Search results for "Bifurcation theory"
showing 10 items of 25 documents
Dynamical Features of the MAP Kinase Cascade
2017
The MAP kinase cascade is an important signal transduction system in molecular biology for which a lot of mathematical modelling has been done. This paper surveys what has been proved mathematically about the qualitative properties of solutions of the ordinary differential equations arising as models for this biological system. It focuses, in particular, on the issues of multistability and the existence of sustained oscillations. It also gives a concise introduction to the mathematical techniques used in this context, bifurcation theory and geometric singular perturbation theory, as they relate to these specific examples. In addition further directions are presented in which the application…
On the number of solutions of a Duffing equation
1991
The exact number of solutions of a Duffing equation with small forcing term and homogeneous Neumann boundary conditions is given. Several bifurcation diagrams are shown.
A strain-difference-based nonlocal elasticity model
2004
Abstract A two-component local/nonlocal constitutive model for (macroscopically) inhomogeneous linear elastic materials (but constant internal length) is proposed, in which the stress is the sum of the local stress and a nonlocal-type stress expressed in terms of the strain difference field, hence identically vanishing in the case of uniform strain. Attention is focused upon the particular case of piecewise homogeneous material. The proposed model is thermodynamically consistent with a suitable free energy potential. It constitutes an improved form of the Vermeer and Brinkgreve [A new effective nonlocal strain measure for softening plasticity. In: Chambon, R., Desrues, J., Vardulakis, I. (E…
Adaptive Fuzzy Control of a Process with Bifurcations
2002
Several fuzzy controllers with different structures have been considered for the control of systems with bifurcation. A mixed feedforward-feedback structure with some additional adaptation mechanisms have been checked by simulation in the control of a non-linear process constituted by a bubble column in which a slow kinetics auto-catalytic reaction takes place. Simulation results show the validity of some of the proposed controllers in avoiding the system reaching bifurcation and instability conditions.
IWCFTA2012 Keynote Speech I - Hidden attractors in dynamical systems: From hidden oscillation in Hilbert-Kolmogorov, Aizerman and Kalman problems to …
2012
Summary form only given. In this survey an attempt is made to reflect the current trends in the synthesis of analytical and numerical methods to develop efficient analytical-numerical methods, based on harmonic linearization, applied bifurcation theory and numerical methods, for searching hidden oscillations.
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
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…
Hidden oscillations in nonlinear control systems
2011
Abstract The method of harmonic linearization, numerical methods, and the applied bifurcation theory together discover new opportunities for analysis of hidden oscillations of control systems. In the present paper new analytical-numerical algorithm for hidden oscillation localization is discussed. Counterexamples construction to Aizerman's conjecture and Kalman's conjecture on absolute stability of control systems are considered.
Modified post-bifurcation dynamics and routes to chaos from double-Hopf bifurcations in a hyperchaotic system
2012
In order to understand the onset of hyperchaotic behavior recently observed in many systems, we study bifurcations in the modified Chen system leading from simple dynamics into chaotic regimes. In particular, we demonstrate that the existence of only one fixed point of the system in all regions of parameter space implies that this simple point attractor may only be destabilized via a Hopf or double Hopf bifurcation as system parameters are varied. Saddle-node, transcritical and pitchfork bifurcations are precluded. The normal form immediately following double Hopf bifurcations is constructed analytically by the method of multiple scales. Analysis of this generalized double Hopf normal form …
Hopf bifurcation at infinity for planar vector fields
2007
We study, from a new point of view, families of planar vector fields without singularities $ \{ X_{\mu}$  :  $-\varepsilon < \mu < \varepsilon\} $ defined on the complement of an open ball centered at the origin such that, at $\mu=0$, infinity changes from repellor to attractor, or vice versa. We also study a sort of local stability of some $C^1$ planar vector fields around infinity.