Numerical investigation of a fractional model of a tumor-immune surveillance via Caputo operator

Fractional calculus has become a potent tool for simulating the complexity of interactions in tumor-immune system dynamics. This paper investigates the existence and uniqueness of its approximation solutions of a fractional model of tumor-immune surveillance. This analysis is essential for proving the validity and dependability of the model and provides a more in-depth understanding of the dynamics of the tumor-immune surveillance system. Second, we examine the fractional model's numerical features. We utilize a numerical technique called the Laplace residual power series method to address the equations' complexity and nonlinearity. By defining the answers as a fractional power series, this method enables us to efficiently approximate the solutions. The use of this technique enables us to investigate the temporal evolution of the tumor-immune system, offering important insights into the stability and behavior of the system. We assess the effectiveness of the Laplace residual power series method in locating approximations through a series of thorough numerical simulations. To ensure the correctness and dependability of our method, we compare the numerical findings whenever possible with well-known analytical solutions

Special Fractional-Order Map and Its Realization

Recent works have focused the analysis of chaotic phenomena in fractional discrete memristor. However, most of the papers have been related to simulated results on the system dynamics rather than on their hardware implementations. This work reports the implementation of a new chaotic fractional memristor map with “hidden attractors”. The fractional memristor map is developed based on a memristive map by using the Grunwald–Letnikov difference operator. The fractional memristor map has flexible fixed points depending on a system’s parameters. We study system dynamics for different values of the fractional orders by using bifurcation diagrams, phase portraits, Lyapunov exponents, and the 0–1 test. We see that the fractional map generates rich dynamical behavior, including coexisting hidden dynamics and initial offset boosting