99 research outputs found
Mixed-mode oscillations in a multiple time scale phantom bursting system
In this work we study mixed mode oscillations in a model of secretion of GnRH
(Gonadotropin Releasing Hormone). The model is a phantom burster consisting of
two feedforward coupled FitzHugh-Nagumo systems, with three time scales. The
forcing system (Regulator) evolves on the slowest scale and acts by moving the
slow nullcline of the forced system (Secretor). There are three modes of
dynamics: pulsatility (transient relaxation oscillation), surge (quasi steady
state) and small oscillations related to the passage of the slow nullcline
through a fold point of the fast nullcline. We derive a variety of reductions,
taking advantage of the mentioned features of the system. We obtain two
results; one on the local dynamics near the fold in the parameter regime
corresponding to the presence of small oscillations and the other on the global
dynamics, more specifically on the existence of an attracting limit cycle. Our
local result is a rigorous characterization of small canards and sectors of
rotation in the case of folded node with an additional time scale, a feature
allowing for a clear geometric argument. The global result gives the existence
of an attracting unique limit cycle, which, in some parameter regimes, remains
attracting and unique even during passages through a canard explosion.Comment: 38 pages, 16 figure
On the approximation of the canard explosion point in singularly perturbed systems without an explicit small parameter
A canard explosion is the dramatic change of period and amplitude of a limit cycle of a system of nonlinear ODEs in a very narrow interval of the bifurcation parameter. It occurs in slow–fast systems and is well understood in singular perturbation problems where a small parameter epsilon defines the time-scale separation. We present an iterative algorithm for the determination of the canard explosion point which can be applied for a general slow–fast system without an explicit small parameter. We also present assumptions under which the algorithm gives accurate estimates of the canard explosion point. Finally, we apply the algorithm to the van der Pol equations, a Templator model for a self-replicating system and a model for intracellular calcium oscillations with no explicit small parameters and obtain very good agreement with results from numerical simulations.<br/
On the use of blow up to study regularizations of singularities of piecewise smooth dynamical systems in
In this paper we use the blow up method of Dumortier and Roussarie
\cite{dumortier_1991,dumortier_1993,dumortier_1996}, in the formulation due to
Krupa and Szmolyan \cite{krupa_extending_2001}, to study the regularization of
singularities of piecewise smooth dynamical systems
\cite{filippov1988differential} in . Using the regularization
method of Sotomayor and Teixeira \cite{Sotomayor96}, first we demonstrate the
power of our approach by considering the case of a fold line. We quickly
recover a main result of Bonet and Seara \cite{reves_regularization_2014} in a
simple manner. Then, for the two-fold singularity, we show that the regularized
system only fully retains the features of the singular canards in the piecewise
smooth system in the cases when the sliding region does not include a full
sector of singular canards. In particular, we show that every locally unique
primary singular canard persists the regularizing perturbation. For the case of
a sector of primary singular canards, we show that the regularized system
contains a canard, provided a certain non-resonance condition holds. Finally,
we provide numerical evidence for the existence of secondary canards near
resonance.Comment: To appear in SIAM Journal of Applied Dynamical System
Sixth International Symposium on Bifurcations and Instabilities in Fluid Dynamics (BIFD2015):Foreword
Codimension three bifurcation of streamline patterns close to a no-slip wall: A topological description of boundary layer eruption
Leukocyte telomere length is associated with elevated plasma glucose and HbA1c in young healthy men independent of birth weight
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