSSS signals,” AEU - International Journal of Electronics and Communications, vol. 99, pp. 293-298, Feb. 2019.

[3] P.S. Gohari, H. Mohammadi and S. Taghvaei, “Using chaotic maps for 3D boundary surveillance by quadrotor robot,” Applied Soft Computing, vol. 76, pp. 68-77, March 2019.

[4] X.Y. Wang and Z.M. Li, “A color image encryption algorithm based on Hopfield chaotic neural network,” Optics and Lasers in Engineering, vol. 115, pp. 107-118, April 2019.

[5] Z. Hua, Y. Zhou and H. Huang, “Cosine-transform-based chaotic system for image encryption,” Information Sciences, vol. 480, pp. 403-419, April 2019.

[6] Y. Naseer, D. Shah and T. Shah, “A novel approach to improve multimedia security utilizing 3D mixed chaotic map,” Microprocessors and Microsystems, vol. 65, pp. 1-6, March 2019.

[7] M. Zarebnia, H. Pakmanesh and R. Parvaz, “A fast multiple-image encryption algorithm based on hybrid chaotic systems for gray scale images,” Optik, vol. 179, pp. 761-773, Feb. 2019.

[8] S. Nasr, H. Mekki and K. Bouallegue, “A multi-scroll chaotic system for a higher coverage path planning of a mobile robot using flatness controller,” Chaos, Solitons & Fractals, vol. 118, pp. 366-375, Jan. 2019.

[9] B. Karakaya, A. Gülten and M. Frasca, “A true random bit generator based on a memristive chaotic circuit: Analysis, design and FPGA implementation,” Chaos, Solitons and Fractals, vol. 119, pp. 143-149, Feb. 2019.

[10] H. Wang and G. Dong, “New dynamics coined in a 4-D quadratic autonomous hyper-chaotic system,” Applied Mathematics and Computation, vol. 346, pp. 272-286, April 2019.

[11] E. Tlelo-Cuautle, L. G. de la Fraga and J. Rangel-Magdaleno, Engineering applications of FPGAs. New York, NY, USA, Springer, 2016.

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