This paper is concerned with the geometry of slow manifolds of a dynamical system with two slow and one fast variable. Specifically, we study the dynamics near a folded node singularity, which is known to give rise to so-called canard solutions. Geometrically, canards are intersection curves of two-dimensional attracting and repelling slow manifolds, and they are a key element of slow-fast dyna...

We study stationary solutions u(x) and their stability of the fourth order Swift-Hohenberg equation on a bounded domain (0, L) with boundary conditions u = 0 and u′′ = 0 at x = 0 and x = L. It is well known that as L increases the set of stationary solutions becomes increasingly complex. Numerical studies have have exhibited two interesting types of structures in the bifurcation diagram for (L,...

The aim of this paper is to study what happens when a slight perturbation a ects the coe cients of a quadratic equation de ning a variety a quadric in R Structurally stable quadrics are those a small perturbation on the coe cients of the equation de ning them does not give rise to a di erent in some sense set of points In particular we characterize structurally stable quadrics and give the bifu...

We present a rigorous analysis of the + J Ising spin-glass model on the Bethe lattice with fixed uncorrelated boundary conditions. Phase diagrams are derived as a function of temperature vs. concentration of ferromagnetic bonds and, for a symmetric distribution of bonds, external field vs. temperature. In this part we characterize the bulk ordered phases using bifurcation theory: we prove the e...

This paper is concerned with bifurcation and stability problems of non-linear systems. The attention is focused on parametrically excited non-linear vibrations. A comparison of C—L method with IHB technique is given on the study of local bifurcations. It is shown that the two methods give qualitatively equivalent bifurcation diagrams. ( 1998 Elsevier Science Ltd. All rights reserved

This study analyzes the chaotic behavior of a micromechanical resonator with electrostatic forces on both sides and investigates the control of chaos. A phase portrait, maximum Lyapunov exponent and bifurcation diagram are used to find the chaotic dynamics of this micro-electro-mechanical system (MEMS). To suppress chaotic motion, a robust fuzzy sliding mode controller (FSMC) is designed to tur...

A simple car-following model that includes the reaction time of drivers is analyzed by numerical continuation techniques for the case of five cars on a ring. A two-dimensional bifurcation diagram is computed that summarizes the dynamics of steady states, oscillating solutions, collisions, and stopping as a function of model parameters. Further, a mechanism is proposed by which different spatial...

The quadratic iteration is mapped using a nondistributive real scator algebra in three dimensions. The bound set S has a rich fractal-like boundary. Periodic points on the scalar axis are necessarily surrounded by off axis divergent magnitude points. There is a one-to-one correspondence of this set with the bifurcation diagram of the logistic map. The three-dimensional S set exhibits selfsimila...