38,785 research outputs found
Some Applications of the Extended Bendixson-Dulac Theorem
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
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
and the total energy 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 , 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
The paper concerns a class of -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
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
Topical observations of the thermosphere at altitudes below 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
Generalized averaging method for linear differential equations with almost periodic coefficient
- …