Multiple bifurcations in an early brain tumor model with piecewise constant arguments

Abstract

In this paper, a differential equation with piecewise constant arguments modeling an early brain tumor growth is considered. The discretization process in the interval t ∈ [n, n+1) leads to two-dimensional discrete dynamical system. By using the Schur–Cohn criterion, stability conditions of the positive equilibrium point of the system are obtained. Choosing appropriate bifurcation parameter, the existence of Neimark–Sacker and flip bifurcations is verified. In addition, the direction and stability of the Neimark–Sacker and flip bifurcations are determined by using the normal form and center manifold theory. Finally, the Lyapunov exponents are numerically computed to characterize the complexity of the dynamical behaviors of the system