20,793 research outputs found

    Number and Amplitude of Limit Cycles emerging from {\it Topologically Equivalent} Perturbed Centers

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    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: dH(x,y)+ϵ(f(x,y)dyg(x,y)dx)=0dH(x,y)+\epsilon(f(x,y)dy-g(x,y)dx)=0, with H(x,y)=1/2(x2+y2)H(x,y)={1/2}(x^2+y^2). This reduction allows us to find the Melnikov function, M(h)=H=hfdygdxM(h)=\int_{H=h}fdy-gdx, associated to each particular problem. We obtain the information on the bifurcation curves of the limit cycles by solving explicitly the equation M(h)=0M(h)=0 in each case.Comment: 17 pages, 0 figure

    Complex oscillations in the delayed Fitzhugh-Nagumo equation

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    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

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    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

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    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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