93 research outputs found
An iterative method for the canard explosion in general planar systems
The canard explosion is the change of amplitude and period of a limit cycle
born in a Hopf bifurcation in a very narrow parameter interval. The phenomenon
is well understood in singular perturbation problems where a small parameter
controls the slow/fast dynamics. However, canard explosions are also observed
in systems where no such parameter is present. Here we show how the iterative
method of Roussel and Fraser, devised to construct regular slow manifolds, can
be used to determine a canard point in a general planar system of nonlinear
ODEs. We demonstrate the method on the van der Pol equation, showing that the
asymptotics of the method is correct, and on a templator model for a
self-replicating system.Comment: Paper presented at the 9th AIMS Conference on Dynamical Systems,
Differential Equations and Applications, Orlando, Florida, USA July 1 - 5,
201
An iterative method for the approximation of fibers in slow-fast systems
In this paper we extend a method for iteratively improving slow manifolds so
that it also can be used to approximate the fiber directions. The extended
method is applied to general finite dimensional real analytic systems where we
obtain exponential estimates of the tangent spaces to the fibers. The method is
demonstrated on the Michaelis-Menten-Henri model and the Lindemann mechanism.
The latter example also serves to demonstrate the method on a slow-fast system
in non-standard slow-fast form. Finally, we extend the method further so that
it also approximates the curvature of the fibers.Comment: To appear in SIAD
Canards in stiction: on solutions of a friction oscillator by regularization
We study the solutions of a friction oscillator subject to stiction. This
discontinuous model is non-Filippov, and the concept of Filippov solution
cannot be used. Furthermore some Carath\'eodory solutions are unphysical.
Therefore we introduce the concept of stiction solutions: these are the
Carath\'eodory solutions that are physically relevant, i.e. the ones that
follow the stiction law. However, we find that some of the stiction solutions
are forward non-unique in subregions of the slip onset. We call these solutions
singular, in contrast to the regular stiction solutions that are forward
unique. In order to further the understanding of the non-unique dynamics, we
introduce a regularization of the model. This gives a singularly perturbed
problem that captures the main features of the original discontinuous problem.
We identify a repelling slow manifold that separates the forward slipping to
forward sticking solutions, leading to a high sensitivity to the initial
conditions. On this slow manifold we find canard trajectories, that have the
physical interpretation of delaying the slip onset. We show with numerics that
the regularized problem has a family of periodic orbits interacting with the
canards. We observe that this family has a saddle stability and that it
connects, in the rigid body limit, the two regular, slip-stick branches of the
discontinuous problem, that were otherwise disconnected.Comment: Submitted to: SIADS. 28 pages, 12 figure
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