Periodic perturbations of quadratic planar polynomial vector fields

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Resonances and subharmonic bifurcations of large amplitude periodic orbits of planar polynomial vector fields

In this work are studied periodic perturbations, depending on two parameters, of planar polynomial vector fields having an annulus of large amplitude periodic orbits, which accumulate on a symmetric infinite heteroclinic cycle. Such periodic orbits and the heteroclinic trajectory can be seen only by the global consideration of the polynomial vector fields on the whole plane, and not by their restriction to any compact set. The global study involving infinity is performed via the Poincare Compactification. It is shown that, for certain types of periodic perturbations, one can seek, in a neighborhood of the origin in the parameter plane, curves C-(m) of subharmonic bifurcations, for which the periodically perturbed system has subharmonics of order m, for any integer m.

PERIODIC PERTURBATION of QUADRATIC SYSTEMS WITH TWO INFINITE HETEROCLINIC CYCLES

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Time-periodic perturbation of a Lienard equation with an unbounded homoclinic loop

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

On the L2 stability of linear multidimensional time varying systems

This paper analyzes the L2 stability of solutions of systems with time-varying coefficients of the form [A + C(t)]x′ = [B + D(t)]x + u, where A, B, C, D are matrices. Following proof of a lemma, the main result is derived, according to which the system is L2 stable if the eigenvalues of the coefficient matrices are related in a simple way. A corollary of the theorem dealing with small periodic perturbations of constant coefficient systems is then proved. The paper concludes with two illustrative examples, both of which deal with the attitude dynamics of a rigid, axisymmetric, spinning satellite in an eccentric orbit, subject to gravity gradient torques.

Relevance of sampling schemes in light of Ruelle's linear response theory

We reconsider the theory of the linear response of non-equilibrium steady states to perturbations. We �rst show that by using a general functional decomposition for space-time dependent forcings, we can de�ne elementary susceptibilities that allow to construct the response of the system to general perturbations. Starting from the de�nition of SRB measure, we then study the consequence of taking di�erent sampling schemes for analysing the response of the system. We show that only a speci�c choice of the time horizon for evaluating the response of the system to a general time-dependent perturbation allows to obtain the formula �rst presented by Ruelle. We also discuss the special case of periodic perturbations, showing that when they are taken into consideration the sampling can be �ne-tuned to make the de�nition of the correct time horizon immaterial. Finally, we discuss the implications of our results in terms of strategies for analyzing the outputs of numerical experiments by providing a critical review of a formula proposed by Reick.

Deformation and tidal evolution of close-in planets and satellites using a Maxwell viscoelastic rheology

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Anundance of phytoplankton at time series station DYNAPROC

A general study of biogeochemical processes (DYNAPROC cruise) was conducted in May 1995 at a time-series station in the open northwestern Mediterranean Sea where horizontal advection was weak. Short-term variations of the vertical distributions of pico- and nanophytoplankton were investigated over four 36-h cycles, along with parallel determinations of metabolic CO2 production rates and amino acid-containing colloid (AACC) concentrations at the chlorophyll maximum depth. The vertical (0-1000-m depth) distributions of (i) AACC, (ii) suspended particles and (iii) metabolic CO2 production rate were documented during the initial and final stages of these 36-h cycles. This study was concerned with diel vertical migration (DVM) of zooplankton, which provided periodic perturbations. Accordingly, the time scale of the experimental work varied from a few hours to a few days. Although all distributions exhibited a periodic behaviour, AACC distributions were generally not linked to diel vertical migrations. In the subsurface layer, Synechococcus made the most abundant population and large variations in concentration were observed both at day and at night. The corresponding integrated (over the upper 90 m) losses of Synechococcus during one night pointed to a potential source of exported organic matter amounting to 534 mg C/m**2. This study stresses the potential importance of organic matter export from the euphotic zone through the daily grazing activity of vertically migrating organisms, which would not be accounted for by measurements at longer time scales. The metabolic CO2 production exhibited a peak of activity below 500 m that was shifted downward, apparently in a recurrent way and independently of the vertical distributions of AACC or of suspended particulate material. To account for this phenomenon, a 'sustained wave train» hypothesis is proposed that combines the effect of the diel superficial faecal pellet production by swarming migrators and the repackaging activity of the nonmigrating midwater populations. Our results confirm the recent finding that the particulate compartment is not the major source of the observed instantaneous remineralisation rate and shed a new light on the fate of organic matter in the aphotic zone.

