## 2 Preliminaries  
### Chaotic Maps  
The logistic map is a mathematical function classified as a quadratic map, renowned for its capacity to generate intricate chaotic behavior. By manipulating a single control parameter, the logistic map showcases a diverse range of dynamic phenomena, encompassing periodicity, bifurcations, and chaotic regimes. [PERSON] and [PERSON] [12] extensively elaborate on these behaviors, providing a comprehensive explanation. In particular, the bifurcations of the logistic map become increasingly pronounced when the control parameter surpasses a critical value of 3.5. The logistic map is formally defined by Equation (1), which is a recursive equation that describes the evolution of a population over time. \[x_{n+1}=rx_{n}(1-x_{n}) \tag{1}\]  
The control parameter \(r>0\) is called the \"biotic potential\", while \(x\) is the initial condition parameter. The dynamics of the logistic map are intricately tied to the value of the parameter \(r\), giving rise to three distinct phenomena: when \(r>1\), the system gradually converges to the stable fixed point at 0; for values of \(1\leq r\leq 3\), the system undergoes a loss of stability at the previous fixed point and a new fixed point emerges at \(x=\frac{1}{r}\); as \(r\) exceeds 3.5, an intriguing phenomenon called bifurcation unfolds, revealing a complex and fascinating diagram. The Henon map is a two-dimensional nonlinear map that exhibits chaotic behavior. It was introduced by [PERSON] [19] in 1976 as a simplified model of the Poincare section of a dynamical system inspired by the Lorenz map. The Henon map comprises two equations that are interrelated and describe the changes in the state variables. Thesecond equation represents a linear relationship with respect to the first state variable, whereas the first equation represents a quadratic relationship involving both state variables. It is demonstrated in Equation (2), which provides a clear depiction of the Henon map's characteristics. The Henon map is known for its sensitivity to initial conditions, which is a hallmark of chaotic systems. Despite its simplicity, the Henon map has been applied to a wide range of fields, including physics, biology, and cryptography, modeling dynamical systems that exhibit chaotic behavior, such as fluid dynamics and celestial mechanics. \[ \begin{array}{c}x_{n+1}=1-{ax_{n}}^{2}+by_{n}\\ y_{n+1}=x_{n}\end{array}  \tag{2}\]  
The equation can also be written as defined by Equation (3):  
\[x_{n+1}=1-{ax_{n}}^{2}+bx_{n-1} \tag{3}\]  
The parameters \(a\) and \(b\) are crucial to the behavior of the map, and by default, the value of parameter a is set to 1.4, while parameter b is set to 0.3. These two parameters play a vital role in determining the behavior of the map and can be adjusted to achieve different outcomes. According to the findings of [PERSON] and [PERSON] [PERSON] and [PERSON] (2018), one of the defining characteristics of the Henon map is the existence of a strange attractor, which is a fractal set that governs the long-term behavior of the system. The shape of the strange attractor is highly dependent on the parameters of the Henon map, with different parameter values leading to distinct attractors [PERSON] and [PERSON] (2018). It has also been used in cryptography as a basis for chaos-based encryption schemes. One of the objectives of this proposed study is to integrate the Henon map and the fuzzy mathematics concept to create a highly chaotic system for image encryption called the fuzzy Henon map. The map exhibits complex behavior that can be utilized for secure image encryption. By utilizing the properties of both concepts, the proposed encryption scheme aims to provide a high level of security for image data. According to [PERSON] et al. [PERSON] et al. (2018), the sine map is a one-dimensional, discrete-time dynamical system that belongs to the class of chaotic maps. The sine map function described in Equation (4) is a fundamental yet powerful tool that maps real numbers onto themselves.