64 research outputs found
From First Lyapunov Coefficients to Maximal Canards
Hopf bifurcations in fast-slow systems of ordinary differential equations can
be associated with surprising rapid growth of periodic orbits. This process is
referred to as canard explosion. The key step in locating a canard explosion is
to calculate the location of a special trajectory, called a maximal canard, in
parameter space. A first-order asymptotic expansion of this location was found
by Krupa and Szmolyan in the framework of a "canard point"-normal-form for
systems with one fast and one slow variable. We show how to compute the
coefficient in this expansion using the first Lyapunov coefficient at the Hopf
bifurcation thereby avoiding use of this normal form. Our results connect the
theory of canard explosions with existing numerical software, enabling easier
calculations of where canard explosions occur.Comment: preprint version - for final version see journal referenc
Hunting French Ducks in a Noisy Environment
We consider the effect of Gaussian white noise on fast-slow dynamical systems
with one fast and two slow variables, containing a folded-node singularity. In
the absence of noise, these systems are known to display mixed-mode
oscillations, consisting of alternating large- and small-amplitude
oscillations. We quantify the effect of noise and obtain critical noise
intensities above which the small-amplitude oscillations become hidden by
fluctuations. Furthermore we prove that the noise can cause sample paths to
jump away from so-called canard solutions with high probability before
deterministic orbits do. This early-jump mechanism can drastically influence
the local and global dynamics of the system by changing the mixed-mode
patterns.Comment: 60 pages, 9 figure
Controlling Canard Cycles
Canard cycles are periodic orbits that appear as special solutions of fast-slow systems (or singularly perturbed ordinary differential equations). It is well known that canard cycles are difficult to detect, hard to reproduce numerically, and that they are sensible to exponentially small changes in parameters. In this paper, we combine techniques from geometric singular perturbation theory, the blow-up method, and control theory, to design controllers that stabilize canard cycles of planar fast-slow systems with a folded critical manifold. As an application, we propose a controller that produces stable mixed-mode oscillations in the van der Pol oscillator
Multiple Time Scale Dynamics With Two Fast Variables And One Slow Variable
This thesis considers dynamical systems that have multiple time scales. The focus lies on systems with two fast variables and one slow variable. The twoparameter bifurcation structure of the FitzHugh-Nagumo (FHN) equation is analyzed in detail. A singular bifurcation diagram is constructed and invariant manifolds of the problem are computed. A boundary-value approach to compute slow manifolds of saddle-type is developed. Interactions of classical invariant manifolds and slow manifolds explain the exponentially small turning of a homoclinic bifurcation curve in parameter space. Mixed-mode oscillations and maximal canards are detected in the FHN equation. An asymptotic formula to find maximal canards is proved which is based on the first Lyapunov coefficient at a singular Hopf bifurcation
A Dynamical Systems Analysis of Movement Coordination Models
In this thesis, we present a dynamical systems analysis of models of
movement coordination, namely the Haken-Kelso-Bunz (HKB) model
and the Jirsa-Kelso excitator (JKE).
The dynamical properties of the models that can describe various phenomena
in discrete and rhythmic movements have been explored in the
models' parameter space. The dynamics of amplitude-phase approximation
of the single HKB oscillator has been investigated. Furthermore, an
approximated version of the scaled JKE system has been proposed and
analysed.
The canard phenomena in the JKE system has been analysed. A combination
of slow-fast analysis, projection onto the Poincare sphere and
blow-up method has been suggested to explain the dynamical mechanisms
organising the canard cycles in JKE system, which have been
shown to have different properties comparing to the classical canards
known for the equivalent FitzHugh-Nagumo (FHN) model. Different
approaches to de fining the maximal canard periodic solution have been
presented and compared.
The model of two HKB oscillators coupled by a neurologically motivated
function, involving the effect of time-delay and weighted self- and
mutual-feedback, has been analysed. The periodic regimes of the model
have been shown to capture well the frequency-induced drop of oscillation
amplitude and loss of anti-phase stability that have been experimentally
observed in many rhythmic movements and by which the development
of the HKB model has been inspired. The model has also been demonstrated
to support a dynamic regime of stationary bistability with the
absence of periodic regimes that can be used to describe discrete movement
behaviours.This work was supported by The Higher Committee For Education Development in Iraq (HCED) and the University of Mosul
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