514 research outputs found
Canards and curvature: nonsmooth approximation by pinching
In multiple time-scale (singularly perturbed) dynamical systems, canards are counterintuitive solutions that evolve along both attracting and repelling invariant manifolds. In two dimensions, canards result in periodic oscillations whose amplitude and period grow in a highly nonlinear way: they are slowly varying with respect to a control parameter, except for an exponentially small range of values where they grow extremely rapidly. This sudden growth, called a canard explosion, has been encountered in many applications ranging from chemistry to neuronal dynamics, aerospace engineering and ecology. Canards were initially studied using nonstandard analysis, and later the same results were proved by standard techniques such as matched asymptotics, invariant manifold theory and parameter blow-up. More recently, canard-like behaviour has been linked to surfaces of discontinuity in piecewise-smooth dynamical systems. This paper provides a new perspective on the canard phenomenon by showing that the nonstandard analysis of canard explosions can be recast into the framework of piecewise-smooth dynamical systems. An exponential coordinate scaling is applied to a singularly perturbed system of ordinary differential equations. The scaling acts as a lens that resolves dynamics across all time-scales. The changes of local curvature that are responsible for canard explosions are then analyzed. Regions where different time-scales dominate are separated by hypersurfaces, and these are pinched together to obtain a piecewise-smooth system, in which curvature changes manifest as discontinuity-induced bifurcations. The method is used to classify canards in arbitrary dimensions, and to derive the parameter values over which canards form either small cycles (canards without head) or large cycles (canards with head)
Bifurcations of piecewise smooth flows:perspectives, methodologies and open problems
In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study
Delay Equations and Radiation Damping
Starting from delay equations that model field retardation effects, we study
the origin of runaway modes that appear in the solutions of the classical
equations of motion involving the radiation reaction force. When retardation
effects are small, we argue that the physically significant solutions belong to
the so-called slow manifold of the system and we identify this invariant
manifold with the attractor in the state space of the delay equation. We
demonstrate via an example that when retardation effects are no longer small,
the motion could exhibit bifurcation phenomena that are not contained in the
local equations of motion.Comment: 15 pages, 1 figure, a paragraph added on page 5; 3 references adde
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On the Gierer-Meinhardt System with Saturation
We consider the following shadow Gierer-Meinhardt system with saturation:
\left\{\begin{array}{l}
A_t=\epsilon^2 \Delta A -A + \frac{A^2}{ \xi (1+k A^2)} \ \ \mbox{in} \ \Omega \times (0, \infty),\\
\tau \xi_t= -\xi +\frac{1}{|\Omega|} \int_\Om A^2\,dx
\ \ \mbox{in} \ (0, +\infty),
\frac{\partial A}{\partial \nu} =0 \ \mbox{on} \ \partial \Omega\times(0,\infty),
\end{array}
\right.
where \ep>0 is a small parameter, k$ for the existence and stability of stable spiky solutions.
In the one-dimensional case we have a complete answer to the stability behavior.
Central to our study are a parameterized ground-state equation
and the associated nonlocal eigenvalue problem (NLEP)
which is solved by functional analysis arguments and the continuation method
An organizing center in a planar model of neuronal excitability
The paper studies the excitability properties of a generalized
FitzHugh-Nagumo model. The model differs from the purely competitive
FitzHugh-Nagumo model in that it accounts for the effect of cooperative gating
variables such as activation of calcium currents. Excitability is explored by
unfolding a pitchfork bifurcation that is shown to organize five different
types of excitability. In addition to the three classical types of neuronal
excitability, two novel types are described and distinctly associated to the
presence of cooperative variables
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