Model parameter

Interannual environmental variability in Peru is dominated by the El Niño Southern Oscillation (ENSO). The most dramatic changes are associated with the warm El Niño (EN) phase (opposite the cold La Niña phase), which disrupts the normal coastal upwelling and affects the dynamics of many coastal marine and terrestrial resources. This study presents a trophic model for Sechura Bay, located at the northern extension of the Peruvian upwelling system, where ENSO-induced environmental variability is most extreme. Using an initial steady-state model for the year 1996, we explore the dynamics of the ecosystem through the year 2003 (including the strong EN of 1997/98 and the weaker EN of 2002/03). Based on support from literature, we force biomass of several non-trophically-mediated 'drivers' (e.g. Scallops, Benthic detritivores, Octopus, and Littoral fish) to observe whether the fit between historical and simulated changes (by the trophic model) is improved. The results indicate that the Sechura Bay Ecosystem is a relatively inefficient system from a community energetics point of view, likely due to the periodic perturbations of ENSO. A combination of high system productivity and low trophic level target species of invertebrates (i.e. scallops) and fish (i.e. anchoveta) results in high catches and an efficient fishery. The importance of environmental drivers is suggested, given the relatively small improvements in the fit of the simulation with the addition of trophic drivers on remaining functional groups' dynamics. An additional multivariate regression model is presented for the scallop Argopecten purpuratus, which demonstrates a significant correlation between both spawning stock size and riverine discharge-mediated mortality on catch levels. These results are discussed in the context of the appropriateness of trophodynamic modeling in relatively open systems, and how management strategies may be focused given the highly environmentally influenced marine resources of the region.

Stabilization of standing waves through time-delay feedback

Standing waves are studied as solutions of a complex Ginzburg-Landau equation subjected to local and global time-delay feedback terms. The onset is described as an instability of the uniform oscillations with respect to spatially periodic perturbations. The solution of the standing wave pattern is given analytically and studied through simulations. © 2013 American Physical Society.

Réponse de la communauté de mollusques aux perturbations physiques et chimiques dans un grand lac fluvial (Lac Saint-Pierre, Fleuve Saint-Laurent, QC)

Les mollusques sont des indicateurs de perturbations anthropiques et environnementales. Ce groupe de macroinvertébrés représente en outre une source importante de nourriture pour les poissons et les oiseaux aquatiques du littoral. Les hypothèses de cette étude sont que la communauté de mollusques est influencée indirectement par les tributaires agricoles et/ou par des variables environnementales (comme la dessiccation et l'exposition aux vagues) puisque ces perturbations sont susceptibles de modifier leurs sources alimentaires et leur habitat. Les indicateurs de la réponse des mollusques aux agents perturbateurs sont la composition, la diversité, la densité, ainsi que la biomasse des espèces. En septembre 2013, des mesures de paramètres physico-chimiques de l'eau ont été réalisées, et des échantillons de mollusques et de végétation aquatique ont été prélevés à 14 sites le long des rives du lac Saint-Pierre (Fleuve Saint-Laurent, Québec, Canada). Le long de la rive nord, les sites fortement exposés à l'action du vent, situés à de plus grandes élévations, affichaient une plus faible densité, biomasse et richesse spécifique de mollusques que les sites de la rive sud, en milieu plus abrité et profond. Les sites physiquement perturbés étaient caractérisés par de faibles biomasses en macrophytes submergés. Les sphaeriidae apparaissent comme des exceptions à ces patrons, montrant une abondance plus élevée aux sites presque dépourvus de macrophytes. Bien que les variables physiques et l'habitat exercent une influence déterminante sur les communautés de mollusques, les gastéropodes et les moules unionidés étaient également affectés par la dégradation de la qualité de l'eau dans le panache des tributaires agricoles. La richesse, la densité et la biomasse des gastéropodes étaient négativement influencées par des teneurs élevées de matières en suspension et de fer dissous. Les résultats de notre étude montrent que la communauté de mollusques du lac Saint-Pierre est directement affectée par l'émersion périodique, l'exposition au vent, et indirectement par l'effet de ces variables physiques sur les macrophytes qui constituent leur habitat.

Periodic window arising in the parameter space of an impact oscillator

In the bi-dimensional parameter space of an impact-pair system, shrimp-shaped periodic windows are embedded in chaotic regions. We show that a weak periodic forcing generates new periodic windows near the unperturbed one with its shape and periodicity. Thus, the new periodic windows are parameter range extensions for which the controlled periodic oscillations substitute the chaotic oscillations. We identify periodic and chaotic attractors by their largest Lyapunov exponents. (C) 2010 Elsevier B.V. All rights reserved.

Stability of periodic optical solitons for a nonlinear Schrodinger system

This paper is concerned with the existence and nonlinear stability of periodic travelling-wave solutions for a nonlinear Schrodinger-type system arising in nonlinear optics. We show the existence of smooth curves of periodic solutions depending on the dnoidal-type functions. We prove stability results by perturbations having the same minimal wavelength, and instability behaviour by perturbations of two or more times the minima period. We also establish global well posedness for our system by using Bourgain`s approach.

Perturbations to Saturn's F-ring strands at their closest approach to Prometheus

The strange morphology of the F ring of Saturn is thought to be caused by the perturbing effects of two close satellites, Prometheus and Pandora. The F ring and the satellites also experience periodic close encounters as a result of differential precession arising from Saturn's oblateness. Using the orbits of the F-ring strands derived by Murray et al. (1997, Icarus 129, 304-316) the behaviour of the ring particles at their closest approach to Prometheus is analysed using numerical simulations. The results show that a gap and a wave are formed in the ring at each encounter with the satellite. However, the gap is expected to have a short lifetime due to keplerian shear. © 2000 Elsevier Science Ltd. All rights reserved.

Transport and dynamical properties for a bouncing ball model with regular and stochastic perturbations

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)