# A Simple Technique for Studying Chaos Using Jerk Equation with Discrete Time Sine Map
## 1 Introduction  
Chaos theory focuses on the fact that a complex system's solutions are not really because of the noise or the irregularities that appear, instead because of the interaction of the system components. Chaos can be loosely defined as a fascinating intermediate state between randomness and steady behavior, and it has been a subject of growing research over the last decade [1]. Chaotic systems are characterized by their sensitive dependency on the initial conditions, which leads to deterministic aperiodic bounded behavior impossible to predict in the long term [2]. These characteristics find their application in various practical systems such as biology, chemistry, secure communications [3], bits generators, cryptography [4], mechanical engineering [5, 6], machine drives [7], and ecology. Being able to mathematically describe a system that negates the scientific viewpoint about the practical systems and can depict chaos is an accomplishment of science [8]. A continuous chaotic system defined in terms of differential equations usually surpass the discrete chaotic system defined in terms of difference equations because of the added complexity and chaotic dynamics. Because of the growing demands of finding a more generalized chaotic system, researchers are comparing the merits and demerits of these systems. Few attempts are made to combine both these classes of system ideas and develop a better solution in terms of easy implementation using simple Op-Amps [9]. The qualitative nature of the solutions of a dynamical system described by a simple ordinary differential equation (ODE) depends on the dynamical system's degree. As the degree increases, the solutions start to arrange themselves on a strange attractor in phase space [10]. The minimum degree required to get a strange attractor is three, as studied in [11]. Although a simple jerk equation with one nonlinear function provides an appropriate framework for visualizing chaotic behavior, the necessary conditions for the onset of chaos in an autonomous dynamical system of ODEs are still unknown [12]. Despite the fact that non-autonomous and autonomous systems have become increasingly straightforward over the last three decades, they have remained predominant models for studying different bifurcation diagrams and aperiodic deterministic behavior in nonlinear dynamics [12; 13]. Many simple nonlinear electronic circuits capable of showing this erratic behavior have been proposed in the literature [14; 15; 16; 17]. These systems can be described in terms of ordinary differential (or difference) equations; examples of such systems are the Colpitts oscillator [18; 19], Chua's circuit [20; 21], Jerk circuit [22; 23], and many more. These circuits use one or more nonlinear elements with an active component for exhibiting chaos while allowing one or more control parameters to control this behavior's intensity. The most prominent of the autonomous chaotic systems is Chua's circuit and Jerk circuit. Initially, the most popular way of generating chaos was to enforce a nonlinear function in the system capable of creating different equilibrium points by varying one of the non-linearity parameters introduced in the system. This method resulted in many practically unrealistic dynamical systems. To ensure the realistic analog/digital implementation of the chaotic circuit, [PERSON] et al. proposed a chaotic circuit where a digitally controlled MOS-transistor acted as a source of non-linearity and was capable of generating multi-scroll attractors [24]. In 2011, [PERSON] et al. replaced the nonlinear part of Colpitts oscillator with a more practical triangular function and validated their proposed circuit numerically and experimentally [25]. In [26], the use of staircase nonlinear term for chaos generation was studied by [PERSON] et al. [PERSON] et al. proposed a saturated nonlinear function series (SNFSs) and modified the generic Chua's circuit to form a chaotic circuit for producing chaotic oscillations at high frequency [27; 28].