Bifurcation and chaotic behavior of a discrete-time Ricardo–Malthus model

Abstract

The dynamics of a discrete-time Ricardo–Malthus model obtained by numerical discretization is investigated, where the step size δ is regarded as a bifurcation parameter. It is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation in the interior of \(R^{2}_{+}\) by using the theory of center manifold and normal form. Numerical simulations are presented not only to illustrate our theoretical results, but also to exhibit the system’s complex dynamical behavior, such as the cascade of period-doubling bifurcation in orbits of period 2, 4, 8 16, period-11, 22, 28 orbits, quasiperiodic orbits, and the chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors.