A New Chaotic System with a Pear-Shaped Equilibrium and Its Circuit Simulation

Aceng Sambas, Sundarapandian Vaidyanathan, Mustafa Mamat, Mohamad Afendee Mohamed , WS Mada Sanjaya 1 Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Indonesia 2 Research and Development Centre, Vel Tech University, Chennai, India 3,4 Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Malaysia 5 Department of Physics, Universitas Islam Negeri Sunan Gunung Djati Bandung, Indonesia


INTRODUCTION
In chaos theory, the key important topics are modeling and applications of nonlinear dynamical systems exhibiting chaotic dynamical behavior. Chaotic systems have generated good interest via various science and engineering applications [1]- [7]. Chaos theory has been also applied for special applications such as voice encryption [8], image encryption [9], robotics [10], secure communication [11]- [13] etc.
In this work, we derive a new chaotic system with a pear-shaped equilibrium curve. Our new system exhibits hidden attractors [29]- [30] as it possesses an infinite number of equilibrium points on a pear-shaped curve. This work makes a new valuable addition to the chaotic systems with closed curves of equilibrium points. We also unveil an electronic circuit simulation of the new chaotic system with a pear-shaped curve of equilibrium points. It is known that checking the feasibility of a chaotic system with electronic circuit realization has practical applications [31]- [35].

A NEW CHAOTIC SYSTEM WITH A PEAR-SHAPED EQUILIBRIUM CURVE
Motivated by the method and structure proposed in [26], we report a new three-dimensional dynamical system given by Which has a total of five nonlinearities in the dynamics. We show that the system (1) is chaotic for the parameter values (a, b, c, d) = (15, 0.02, 6, 6).
The Lyapunov chaos exponents are determined as (L 1 , L 2 , L 3 ) = (0.0073, 0, -0.0084). Since L 1 > 0, the new system (1) is chaotic. By adding L 1 , L 2 and L 3 , we get the sum as -0.0011, which is negative. This shows that the new system (1) is dissipative. The Kaplan-Yorke dimension is determined as which is a high value showing the complexity of the new system. The equilibrium points of the new system (1) are tracked by solving the following system:  The phase portraits of the new chaotic system (1) with pear-shaped equilibrium curve are displayed in Figure 2. The Lyapunov chaos exponents of the new chaotic system (1) are displayed in Figure 2

CIRCUIT IMPLEMENTATION OF THE NEW CHAOTIC SYSTEM
In order to prove the chaotic behaviors of system (1), a simulation circuit is constructed in this study, which is shown in Figure 3. The new chaotic system (1) can be implemented by the resistance, capacitance, operational amplifier and analog multiplier. Here the variables x, y, z of new chaotic system (1) are the voltages across the capacitor C 1 , C 2 and C 3 , respectively.
The Multisim simulation oscilloscope outputs (phase portraits) of circuitry of the re-scaled new chaotic system, for parameters (a, b, c, d) = (15, 0.02, 6, 6) are seen in

CONCLUSION
Discovering chaotic systems with infinite number of equilibrium points such as curve equilibrium is an active topic of research in the chaos literature. In this work, we reported a new chaotic system with a pearshaped equilibrium curve. We also showed an electronic circuit simulation of the new chaotic system to check the feasibility of the chaotic system model.