We study a simple map as a minimal model of excitable cells. The map has two fast variables which mimic the behavior of class I neurons, undergoing a sub-critical Hopf bifurcation. Adding a third slow variable allows the system to present bursts and other interesting biological behaviors. Bifurcation lines which locate the excitability region are obtained for different planes in parameter space.

In this work, a bidimensional continuous-time differential equations system is analyzed which is derived of Leslie-type predator-prey schemes by considering a nonmonotonic functional response and Allee effect on population prey. For the system obtained we describe the bifurcation diagram of limit cycles that appears in the first quadrant, the only quadrant of interest for the sake of realism. W...

Computational investigations have been performed into the behavior of an incompressible fluid flow in the vicinity of a plane symmetric channel contraction. Our aim is to determine the critical Reynolds number, above which the flow becomes asymmetric with respect to the channel geometry using the bifurcation diagram. Three channels, which are characterized by the contraction ratio, are studied ...

This paper discusses feed-forward chains near points of synchrony-breaking Hopf bifurcation. We show that at synchrony-breaking bifurcations the center manifold inherits a feed-forward structure and use this structure to provide a simplified proof of the theorem of Elmhirst and Golubitsky that there is a branch of periodic solutions in such bifurcations whose amplitudes grow at the rate of λ 1 ...

We develop an algorithm to identify invariant curves efficiently. In dynamical systems, it helps do continuations of branches of quasi-periodic solutions smoothly in bifurcation diagrams. Several examples are demonstrated to provide the numerical evidence for the versatility of the algorithm. © 2001 Elsevier Science B.V. All rights reserved.

Perturbing an integrable 3 degree of freedom (d.o.f.) Hamiltonian system containing a normally parabolic 2-torus which is m-resonant (m = 1 or 2) creates a parabolic m-resonance (m-PR). PRs of different types are either persistent or of low co-dimension, hence they appear robustly in many applications. Energy–momenta bifurcation diagram is constructed as a tool for studying the global structure...

The equations for geodesic flow on the ellipsoid are well known, and were first solved by Jacobi in 1838 by separating the variables of the Hamilton-Jacobi equation. In 1979 Moser investigated the case of the general ellipsoid with distinct semi-axes and described a set of integrals which weren’t know classically. After reviewing the properties of geodesic flow on the three dimensional ellipsoi...

An important challenge in theoretical ecology is to find good-coarse grained representations of complex food webs. Here we use the approach of generalized modeling to show that it may be possible to formulate a coarse-graining algorithm that conserves the local dynamics of the model exactly. We show examples of food webs with a different number of species that have exactly identical local bifur...

We study the local Hopf bifurcations of codimension one and two which occur in the Shimizu-Morioka system. This system is a simplified model proposed for studying the dynamics of the well known Lorenz system for large Rayleigh numbers. We present an analytic study and their bifurcation diagrams of these kinds of Hopf bifurcation, showing the qualitative changes in the dynamics of its solutions ...

We consider the following semilinear elliptic equation:    −∆u = λeup in B1, u = 0 on ∂B1, (0.1) where B1 is the unit ball in R, d ≥ 3, λ > 0 and p > 0. First, following Merle and Peletier [13], we show that there exists a unique eigenvalue λp,∞ such that (0.1) has a solution (λp,∞,Wp) satisfying lim|x|→0 Wp(x) = ∞. Secondly, we study a bifurcation diagram of regular solutions to (0.1). It f...