20,793 research outputs found
Number and Amplitude of Limit Cycles emerging from {\it Topologically Equivalent} Perturbed Centers
We consider three examples of weekly perturbed centers which do not have {\it
geometrical equivalence}: a linear center, a degenerate center and a
non-hamiltonian center. In each case the number and amplitude of the limit
cycles emerging from the period annulus are calculated following the same
strategy: we reduce of all of them to locally equivalent perturbed integrable
systems of the form: , with
. This reduction allows us to find the Melnikov
function, , associated to each particular problem. We
obtain the information on the bifurcation curves of the limit cycles by solving
explicitly the equation in each case.Comment: 17 pages, 0 figure
Complex oscillations in the delayed Fitzhugh-Nagumo equation
Motivated by the dynamics of neuronal responses, we analyze the dynamics of
the Fitzhugh-Nagumo slow-fast system with delayed self-coupling. This system
provides a canonical example of a canard explosion for sufficiently small
delays. Beyond this regime, delays significantly enrich the dynamics, leading
to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a
delay-induced subcritical Bogdanov-Takens instability arising at the fold
points of the S-shaped critical manifold. Underlying the transition from
canard-induced to delay-induced dynamics is an abrupt switch in the nature of
the Hopf bifurcation
Multi-stabilities and symmetry-broken one-colour and two-colour states in closely coupled single-mode lasers
We theoretically investigate the dynamics of two mutually coupled identical
single-mode semi-conductor lasers. For small separation and large coupling
between the lasers, symmetry-broken one-colour states are shown to be stable.
In this case the light output of the lasers have significantly different
intensities while at the same time the lasers are locked to a single common
frequency. For intermediate coupling we observe stable symmetry-broken
two-colour states, where both lasers lase simultaneously at two optical
frequencies which are separated by up to 150~GHz. Using a five dimensional
model we identify the bifurcation structure which is responsible for the
appearance of symmetric and symmetry-broken one-colour and two-colour states.
Several of these states give rise to multi-stabilities and therefore allow for
the design of all-optical memory elements on the basis of two coupled
single-mode lasers. The switching performance of selected designs of optical
memory elements is studied numerically.Comment: 12 pages, 15 figure
Oscillators and relaxation phenomena in Pleistocene climate theory
Ice sheets appeared in the northern hemisphere around 3 million years ago and
glacial-interglacial cycles have paced Earth's climate since then. Superimposed
on these long glacial cycles comes an intricate pattern of millennial and
sub-millennial variability, including Dansgaard-Oeschger and Heinrich events.
There are numerous theories about theses oscillations. Here, we review a number
of them in order to draw a parallel between climatic concepts and dynamical
system concepts, including, in particular, the relaxation oscillator,
excitability, slow-fast dynamics and homoclinic orbits. Namely, almost all
theories of ice ages reviewed here feature a phenomenon of synchronisation
between internal climate dynamics and the astronomical forcing. However, these
theories differ in their bifurcation structure and this has an effect on the
way the ice age phenomenon could grow 3 million years ago. All theories on
rapid events reviewed here rely on the concept of a limit cycle in the ocean
circulation, which may be excited by changes in the surface freshwater surface
balance. The article also reviews basic effects of stochastic fluctuations on
these models, including the phenomenon of phase dispersion, shortening of the
limit cycle and stochastic resonance. It concludes with a more personal
statement about the potential for inference with simple stochastic dynamical
systems in palaeoclimate science.
Keywords: palaeoclimates, dynamical systems, limit cycle, ice ages,
Dansgaard-Oeschger eventsComment: Published in the Transactions of the Philosophical Transactions of
the Royal Society (Series A, Physical Mathematical and Engineering Sciences),
as a contribution to the Proceedings of the workshop on Stochastic Methods in
Climate Modelling, Newton Institute (23-27 August). Philosophical
Transactions of the Royal Society (Series A, Physical Mathematical and
Engineering Sciences), vol. 370, pp. xx-xx (2012); Source codes available on
request to author and on http://www.uclouvain.be/ito
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