Stability and Cycles in a Cobweb Model with Heterogeneous Expectations

We investigate the dynamics of a cobweb model with heterogeneous beliefs, generalizing the example of Brock and Hommes (1997). We examine situations where the agents form expectations by using either rational expectations, or a type of adaptive expectations with limited memory defined from the last two prices. We specify conditions that generate cycles. These conditions depend on a set of factors that includes the intensity of switching between beliefs and the adaption parameter. We show that both Flip bifurcation and Neimark-Sacker bifurcation can occur as primary bifurcation when the steady state is unstable.Bounded rationality, Cobweb model, Flip bifurcation, Neimark-Sacker bifurcation.

Period-doubling bifurcation and Neimark-Sacker bifurcation of a discrete predator-prey model with Allee effect and cannibalism

In this paper, a discrete predator-prey model incorporating Allee effect and cannibalism is derived from its continuous version by semidiscretization method. Not only the existence and local stability of fixed points of the discret system are investigated, but more important, the sufficient conditions for the occurrence of its period-doubling bifurcation and Neimark-Sacker bifurcation are obtained using the center manifold theorem and local bifurcation theory. Finally some numerical simulations are given to illustrate the existence of Neimark-Sacker bifurcation. The outcome of the study reveals that this discrete system undergoes various bifurcations including period-doubling bifurcation and Neimark-Sacker bifurcation

Coexisting patterns of population oscillations: the degenerate Neimark Sacker bifurcation as a generic mechanism

We investigate a population dynamics model that exhibits a Neimark Sacker bifurcation with a period that is naturally close to 4. Beyond the bifurcation, the period becomes soon locked at 4 due to a strong resonance, and a second attractor of period 2 emerges, which coexists with the first attractor over a considerable parameter range. A linear stability analysis and a numerical investigation of the second attractor reveal that the bifurcations producing the second attractor occur naturally in this type of system.Comment: 8 pages, 3 figure

Neimark-Sacker Bifurcation in a Discrete-Time Financial System

A discrete-time financial system is proposed by using forward Euler scheme. Based on explicit Neimark-Sacker bifurcation (also called Hopf bifurcation for map) criterion, normal form method and center manifold theory, the system's existence, stability and direction of Neimark-Sacker bifurcation are studied. Numerical simulations are employed to validate the main results of this work. Some comparison of bifurcation between the discrete-time financial system and its continuous-time system is given