This chapter announces a new four-dimensional hyperchaotic system having two positive Lyapunov exponents, a zero Lyapunov exponent, and a negative Lyapunov exponent. Since the sum of the Lyapunov exponents of the new hyperchaotic system is shown to be negative, it is a dissipative system. The phase portraits of the new hyperchaotic system are displayed with both two-dimensional and three-dimensional phase portraits. Next, the qualitative properties of the new hyperchaotic system are dealt with in detail. It is shown that the new hyperchaotic system has three unstable equilibrium points. Explicitly, it is shown that the equilibrium at the origin is a saddle-point, while the other two equilibrium points are saddle-focus equilibrium points. Thus, it is shown that all three equilibrium points of the new hyperchaotic system are unstable. Numerical simulations with MATLAB have been shown to validate and demonstrate all the new results derived in this chapter. Finally, a circuit design of the new hyperchaotic system is implemented in MultiSim to validate the theoretical model.
Top1. Introduction
The first experimental chaotic system was discovered by the American meteorologist E.N. Lorenz when he was studying weather patterns with a 3-D model (Lorenz, 1963). Characteristics of a chaotic system are: (1) dynamical instability, (2) topological mixing, and (3) dense periodic orbits (Hasselblatt & Katok, 2003). Dynamical instability is also known as the “butterfly effect”, which means that a small change in initial conditions of system can create significant differences. This characteristic makes the chaotic system highly sensitive to initial conditions (Strogatz, 1994). Topologically mixing for a chaotic system refers to stretching and folding of the phase space, which means that the chaotic trajectory at the phase space will evolve in time so that each given area of this trajectory will eventually cover part of any particular region. Having dense periodic orbits means that the trajectory of a chaotic system can come arbitrarily close every possible asymptotic state. Hence, the future behavior of a chaotic system is quite complex and also unpredictable.
Chaos theory has several applications in science and engineering (Strogatz, 1994; Azar & Vaidyanathan, 2015a,b,c, 2016; Vaidyanathan & Volos, 2016; Azar et al., 2017a,b,c; Boulkroune et al, 2016a,b; Vaidyanathan et al, 2015a,b,c; Wang et al., 2017; Soliman et al., 2017; Tolba et al., 2017; Grassi et al., 2017; Ouannas et al., 2016a,b, 2017a,b,c,d,e,f,g,h,I,j; Singh et al., 2017; Vaidyanathan & Azar, 2015a,b,c,d, 2016a,b,c,d,e,f,g, 2017a,b,c). Some important applications of chaotic systems can be described as lasers (Donati & Hwang, 2012; Yuan, Zhang & Wang, 2013; Fen, 2017), neural networks (Potapov & Ali, 2000; Yang & Yuan, 2005; Kaslik & Sivasundaram, 2012; Akhmet & Fen, 2014; Przystalka & Moczulski, 2015; Vaidyanathan, 2015r; Fen & Fen, 2017; Bouallegue, 2017), memristors (Raj & Vaidyanathan, 2017; Pham et al., 2017; Vaidyanathan & Volos, 2017; Vaidyanathan, 2017b), chemical reactors (Vaidyanathan, 2015a,b,d,f,l), dynamo systems (Vaidyanathan, 2015c,e,p), fuzzy logic models (Yau & Shieh, 2008; Pai, Yau & Kuo, 2010; Li, 2011), biological systems (Vaidyanathan 2015g,j,k,n, 2017j), convection systems (Vaidyanathan, 2015i,o), DC motors (Rajagopal et al., 2017), mechanical systems (Vaidyanathan, 2017a,c,d,e,i), Tokamak systems (Vaidyanathan, 2015q), cardiology (Witte & Witte, 1991; Bozoki, 1997), oscillators (Vaidyanathan, 2015m, 2016a, 2016b, 2016c, 2016d, 2016e, 2016f, 2016g, 2017f, 2017g, 2017h; Vaidyanathan & Volos, 2015; Vaidyanathan & Boulkroune, 2016; Vaidyanathan & Pakiriswamy, 2016; Vaidyanathan & Rajagopal, 2016, 2017), voice encryption (Vaidyanathan et al., 2017), etc.