2,297 research outputs found
Non-integrability of the Armbruster-Guckenheimer-Kim quartic Hamiltonian through Morales-Ramis theory
We show the non-integrability of the three-parameter
Armburster-Guckenheimer-Kim quartic Hamiltonian using Morales-Ramis theory,
with the exception of the three already known integrable cases. We use
Poincar\'e sections to illustrate the breakdown of regular motion for some
parameter values.Comment: Accepted for publication in SIAM Journal on Applied Dynamical
Systems. Adapted version for arxiv with 19 pages and 11 figure
Quiver representations and dimension reduction in dynamical systems
Dynamical systems often admit geometric properties that must be taken into
account when studying their behaviour. We show that many such properties can be
encoded by means of quiver representations. These properties include classical
symmetry, hidden symmetry and feedforward structure, as well as subnetwork and
quotient relations in network dynamical systems. A quiver equivariant dynamical
system consists of a collection of dynamical systems with maps between them
that send solutions to solutions. We prove that such quiver structures are
preserved under Lyapunov-Schmidt reduction, center manifold reduction, and
normal form reduction.Comment: Revised version, accepted in the SIAM Journal on Applied Dynamical
Systems; 43 pages, 10 figure
The effects of delay on the HKB model of human motor coordination
Understanding human motor coordination holds the promise of developing
diagnostic methods for mental illnesses such as schizophrenia. In this paper,
we analyse the celebrated Haken-Kelso-Bunz (HKB) model, describing the dynamics
of bimanual coordination, in the presence of delay. We study the linear
dynamics, stability, nonlinear behaviour and bifurcations of this model by both
theoretical and numerical analysis. We calculate in-phase and anti-phase limit
cycles as well as quasi-periodic solutions via double Hopf bifurcation analysis
and centre manifold reduction. Moreover, we uncover further details on the
global dynamic behaviour by numerical continuation, including the occurrence of
limit cycles in phase quadrature and 1-1 locking of quasi-periodic solutions.Comment: Submitted to the SIAM Journal on Applied Dynamical Systems. 27 pages,
8 figure
The large core limit of spiral waves in excitable media: A numerical approach
We modify the freezing method introduced by Beyn & Thuemmler, 2004, for
analyzing rigidly rotating spiral waves in excitable media. The proposed method
is designed to stably determine the rotation frequency and the core radius of
rotating spirals, as well as the approximate shape of spiral waves in unbounded
domains. In particular, we introduce spiral wave boundary conditions based on
geometric approximations of spiral wave solutions by Archimedean spirals and by
involutes of circles. We further propose a simple implementation of boundary
conditions for the case when the inhibitor is non-diffusive, a case which had
previously caused spurious oscillations.
We then utilize the method to numerically analyze the large core limit. The
proposed method allows us to investigate the case close to criticality where
spiral waves acquire infinite core radius and zero rotation frequency, before
they begin to develop into retracting fingers. We confirm the linear scaling
regime of a drift bifurcation for the rotation frequency and the core radius of
spiral wave solutions close to criticality. This regime is unattainable with
conventional numerical methods.Comment: 32 pages, 17 figures, as accepted by SIAM Journal on Applied
Dynamical Systems on 20/03/1
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