38,785 research outputs found

    Some Applications of the Extended Bendixson-Dulac Theorem

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    During the last years the authors have studied the number of limit cycles of several families of planar vector fields. The common tool has been the use of an extended version of the celebrated Bendixson-Dulac Theorem. The aim of this work is to present an unified approach of some of these results, together with their corresponding proofs. We also provide several applications.Comment: 19 pages, 3 figure

    Stability Properties of 1-Dimensional Hamiltonian Lattices with Non-analytic Potentials

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    We investigate the local and global dynamics of two 1-Dimensional (1D) Hamiltonian lattices whose inter-particle forces are derived from non-analytic potentials. In particular, we study the dynamics of a model governed by a "graphene-type" force law and one inspired by Hollomon's law describing "work-hardening" effects in certain elastic materials. Our main aim is to show that, although similarities with the analytic case exist, some of the local and global stability properties of non-analytic potentials are very different than those encountered in systems with polynomial interactions, as in the case of 1D Fermi-Pasta-Ulam-Tsingou (FPUT) lattices. Our approach is to study the motion in the neighborhood of simple periodic orbits representing continuations of normal modes of the corresponding linear system, as the number of particles NN and the total energy EE are increased. We find that the graphene-type model is remarkably stable up to escape energy levels where breakdown is expected, while the Hollomon lattice never breaks, yet is unstable at low energies and only attains stability at energies where the harmonic force becomes dominant. We suggest that, since our results hold for large NN, it would be interesting to study analogous phenomena in the continuum limit where 1D lattices become strings.Comment: Accepted for publication in the International Journal of Bifurcation and Chao

    Asymptotic behaviour for a class of non-monotone delay differential systems with applications

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    The paper concerns a class of nn-dimensional non-autonomous delay differential equations obtained by adding a non-monotone delayed perturbation to a linear homogeneous cooperative system of ordinary differential equations. This family covers a wide set of models used in structured population dynamics. By exploiting the stability and the monotone character of the linear ODE, we establish sufficient conditions for both the extinction of all the populations and the permanence of the system. In the case of DDEs with autonomous coefficients (but possible time-varying delays), sharp results are obtained, even in the case of a reducible community matrix. As a sub-product, our results improve some criteria for autonomous systems published in recent literature. As an important illustration, the extinction, persistence and permanence of a non-autonomous Nicholson system with patch structure and multiple time-dependent delays are analysed.Comment: 26 pages, J Dyn Diff Equat (2017

    Hopf bifurcations in time-delay systems with band-limited feedback

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    We investigate the steady-state solution and its bifurcations in time-delay systems with band-limited feedback. This is a first step in a rigorous study concerning the effects of AC-coupled components in nonlinear devices with time-delayed feedback. We show that the steady state is globally stable for small feedback gain and that local stability is lost, generically, through a Hopf bifurcation for larger feedback gain. We provide simple criteria that determine whether the Hopf bifurcation is supercritical or subcritical based on the knowledge of the first three terms in the Taylor-expansion of the nonlinearity. Furthermore, the presence of double-Hopf bifurcations of the steady state is shown, which indicates possible quasiperiodic and chaotic dynamics in these systems. As a result of this investigation, we find that AC-coupling introduces fundamental differences to systems of Ikeda-type [Ikeda et al., Physica D 29 (1987) 223-235] already at the level of steady-state bifurcations, e.g. bifurcations exist in which limit cycles are created with periods other than the fundamental ``period-2'' mode found in Ikeda-type systems.Comment: 32 pages, 5 figures, accepted for publication in Physica D: Nonlinear Phenomen

    Aerodynamic Stability of Satellites in Elliptic Low Earth Orbits

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    Topical observations of the thermosphere at altitudes below 200km200 \, km are of great benefit in advancing the understanding of the global distribution of mass, composition, and dynamical responses to geomagnetic forcing, and momentum transfer via waves. The perceived risks associated with such low altitude and short duration orbits has prohibited the launch of Discovery-class missions. Miniaturization of instruments such as mass spectrometers and advances in the nano-satellite technology, associated with relatively low cost of nano-satellite manufacturing and operation, open an avenue for performing low altitude missions. The time dependent coefficients of a second order non-homogeneous ODE which describes the motion have a double periodic shape. Hence, they will be approximated using Jacobi elliptic functions. Through a change of variables the original ODE will be converted into Hill's ODE for stability analysis using Floquet theory. We are interested in how changes in the coefficients of the ODE affect the stability of the solution. The expected result will be an allowable range of parameters for which the motion is dynamically stable. A possible extension of the application is a computational tool for the rapid evaluation of the stability of entry or re-entry vehicles in rarefied flow regimes and of satellites flying in relatively low orbits.Comment: 18 pages, 16 figure

    A generalized averaging method for linear differential equations with almost periodic coefficients

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    Generalized averaging method for linear differential equations with almost periodic coefficient
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