ON THE CHAOTIC NATURE OF A CAPUTO FRACTIONAL MATHEMATICAL MODEL OF CANCER AND ITS CROSSOVER BEHAVIORS

Abstract

In this paper, a three-dimensional mathematical model of cancer, incorporating tumor cells, host normal cells, and effector immune cells, is examined. The chaotic nature of this cancer model is explored within the framework of Caputo fractional calculus. The analysis employs local stability assessment using the Matignon criteria, Lyapunov exponents (LEs) for various fractional orders of derivatives, bifurcation plots reflecting changes in the fractional derivative order, simulation outcomes of a novel chaotic attractor, Kaplan–Yorke dimension, and sensitivity to initial conditions. The results demonstrate that the Caputo fractional cancer model exhibits chaotic behavior for selected parameters. Moreover, the fractional derivative orders significantly influence the extent of chaotic behavior. Specifically, as the fractional derivative order increases from zero to one, the intensity of chaos diminishes, evidenced by a decrease in both the LE and the Kaplan–Yorke dimension. A piecewise modeling approach is applied to the cancer model, incorporating deterministic, stochastic, and power-law behaviors. Three distinct scenarios are developed within this framework, revealing several novel crossover behaviors through simulation. Notably, it is observed that when cancer dynamics initially follow a power-law behavior and subsequently transition into a stochastic process, the disease dynamics can stabilize, suggesting a positive outlook for disease control. Conversely, if the cancer dynamics begin with a deterministic process and then shift to a stochastic process, the system may enter a chaotic state, complicating disease management efforts.