Search results for "bifurcation"

Pattern formation and transition to chaos in a chemotaxis model of acute inflammation

2021

We investigate a reaction-diffusion-chemotaxis system that describes the immune response during an inflammatory attack. The model is a modification of the system proposed in Penner, Ermentrout, and Swigon [SIAM J. Appl. Dyn. Syst., 11 (2012), pp. 629-660]. We introduce a logistic term in the immune cell dynamics to reproduce the macrophages' activation, allowing us to describe the disease evolution from the early stages to the acute phase. We focus on the appearance of pattern solutions and their stability. We discover steady-state (Turing) and wave instabilities and classify the bifurcations deriving the corresponding amplitude equations. We study stationary radially symmetric solutions an…

Bifurcations of cuspidal loops

1997

A cuspidal loop for a planar vector field X consists of a homoclinic orbit through a singular point p, at which X has a nilpotent cusp. This is the simplest non-elementary singular cycle (or graphic) in the sense that its singularities are not elementary (i.e. hyperbolic or semihyperbolic). Cuspidal loops appear persistently in three-parameter families of planar vector fields. The bifurcation diagrams of unfoldings of cuspidal loops are studied here under mild genericity hypotheses: the singular point p is of Bogdanov - Takens type and the derivative of the first return map along the orbit is different from 1. An analytic and geometric method based on the blowing up for unfoldings is propos…

Adaptive control of a seven mode truncation of the Kolmogorov flow with drag

2009

Abstract We study a seven dimensional nonlinear dynamical system obtained by a truncation of the Navier–Stokes equations for a two dimensional incompressible fluid with the addition of a linear term modelling the drag friction. We show the bifurcation sequence leading from laminar steady states to chaotic solutions with increasing Reynolds number. Finally, we design an adaptive control which drives the state of the system to the equilibrium point representing the stationary solution.

On the time function of the Dulac map for families of meromorphic vector fields

2003

Given an analytic family of vector fields in Bbb R2 having a saddle point, we study the asymptotic development of the time function along the union of the two separatrices. We obtain a result (depending uniformly on the parameters) which we apply to investigate the bifurcation of critical periods of quadratic centres.

On the construction of lusternik-schnirelmann critical values with application to bifurcation problems

1987

An iterative method to construct Lusternik-Schnirelmann critical values is presented. Examples of its use to obtain numerical solutions to nonlinear eigenvalue problems and their bifurcation branches are given

Branches of index-preserving solutions to systems of second order ODEs

2009

We investigate the existence of a continuum of index-preserving solutions to a Dirichlet problem associated with a parameter-dependent system of second order ordinary differential equations, developing a detailed analysis on the behaviour of the branches of nontrivial solutions. Our approach is based on the Rabinowitz global bifurcation Theorem combined with the notion of index and nullity of suitable linear boundary value problems. An application of the result to the study of branches of odd, periodic solutions for suitable systems of two linearly coupled pendulums of lenghts variables is also analyzed.

Generic unfoldings with the same bifurcation diagram which are not (C0, C0)— equivalent

1997

Up and Down States During Slow Oscillations in Slow-Wave Sleep and Different Levels of Anesthesia

2021

Slow oscillations are a pattern of synchronized network activity generated by the cerebral cortex. They consist of Up and Down states, which are periods of activity interspersed with periods of silence, respectively. However, even when this is a unique dynamic regime of transitions between Up and Down states, this pattern is not constant: there is a range of oscillatory frequencies (0.1–4 Hz), and the duration of Up vs. Down states during the cycles is variable. This opens many questions. Is there a constant relationship between the duration of Up and Down states? How much do they vary across conditions and oscillatory frequencies? Are there different sub regimes within the slow oscillation…

Application of a non linear local analysis method for the problem of mixed convection instability

2007

Abstract We consider the problem of laminar mixed convection flow between parallel, vertical and uniformly heated plates where the governing dimensionless parameters are the Prandtl, Rayleigh and Reynolds numbers. Using the method based on the centre manifold theorem which was derived from the general theory of dynamical systems, we reduce a three-dimensional simplified model of ordinary differential amplitude equations emanating from the original Navier-Stokes system of the problem in the vicinity of a trivial stationary solution. We have found that when the forcing parameter, the Rayleigh number, increases beyond the critical value Ra s , the stationary solution is a pitchfork bifurcation…

Analytical properties of horizontal visibility graphs in the Feigenbaum scenario

2012

Time series are proficiently converted into graphs via the horizontal visibility (HV) algorithm, which prompts interest in its capability for capturing the nature of different classes of series in a network context. We have recently shown [1] that dynamical systems can be studied from a novel perspective via the use of this method. Specifically, the period-doubling and band-splitting attractor cascades that characterize unimodal maps transform into families of graphs that turn out to be independent of map nonlinearity or other particulars. Here we provide an in depth description of the HV treatment of the Feigenbaum scenario, together with analytical derivations that relate to the degree di…