Arnold's CAT Map (ACM) is a chaotic transformation of the 2-dimensional toral automorphism T 2 defined by the mapping Γ ∶ T 2 → T 2 . There are many applications of ACM in various research areas such as: steganography, encryption of images, texts and watermarks. The transformation of an image is achieved by the randomized order of pixels. After a finite number of repetitions of the transformation, the original image reappears. In this study, encryption of two images is demonstrated together with a proposed algorithm. Moreover, the periodicity of ACM is discussed and an algorithm to change the period of ACM is suggested. The resultant period obtained from the new algorithm is compared with the period obtained from the usual ACM. The results show that the period of the proposed algorithm grows exponentially while the period of ACM has an upper bound.

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In this study, the first order nonlinear Volterra type integro-differential equations are used in order to identify approximate solutions concerning Euler polynomials of a matrix method based on collocation points. This method converts the mentioned nonlinear integro-differential equation into the matrix equation with the utilization of Euler polynomials along with collocation points. The matrix equation is a system of nonlinear algebraic equations with the unknown Euler coefficients. Additionally, this approach provides analytic solutions, if the exact solutions are polynomials. Furthermore, some illustrative examples are presented with the aid of an error estimation by using the Mean-Value Theorem and residual functions. The obtained results show that the developed method is efficient and simple enough to be applied. And also, convergence of the solutions of the problems were examined. In order to obtain the matrix equations and solutions for the selected problems, code was developed in MATLAB.

In order to solve mth-order linear differential difference equations with variable coefficients under mixed conditions, this research suggests a combined operational matrix approach based on Lucas polynomials. The simplicity of the proposed method’s application is a benefit. The technique simplifies the provided problem by turning it into a matrix equation. Absolute errors are used to validate illustrative examples. The solutions are enhanced by residual error estimates. The results, which are displayed in graphs and tables, are contrasted with the accepted approaches in the literature.

In this study, linear Volterra-Fredholm integral equations are approximatively solved in terms of Lucas polynomials about any point in this study using a practical matrix approach. This technique uses collocation points and Lucas polynomials to transform the aforementioned linear Volterra-Fredholm integral problem into a matrix equation. Lucas coefficients are unknown in the system of linear algebraic equations. With the use of an error estimation, some illustrated examples are also provided. The outcomes demonstrate how effective and practical the suggested methodology is. Code was created in MATLAB to acquire the matrix equations and answers for the chosen issues.

In this study, the dynamical behaviors of a prey–predator model with multiple strong Allee effect are investigated. The fixed points of the model are examined for existence and topological classification. By selecting as the bifurcation parameter $\beta$, it is demonstrated that the model can experience a Neimark-Sacker bifurcation at the unique positive fixed point. Bifurcation theory is used to present the Neimark-Sacker bifurcation conditions of existence and the direction of the bifurcation. Additionally, some numerical simulations are provided to back up the analytical result. Following that, the model's bifurcation diagram and the triangle-shaped stability zone are provided.

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In this paper, a useful matrix approach for high-order linear Fredholm integro-differential equations with initial boundary conditions expressed as Lucas polynomials is proposed. Through the use of a matrix equation which is equivalent to a set of linear algebraic equations—the method transforms the integro differential equation. When compared to other methods that have been proposed in the literature, the numerical results from the suggested technique reveal that it is effective and promising, and error estimation of the scheme was derived. These results were compared with the exact solutions to the tested issues.