Fractional Dynamics and Recurrence Analysis in Cancer Model

In this work, we analyze the effects of fractional derivatives in the chaotic dynamics of a cancer model. We begin by studying the dynamics of a standard model, i.e. with integer derivatives. We study the dynamical behavior by means of the bifurcation diagram, Lyapunov exponents, and recurrence quantification analysis (RQA), such as the recurrence rate (RR), the determinism (DET), and the recurrence time entropy (RTE). We find a high correlation coefficient between the Lyapunov exponents and RTE. Our simulations suggest that the tumor growth parameter (rho(1)) is associated with a chaotic regime. Our results suggest a high correlation between the largest Lyapunov exponents and RTE. After understanding the dynamics of the model in the standard formulation, we extend our results by considering fractional operators. We fix the parameters in the chaotic regime and investigate the effects of the fractional order. We demonstrate how fractional dynamics can be properly characterized using RQA measures, which offer the advantage of not requiring knowledge of the fractional Jacobian matrix. We find that the chaotic motion is suppressed as alpha decreases, and the system becomes periodic for alpha (sic) 0.9966. We observe limit cycles for alpha is an element of(0.9966, 0.899) and fixed points for alpha < 0.899. The fixed point is determined analytically for the considered parameters. Finally, we discover that these dynamics are separated by an exponential relationship between alpha and rho(1). Also, the transition depends on a supper transient which obeys the same relationship.