2,136 research outputs found
Bifurcations of Periodic Orbits in the Generalised Nonlinear Schr\"{o}dinger Equation
We focus on the existence and persistence of families of saddle periodic
orbits in a four-dimensional Hamiltonian reversible ordinary differential
equation derived using a travelling wave ansatz from a generalised nonlinear
Schr{\"o}dinger equation (GNLSE) with quartic dispersion. In this way, we are
able to characterise different saddle periodic orbits with different signatures
that serve as organising centres of homoclinic orbits in the ODE and solitons
in the GNLSE. To achieve our objectives, we employ numerical continuation
techniques to compute these saddle periodic orbits, and study how they organise
themselves as surfaces in phase space that undergo changes as a single
parameter is varied. Notably, different surfaces of saddle periodic orbits can
interact with each other through bifurcations that can drastically change their
overall geometry or even create new surfaces of periodic orbits. Particularly
we identify three different bifurcations: symmetry-breaking, period-
multiplying, and saddle-node bifurcations. Each bifurcation exhibits a
degenerate case, which subsequently gives rise to two bifurcations of the same
type that occurs at particular energy levels that vary as a parameter is
gradually increased.
Additionally, we demonstrate how these degenerate bifurcations induce
structural changes in the periodic orbits that can support homoclinic orbits by
computing sequences of period- multiplying bifurcations
Homoclinic orbits and chaos in a pair of parametrically-driven coupled nonlinear resonators
We study the dynamics of a pair of parametrically-driven coupled nonlinear
mechanical resonators of the kind that is typically encountered in applications
involving microelectromechanical and nanoelectromechanical systems (MEMS &
NEMS). We take advantage of the weak damping that characterizes these systems
to perform a multiple-scales analysis and obtain amplitude equations,
describing the slow dynamics of the system. This picture allows us to expose
the existence of homoclinic orbits in the dynamics of the integrable part of
the slow equations of motion. Using a version of the high-dimensional Melnikov
approach, developed by Kovacic and Wiggins [Physica D, 57, 185 (1992)], we are
able to obtain explicit parameter values for which these orbits persist in the
full system, consisting of both Hamiltonian and non-Hamiltonian perturbations,
to form so-called Shilnikov orbits, indicating a loss of integrability and the
existence of chaos. Our analytical calculations of Shilnikov orbits are
confirmed numerically
Kneadings, Symbolic Dynamics and Painting Lorenz Chaos. A Tutorial
A new computational technique based on the symbolic description utilizing
kneading invariants is proposed and verified for explorations of dynamical and
parametric chaos in a few exemplary systems with the Lorenz attractor. The
technique allows for uncovering the stunning complexity and universality of
bi-parametric structures and detect their organizing centers - codimension-two
T-points and separating saddles in the kneading-based scans of the iconic
Lorenz equation from hydrodynamics, a normal model from mathematics, and a
laser model from nonlinear optics.Comment: Journal of Bifurcations and Chaos, 201
Spiral attractors as the root of a new type of "bursting activity" in the Rosenzweig-MacArthur model
We study the peculiarities of spiral attractors in the Rosenzweig-MacArthur
model, that describes dynamics in a food chain "prey-predator-superpredator".
It is well-known that spiral attractors having a "teacup" geometry are typical
for this model at certain values of parameters for which the system can be
considered as slow-fast system. We show that these attractors appear due to the
Shilnikov scenario, the first step in which is associated with a supercritical
Andronov-Hopf bifurcation and the last step leads to the appearance of a
homoclinic attractor containing a homoclinic loop to a saddle-focus equilibrium
with two-dimension unstable manifold. It is shown that the homoclinic spiral
attractors together with the slow-fast behavior give rise to a new type of
bursting activity in this system. Intervals of fast oscillations for such type
of bursting alternate with slow motions of two types: small amplitude
oscillations near a saddle-focus equilibrium and motions near a stable slow
manifold of a fast subsystem. We demonstrate that such type of bursting
activity can be either chaotic or regular
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