One of the eye movements is the saccade, which has led to the introduction of the saccadic model. This study is based on the part of the saccadic model, which means the burst neurons and resettable integrator model. The principal limitation of the original model is the lack of differentiability at the equilibrium point. By using the Naka–Rushton function, we introduce a new model in place of the original one, so that the equilibrium point of the system becomes a differentiable point in the modified model. Our focus in this work is to investigate the fundamental properties of the discrete model of our novel system. We apply the forward Euler method to transform the new model to a discrete model. With the utilization of the center manifold theory, we describe some of its dynamical features, such as stability, instability, and bifurcation at a fixed point. Finally, both analytical and numerical simulations are used to continue investigating the period-doubling bifurcation according to the numerical parameters in the saccadic model.
Abstract:In this paper we consider the metric entropies of the maps of an iterated function system deduced from a black hole which are known the Bekenstein-Hawking entropies and its subleading corrections. More precisely, we consider the recent model of a Bohr-like black hole that has been recently analysed in some papers in the literature, obtaining the intriguing result that the metric entropies of a black hole are created by the metric entropies of the functions, created by the black hole principal quantum numbers, i.e., by the black hole quantum levels. We present a new type of topological entropy for general iterated function systems based on a new kind of the inverse of covers. Then the notion of metric entropy for an Iterated Function System (IFS) is considered, and we prove that these definitions for topological entropy of IFS's are equivalent. It is shown that this kind of topological entropy keeps some properties which are hold by the classic definition of topological entropy for a continuous map. We also consider average entropy as another type of topological entropy for an IFS which is based on the topological entropies of its elements and it is also an invariant object under topological conjugacy. The relation between Axiom A and the average entropy is investigated.
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