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Top1. Introduction
By chaos and chaotic systems, we generally refer to dynamical systems whose behavior is highly sensitive to initial conditions. That is, small changes in the initial conditions of such a system will lead to completely different trajectories in the solution space, a phenomenon that is widely referred to as the butterfly effect, see for example Alligood et al. (1996). Thus, despite such systems being deterministic in nature with no stochastic parameters, their complex nature constitutes their behavior hard to predict and control. This happens due to the presence of nonlinear terms in the differential equations that describe the system,
Chaotic systems were introduced in the seminal work of Lorenz (1963), that introduced a three dimensional weather model that described forced dissipative hydrodynamic flow. Since then, chaos theory as a discipline has vastly expanded and chaotic systems have been extensively studied over the last 40 years. The advancements in this area have been greatly influenced by the advancement of technology and personal computers, which makes it easier to follow the trajectories of chaotic systems for a longer time period and with a higher precision. Thus, the theoretical analysis of chaos has been assisted by the numerical simulations of such chaotic systems, made possible by the use of computers.
Out of the many chaotic systems, the ones that seem to be the most interesting are those that exhibit an attracting behavior. By that, we refer to systems whose trajectories converge with time to a set of values, called an attractor. This can be a single point, a curve, a surface or a manifold. For example, in a system with a spherical bowl and a rolling marble, the bottom of the bowl is called a fixed point attractor. The attractor is called strange if it is a set with fractal structure, see Grassberger and Procaccia (2004); Ott (1981). Examples include the aforementioned Lorenz system, as well as the Hénon, Lozi and Tinkerbell maps that will be introduced in the next sections. It should also be noted that a system may have different attractors depending on the choice of initial conditions.
Chaotic systems have found their place in applications spanning a wide variety of natural and artificial systems. These include oscillators (Heagy, 1992; Kengne et al. 2012; Sharma et al. 2012), secure communications (Murali & Lakshmanan, 1998; Zaher, & Abu-Rezq, 2011), controller design (dos Santos Coelho & Mariani, 2012), lasers (Li et al. 2014; Yuan et al. 2014), chemical reactions (Gaspard, 1999), pseudo-random number generation (Stoyanov & Kordov, 2015), optical ring phase resonators (Aboites et al. 2009), biology (Das et al. 2014; Kyriazis, 1991) medicine (Qu, 2011, Witte & Witte, 1991), finance (Guegan, 2009) and more, see also Azar & Vaidyanathan (2014, 2015a, 2015b, 2016); Azar & Zhu (2015); Elhadj (2013); Zhu & Azar (2015) and the references therein.
A major task in the control of chaotic systems, is the one of finding a feedback control of the general form u(t)=k(x(t)), where u the input, x the state of the system and k the feedback law, in order to stabilize the system around its unstable equilibrium. Techniques include active control, adaptive control, sliding mode control, fuzzy logic control and more, see Azar & Vaidyanathan, (2014, 2015a, 2015b, 2016); Azar & Zhu, (2015); Vaidyanathan & Azar (2016a), Zhu & Azar, (2015) and the references therein.