A New Six-Term 3D Unified Chaotic System
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RESEARCH PAPER
A New Six-Term 3D Unified Chaotic System Engin Can1
•
Ug˘ur Erkin Kocamaz2 • Yılmaz Uyarog˘lu3
Received: 8 October 2018 / Accepted: 1 February 2020 Shiraz University 2020
Abstract In this study, four different 3D five-term chaotic flows are unified and a novel six-term 3D unified chaotic system with three nonlinearities is introduced. Firstly, the theoretical system via an electronic circuit is realized, and then the basic dynamical properties of the proposed unified chaotic system are numerically and analytically analyzed, i.e., sensitivity to initial conditions, equilibrium points, eigenvalues, Kaplan–Yorke dimensions, dissipativity, Lyapunov exponents and bifurcation diagrams. Investigation results clearly present that this is a new unified chaotic system and earns further detailed disquisition. Keywords Sprott B chaotic flow Sprott C chaotic flow van der Schrier–Mass chaotic system Munmuangsaen– Srisuchinwong chaotic system Unified chaotic system
1 Introduction Chaos theory is used to explain the apparently complex behaviors of simple, linear and well-behaved mathematically defined nonlinear systems. The essential characteristics of chaotic systems are sensitive dependence on initial values and have infinite number of different periodic responses (Miladi et al. 2015). Nonlinear dynamical systems including simple mathematical equations can exhibit chaos that has rich signal trajectories. Because of extreme sensitivity to initial conditions, the chaotic behaviors occur over long-term unpredictability (Stollenwerk et al. 2015). & Engin Can [email protected] Ug˘ur Erkin Kocamaz [email protected] Yılmaz Uyarog˘lu [email protected] 1
Department of Basic Sciences of Engineering, Faculty of Technology, Sakarya University of Applied Sciences, 54187 Serdivan, Sakarya, Turkey
2
Department of Computer Technologies, Vocational School of Karacabey, Bursa Uludag˘ University, 16700 Karacabey, Bursa, Turkey
3
Department of Electrical and Electronics Engineering, Faculty of Engineering, Sakarya University, 54187 Serdivan, Sakarya, Turkey
Therefore, chaos is defined as the sufficient conditions of unstable behavior in deterministic dynamical systems. The researchers have discovered many chaotic systems after the first chaotic attractor was presented by Lorenz (1963). The differential equations of the Lorenz system come from a simplified model of atmospheric convection, and it shows two-scroll graphics. A simple continuous-time three-dimensional chaotic system, which constitutes chemical reaction, was pointed out by Ro¨ssler (1976) and a four-dimensional hyperchaotic system by Ro¨ssler (1979). A double-scroll attractor was determined from an electronic circuit named Chua’s circuit by Matsumoto (1984). Sprott (1994) explored 19 simpler third-order chaotic flows, which have either two nonlinearities with five terms (Sprott A–E) or one nonlinearity with six terms (Sprott F– S). Chen and Ueta (1999) proposed a novel three-dimensional chaotic attractor, which is called as Chen chaotic syste
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